XXI Brazilian Topology Meeting Abstract Booklet

Niter´oi 12th–18th August 2018 ii General Info

Welcome at Niter´oi!

In order to provide as much information as possible about the conference, we are providing this booklet which encloses all the abstratcs of the talks. The additional information and references given by the speakers are also included. They are ordered according to the schedule of the conference, this is also displayed at the margin of each abstract.

Recall that the two mini courses are given in parallel to some contributed talks. Mini courses will be taught at the room 205 in the second floor of the Block G of Gragoat´acampus where the Instituto de Matem´aticae Estat´ısticastands. All of the plenary and contributed talks will be held at the Auditorium of the same Block G.

There will be two poster sessions on tuesday and thursday, don’t miss them.

iii iv Contents

Welcome iii

Abstracts 1 Mini Curso: Introdu¸c˜aoElementar `aAn´aliseTopol´ogicade Dados (Wash- ington Mio)...... 1 Mini Course: Introduction to Geometric Group Theory (Francesco Matucci) 1 The topology of singular points of real analytic curves (Etienne´ Ghys) . . . 2 An exotic non-leaf in C1,0 regularity (Carlos Meni˜noCot´on) ...... 2 Finiteness Properties of Nekrashevych Groups (Rachel Skipper) ...... 4 The fixed points of multimaps on surface with application to the torus- a Braid approach (Daciberg Lima Gon¸calves)...... 4 Localization of Chern-Simons Type Invariants of Riemannian Foliations (Dirk T¨oben) ...... 5 Positively curved Killing foliations via deformations (Francisco C. Caramello Jr.)...... 6 Torus is a Wecken space for Nielsen-Borsuk-Ulam theory (Daniel Vendr´uscolo) 6 N-expansive measures (Carlos Morales) ...... 7 Minimal triangulations of the homotopy types of surfaces (Eugenio Borghini) 7 Compact boundaries of groups (Ross Geoghegan) ...... 8 The posets of p-subgroups of a nite group as nite topological spaces (Kevin Ivan Piterman)...... 8 Weingarten surfaces with constant second fundamental form (Alexandre Paiva Barreto) ...... 9 Fixed points of diffeomorphisms on nilmanifolds with a free nilpotent fun- damental group (Karel Dekimpe) ...... 9 Homological representations of punctured torus mapping class groups (Julien Korinman) ...... 10 A trace formula for foliated flows (Jes´usA. Alvarez´ L´opez) ...... 10 Limit aperiodic colorings (Ram´onBarral Lij´o) ...... 12 A conjecture on the BNS-invariant for Artin Groups (Kisnney Almeida) . 14 Concluding the classification of Handel-Thurston examples (Thierry Barbot) 15 Deformations of compact Hausdorff foliations (Matias L. del Hoyo) . . . . 16 Taut foliations on surface bundles over S1 (Hiraku Nozawa) ...... 17 Reeb graph: from the analysis through topology to the group theory and conversly (Wac law Marzantowicz)...... 20

v A Combinatorial/Algebraic Topological Approach to Nonlinear Dynamics (Konstantin Mischaikow)...... 20 Existence of common zeros for commuting vector fields on three manifolds (Bruno Santiago)...... 21 Almost-crystallographic groups as quotients of Artin braid groups (Oscar Ocampo)...... 22 Thickness of skeletons of hyperbolic orbifolds (Mikhail Belolipetsky) . . . . 22 The Borsuk-Ulam property for homotopy class in fibrations with basis S1 and fiber torus (Vinicius C. Laass)...... 22 Closure of singular foliations: the proof of Molino’s conjecture (Marcos M. Alexandrino) ...... 23 Meridional rank of knots whose exterior is a graph manifold (Ederson R. F. Dutra) ...... 23 Quotients of the torus braid groups and crystallographic groups (Carolina de Miranda e Pereiro) ...... 25 Stable Signatures for Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence (Facundo M´emoli)...... 25 Deficient and multiple points of maps into a manifold (Tha´ısF. M. Monis) 26 Limiting Actions of Infinite Surface Groups (Marcel Vinhas Bertolini) . . . 27 Retracts of free groups (Slobodan Tanushevski) ...... 28 Topological data analysis on materials science and several problems in random topology (Yasuaki Hiraoka)...... 29 On the construction of a covering map (Samson Saneblidze) ...... 30 Topology of the leaves of hyperbolic surface laminations (S´ebastien Alvarez´ ) 31

Author Index 33

vi Abstracts

Mini Curso: Introdu¸c˜aoElementar `aAn´aliseTopol´ogicade Dados Monday, Tuesday & Thursday Washington Mio 11:20–12:20 Florida State University (USA) Wednesday [email protected] 09:50–10:50

A an´alisetopol´ogicade dados tem suas ra´ızesem trabalhos publicados na d´ecadade 90, adquirindo novo ´ımpeto mais recentemente tanto em seus aspectos te´oricoscomo em aplica¸c˜oes.Este mini-curso tem como objetivos principais: (i) introduzir, a n´ıvel elementar, alguns conceitos b´asicosna disciplina e (ii) ilustrar o uso das t´ecnicasem an´alisede dados complexos.

Mini Course: Introduction to Geometric Group Theory Monday, Tuesday & Thursday Francesco Matucci 14:30–15:30 Universidade Estadual de Campinas (Brazil) [email protected]

Geometric Group Theory sees groups as symmetries of a space on which they act on. The goal is to find connections between the algebraic structure of groups, the geometry and topology of the associated space and the dynamics of the action. Starting from an abstract group, we can always associate a geometric object (known as the Cayley graph) which realizes this natural symmetry. The goal of these talks is to introduce some concepts used to analyze groups (such as that of quasi-isometry) and some interesting classes of groups whose definition or proof methods border in other areas of Math (such as geometry, combinatorics, computer science, analysis, topology and more).

[1] Martin Bridson.Geometric and Combinatorial Group Theory. The Princeton Companion to Mathematics, edited by Timothy Gowers, June Barrow-Green, Imre Leader, 431–448, Princeton University Press, 2008. [2] Office Hours with a Geomet- ric Group Theorist, edited by Matt Clay and Dan Margalit, 456 pages, Princeton University Press, 2017. [3] Clara Loh¨ . Geometric Group Theory: An Introduction. 389 pages, Springer Universitext, 2017.

1 Monday The topology of singular points of real analytic curves 09:30–10:20 Plenary Talk Etienne´ Ghys Ecole´ Normale Sup´erieurede Lyon (France) [email protected]

In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram: an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this talk is to give an complete description of those “ana- lytic” chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

Monday An exotic non-leaf in C1,0 regularity 10:30–10:55 Carlos Meni˜noCot´on Universidade Federal Fluminense (Brazil) carlos [email protected]

We show a smoothing of a manifold homeomorphic to R4 punctured along an infinite closed and discrete set which is not diffeomorphic to any leaf of a C1,0 codimension one foliation on a compact manifold. In the process of the proof it will be shown some smoothings of R4 that cannot be diffeomorphic to leaves of C2 codimension one foliations. This is a joint work with Paul A. Schweitzer.

Introduction

An exotic non-leaf is a topological manifold which is homemorphic to a leaf of a foliation on a compact manifold but admits at least one smoothing which is not diffeomorphic to any leaf. In this talk, some results contained in [5] and [6] will be presented. The moti- vation of this work is a long standing conjecture in foliation theory which says that the known family of R4 exotica should not be diffeomorphic to leaves of foliations on compact manifolds. We obtained in [5] a positive answer for this conjecture for some especific families of smoothings and in the particular case of C2 codimension one foliations. In a recent development, we show how to reduce the regularity of the foliation to C1,0 (i.e. tangent to a continuous plane distribution). There is a price to pay: the number of ends increases from 1 to ∞. The topology of the manifold is still simple enough (just R4 punctured along an infinite closed and discrete set) to be realized topologically as a leaf. Another interesting fact is that these pathological smoothings involve all the clas- sical qualitative results on exotic R4’s and global 4 dimensional differential topology.

2 Construction

The smoothings of R4 which are shown to be C2 exotic non-leaves in [5] are certain classes of exotic R4’s (called semi-simple-definite in [8]). They are examples of large exotica, i.e., exotic R4’s which cannot be smoothly embedded in S4. Our example will be obtained by performing an infinite connected sum with a suitable family of small exotica (those which can be smoothly embedded in S4) on one of the above considered large exotica.

Idea of proof

We follow the same reasoning used in [3] where the first examples of topological non- leaves were exhibited. The basis of the proof relies in the existence of infinitely many pairwise disjoint compact codimension zero submanifolds (with boundary), called blocks, which are topologically rigid, i.e. any other submanifold homeomorphic to a block must meet it. In our given manifold there exists a similar notion of rigid block: there are 4- dimensional exotic pair of pants which are rigid but only in the smooth category. It is interesting to note that the rigidity is a consequence of the celebrated theorems of Donaldson [2], Furuta [4] and Taubes [7]. This fact, jointly with some properties of aperiodicity for these exotic R4’s (see [1] and [7]), allows to accomplish the task and show that our presented manifold is an exotic non-leaf in any regularity. We also observe that there is a lot of freedom in the construction of this exotic non-leaf, in fact there exists uncountably many pairwise not diffeomorphic smooth- ings with similar properties.

[1] S. De Michelis, M.H. Freedman. Uncountably many exotic R4’s in standard 4- space. J. Diff. Geom. 35, 219–254, 1992. [2] S. K. Donaldson. An application of gauge theory to four-dimensional topology. J. Diff. Geom. 18-2, 279–315, 1983. [3] E. Ghys. Une vari´et´equi n’est pas une feuille. Topology 24-1, 67–73, 1985. [4] M. Furuta. Monopole equation and the 11/8-conjecture. Math. Res. Lett. 8-3, 279–291 (1983). [5] C. Meni˜noCot´on,P.A. Schweitzer, S.J. Exotic open 4-manifolds which are non-leaves. Geom. Topol. (to appear), 2018. https://arxiv.org/abs/1410.8182 [6] C. Meni˜noCot´on,P.A. Schweitzer, S.J. Exotic non-leaves with infinitely many ends. Preprint, to appear, 2018. [7] C.H. Taubes. Gauge theory on asymptotically periodic 4-manifolds. J. Diff. Geom. 25, 363–430, 1987. [8] L. Taylor. An invariant of smooth 4-manifolds. Geom. Topol. 1, 71–89, 1997.

3 Monday Finiteness Properties of Nekrashevych Groups 11:45–12:10 Rachel Skipper Binghamton University (USA) [email protected]

A group is of type Fn if it admits a classifying space with compact n-skeleton. We discuss some recent results about finiteness properties of Nekrashevych groups, a class of groups whose building blocks are self-similar groups and Higman-Thompson groups. Since these groups are often virtually simple and since finiteness properties are a quasi-isometric invariant, we use these results to build a new infinite family of pairwise non-quasi-isometric simple groups. This is a joint work with Stefan Witzel and Matthew C. B. Zaremsky.

The fixed points of multimaps on surface with application Monday to the torus- a Braid approach 12:10–13:00 Plenary Talk Daciberg Lima Gon¸calves Universidade de S˜aoPaulo (Brazil) [email protected]

Let ϕ : S ( S be a n-valued continuous multimap on some compact surface S. First we classify the homotopy classes of multimaps where for most of the surfaces the classification is given in terms of the braids on n-strings of the surface S. Then we give an algebraic criterion to decide which homotopy classes contains a multimap which is fixed point free. We will focus on the cases where S is a closed surface of Euler characteristic ≤ 0. Despite the fact that the algebraic criterion is quite hard, we perform some specific calculations for the case where S is the torus. Further, we provide some explict calculation in the case of 2-vallued map, which shows the level of difficult of the general problem. The concept of Nielsen number for a surface has been developed. Then I explain the status of the Wecken property for multimaps on the torus. In fact it is an open question if there is an example of a valued map which has Nielsen number zero but it can not be deformed to fixed point free. Finally a brief exposition about the case of the projective plane should be presented if time allows. Below are some of the relevant references for our purpose. This is a joint work with John Guaschi.

[1] P. Bellingeri. On presentations of surface braid groups. J. Algebra, 274, 2004, 543-563. [2] J. Better. A Wecken theorem for n-valued maps. Top. Applic., 159, 2012, 3707-3715. [3] Robert F. Brown. Fixed points of n-valued multimaps of the circle. Bull. Pol. Acad. Sci. Math., 54 no. 2, 2006, 153-162. [4] Robert F. Brown. The Lefschetz number of an n-valued multimap. JP J. Fixed Point Theory Appl., 2 no. 1, 2007, 53-60. [5] Robert F. Brown. Nielsen numbers of n-valued fiber maps. J. Fixed Point Theory Appl., 4 no. 2, 2008, 183-201.

4 [6] Robert F. Brown, Lo Kim Lin, Jon T.. Coincidences of projections and linear n-valued maps of tori. Topology Appl., 157 no. 12, 2010, 1990-1998. [7] D. L. Gonc¸alves, J. Guaschi. The Borsuk-Ulam theorem for maps into a surface. Topology and its Applications, 157, 2010, 1742-1759. [8] H. Schirmer. An index and a Nielsen number for n-valued multifunctions. Fund. Math., 124, 1984, 207-219. [9] H. Schirmer. A minimum theorem for n-valued multifunctions. Fund. Math., 126, 1985, 83-92. [10] G. P. Scott. Braid groups and the group of homeomorphisms of a surface. Proc. Camb. Phil. Soc., 68, 1970, 605-617.

Localization of Chern-Simons Type Invariants of Riemannian Foliations Monday 14:30–14:55 Dirk T¨oben Universidade Federal de S˜aoCarlos (Brazil) [email protected]

We prove an Atiyah-Bott-Berline-Vergne type localization formula for certain Rie- mannian foliations in the context of equivariant basic cohomology. As an application, we localize the volume of Sasakian manifolds and secondary characteristic classes of Riemannian foliations to the union of closed leaves. Various examples are given to illustrate our method. This is a joint work with O. Goertsches and H. Nozawa.

Additional Information

The talk will be about the results of [5], the last of a series of five papers ([1]-[5]) using the concept of equivariant basic cohomology introduced in [1]. This notion allows one to transfer methods from compact transformation groups, in particular localization of cohomology, to the theory of Riemannian foliations.

[1] O. Goertsches, D. Toben¨ . Equivariant basic cohomology of Riemannian foliations. To appear in J. Reine Angew. Math. [2] O. Goertsches, H. Nozawa, D. Toben¨ . Equivariant cohomology of K- contact manifolds. Mathematische Annalen, v. 354, p. 1555-1582, 2012. [3] O. Goertsches, H. Nozawa, D. Toben¨ . Rigidity and vanishing of basic Dolbeault cohomology of Sasakian manifolds. Journal of Symplectic Geometry, v. 14, p. 31-70, 2016. [4] D. Toben¨ . Localization of basic characteristic classes. Annales de l’Institut Fourier, v. 64, p. 537-570, 2014. [5] O. Goertsches, H. Nozawa, D. Toben¨ . Localization of Chern-Simons type invariants of Riemannian foliations. To appear in Israel Journal of Mathematics.

5 Monday Positively curved Killing foliations via deformations 15:00–15:25 Francisco C. Caramello Jr. Universidade Federal S˜aoCarlos (Brazil)

We will present some results on Killing foliations, a class of Riemannian foliations, obtained during the speakers doctorate research under the supervision and with the collaboration of Prof. Dirk T¨oben (UFSCar). We show that a manifold admitting a Killing foliation with positive transverse curvature and maximal transverse sym- metry rank fibers over finite quotients of spheres or weighted complex projective spaces. This and other similar results are obtained by deforming the foliation into a generalized Seifert fibration while maintaining its transverse geometry, which al- lows us to apply results from the Riemannian geometry of orbifolds to the space of leaves. In this vein, we also obtain a transverse version of Chengs sphere theorem for positively curved Killing foliations with maximal transverse diameter, as well as a generalization of the transverse analog of Bergers theorem for Killing foliations. Fi- nally, we show that the basic Euler characteristic is preserved by such deformations, which provides some topological obstructions for Riemannian foliations.

Monday Torus is a Wecken space for Nielsen-Borsuk-Ulam theory 15:50–16:15 Daniel Vendr´uscolo universidade Federal de S˜aoCarlos (Brazil) [email protected]

In some recent works it was used Nielsen theory to study Borsuk-Ulam coinci- dence problems. In particular in [1] it was defined a Nilsen-Borsuk-Ulam number with the usual properties of a Nielsen number type. In [2] it was studied (using braids groups) the homotopies classes of maps between surfaces (and classes of free involutions) for that the Borsuk-Ulam is true (in the sense that there always is a point x such that the f(x) = f(τ(x)) were τ is the free involution and f is any representative of the homotopy class studied). In this work we compute the Nielsen-Borsuk-Ulam numbers for some maps be- tween surfaces showing that the 2-dimentonal torus is a Wecken space for such theory. This is a joint work with Givanildo Donizeti de Melo.

[1] F. S. Cotrim & D. Vendruscolo´ . The Nielsen Borsuk-Ulam number. Bull. Belg. Math. Soc. Simon Stevin 24(4), 613619, 2017. [2] D. L. Gonc¸alves, J. Guaschi & V. C. Laass. The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero. arXiv:1608.00397

6 N-expansive measures Monday 16:25–16:50 Carlos Morales Chungnam National University (Republic of Korea) [email protected]

We merge the concepts of N-expansive homeomorphism [2] and expansive measure [3] into the notion of N-expansive measure. We obtain some properties for the N-expansive measures including: characterization of N-expansive homeomorphisms and strongly measure expansive homeomorphisms (as in [1]), characterization as nite convex combination of Dirac measures (in the equicontinuous case), 1- expansive measures on complete separable metric spaces and characterization in one-dimensional dynamics. This is a joint work with Keonhee Lee.

[1] Cordeiro, W., Denker, M., Zhang, X.. On specfication and measure expansive- ness. Discrete Contin. Dyn. Syst. 37 (2017), no. 4, 1941–1957. [2] Morales, C.A..A generalization of expansivity. Discrete Contin. Dyn. Syst. 32 (2012), no. 1, 293–301. [3] Morales, C.A., Sirvent, V.F.. Expansive measures. Pub- lica¸c˜oesMatem´aticas do IMPA. 29o Col´oquio Brasileiro de Matem´atica. Instituto Nacional de Matem´aticaPura e Aplicada (IMPA), Rio de Janeiro, 2013.

Minimal triangulations of the homotopy types of surfaces Monday 16:50–17:15 Eugenio Borghini FCEN, Universidad de Buenos Aires (Argentina) [email protected]

The covering type of a topological space is a notion introduced by Karoubi and Weibel to measure the complexity of a space in terms of good coverings [1]. For a space X of the homotopy type of a compact CW-complex, the covering type coincides with the minimum number of vertices of a simplicial complex homotopy equivalent to X. In this talk I will show, by exploiting the multiplicative structure of the cohomology ring of surfaces, that the covering type of a closed surface S -with the sole exception of the double torus- is attained at a minimal triangulation of S [2].

[1]M. Karoubi, Ch. Weibel. On the covering type of a space. Enseign. Math. 62, 457–474, 2016. [2]E. Borghini, E.G.Minian. The covering type of closed surfaces and minimal trian- gulations. Preprint, 2017. Available at ArXiv: https://arxiv.org/abs/1712.02833.

7 Tuesday Compact boundaries of groups 09:30–10:20 Plenary Talk Ross Geoghegan Binghamton University (USA) [email protected]

Proper CAT (0) spaces are natural generalizations of simply connected complete manifolds on non-positive sectional curvature. CAT (0) spaces have a simple and beautiful metric geometry, and their boundaries at infinity are metric compacta whose properties involve classical point-set topology. When a group G acts properly discontinuously and cocompactly on a proper CAT (0) space (and this happens in a variety of important mathematical situations) G is called a “CAT (0) group”. This is because the topology of the compact boundary closely reflects algebraic properties of the group. So this part of geometric group theory is a natural meeting place for algebra and topology. I will discuss the “semistability problem for CAT (0) groups”, a still-open problem which nicely illustrates this meeting place.

The posets of p-subgroups of a Tuesday nite group as nite topological spaces 10:30–10:55 Kevin Ivan Piterman FCEN - Universidad de Buenos Aires (Argentina) [email protected]

For a nite group G and a prime number p dividing its order, let Sp(G) and Ap(G) denote respectively the poset of nontrivial p-subgroups of G and nontrivial elementary abelian p- subgroups of G. They have been first studied by K. Brown [2] and D. Quillen [4] by means of the topological properties of their associated simplicial complexes K(Sp(G)) and K(Ap(G)). Quillen’s conjecture asserts that G has a nontrivial normal p-subgroup if and only if K(Sp(G)) is contractible. This conjecture has been widely studied in the last four decades by group theorists and algebraic topologists (see for example [1]). In [5] Stong studied the posets Sp(G) and Ap(G) as nite topological spaces (with an intrinsic topology). With this topology, they are not in general homotopy equivalent, although their associated simplicial complexes K(Sp(G)) and K(Ap(G)) always are. In this talk I will show that the nite space Ap(G) can be homotopically trivial but not contractible (this answers a question posted by Stong), and I will describe the contractibility of Sp(G) and Ap(G) as nite spaces in terms of algebraic properties of the group. These results recently appeared in [3]. I will also analyze a stronger formulation of a conjecture of P. Webb [6] in terms of nite spaces (the original conjecture was proved by P. Symonds), and prove particular cases of the strong conjecture using a combination of topological methods and fusion.

[1] M. Aschbacher, S. D. Smith. On Quillen’s conjecture for the p-groups com- plex. Ann. of Math. 137 (2), no. 3, 473–529, 1993. [2] K. Brown. Euler characteristics of groups: The p-fractional part. Invent. Math. 29, no. 1, 1–5, 1975.

8 [3] E.G. Minian, K.I. Piterman. The homotopy types of the posets of p-subgroups of a nite group. Adv. Math. 328, 1217–1233, 2018. [4] D. Quillen. Homotopy properties of the poset of nontrivial p-subgroups of a group. Adv. Math. 28, 101–128, 1978. [5] R. E. Stong. Group actions on nite spaces. Discrete Math. 40, 95–100, 1984. [6] P. J. Webb. Subgroup complexes. The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986). Proc. Sympos. Pure Math., 47, Amer. Math. Soc., Providence, RI, 1987, 349–365.

Weingarten surfaces with constant second fundamental form Tuesday 11:45–12:10 Alexandre Paiva Barreto Universidade Federal S˜aoCarlos (Brazil) [email protected]

A surface S in the 3-dimensional Euclidean space is called a Weingarten surface if there exists some relation W (λ1, λ2) = 0 , among its principal curvatures λ1 and λ2. Weingarten surfaces is a classical topic in Differential Geometry that began with the works of Weingarten in the middle of the 19th century. The complete classification of Weingarten surfaces,even in the case where W is linear, is far from be- ing achieved. In this work we study surfaces in the 3-dimensional Euclidean space whose shape operators have constant norm. In other words, we study Weingarten surfaces that satisfy the non-linear relation

2 2 W (λ1, λ2) = (λ1) + (λ2) = c for some c > 0. A general approach for all of these surfaces seems not to be possible, then we will focus our attention on the class of regular surfaces obtained by translation of the graph of a C2-function f : R → R over a spatial curve.

Fixed points of diffeomorphisms on nilmanifolds with a free nilpotent fundamental group Tuesday 12:10–13:00 Karel Dekimpe Plenary Talk Katholieke Universiteit Leuven (Belgium) [email protected]

Let M be a nilmanifold with a fundamental group which is free 2-step nilpotent on at least 4 generators. We will show that for any nonnegative integer n there exists a self-diffeomorphism hn of M such that hn has exactly n fixed points and any self- map f of M which is homotopic to hn has at least n fixed points. We will also shed some light on the situation for less generators and also for higher nilpotency classes. This is a joint work with Sam Tertooy and Antonio R. Vargas.

9 Homological representations of punctured torus mapping Tuesday class groups 14:30–14:55 Julien Korinman Universidade Federal de S˜aoCarlos (Brazil) [email protected]

At the begining of the 90th, Lawrence introduced new families of representations of the braid groups and the punctured sphere mapping class groups using homology of local systems on configuration spaces. These representations are closely related to construction arising in quantum groups and conformal field theory. Their homolog- ical nature permited to Krammer and Bigelow to prove they were faithfull solving a long standing conjecture on the linearity of braids groups. Their result also implies the linearity of the genus 2 mapping class groups. In this talk we will generalize Lawrences construction to produce homological representations of the mapping class group of a punctured genus one surface. Our construction shares many ressemblence with representations arising in CFT and quantum Teichmuller theory. We naturally conjecture they are faithfull and will present arguments in that direction. This is a joint work with Edivaldo Lopes dos Santos.

Tuesday A trace formula for foliated flows 15:00–15:25 Jes´usA. Alvarez´ L´opez University of Santiago de Compostela (Spain) [email protected]

We consider codimension one foliation F on a compact manifold M admitting a foliated flow φt, whose closed orbits and preserved leaves are simple. In this case, there are finitely many preserved leaves, which are compact, forming a compact subset M 0. A version of the reduced leafwise cohomology, HIb (F), is defined by using leafwise differential forms which may be singular at M 0, but with nice singularities; namely, leafwise differential forms conormal to M 0. The talk will be about our progress to define distributional traces of the induced action of φt on Hb rI(F), for every degree r, and to prove a corresponding Leftchetz trace formula involving the closed orbits and leaves preserved by φt. The formula also involves a version of the η-invariant of M 0. This kind of distributional trace formula was conjectured by Christopher Deninger, and it was proved by the first two authors when no leaf is preserved by φt. This is a joint work with Yuri Kordyukov and Eric Leichtnam.

Foliated flows

Let F be a codimension one foliation F on a compact manifold M. A flow φt on M is called foliated if it maps leaves to leaves. Then φt induces a local flow φ¯t on local transversals, and the leaves preserved by φt correspond to the fixed points of φ¯t. The leaves preserved by φt are called simple if they correspond to simple fixed points of

10 φ¯t; i.e., the eigenvalues of its tangent map are different of one. Simple closed orbits of φt are similarly defined using the tangent map of its restriction to the leaves. In this case, there is a finite number of preserved leaves, which are compact, forming a compact subset M 0.

Limit conormal reduced leafwise cohomology

Leafwise forms are the smooth sections on M of the exterior bundle V T ∗F of the leaves. We may also consider its distributional sections, called distributional leafwise forms. A distributional leafwise form is called conormal to M 0 if all of its Lie derivatives with respect to infinitesimal transformations are in a fixed Sobolev space of order s. They form a topological differential complex I[s](F) with the [s] leafwise de Rham derivative dF . Its cohomology is denoted by HI (F), and its reduced cohomology by HI[s](F). We may also consider the topological differential S [s] complex I(F) = s I (F) with dF , whose cohomology is denoted by HI(F), and its reduced cohomology by HI(F). The inductive limit of the spaces HI[s](F) as s → −∞, denoted by HIb (F), is called the limit conormal reduced leafwise cohomology. The restrictions of conormal leafwise forms in I[s](F) to the open saturated set M 1 = M \M 0 form another topological differential complex J [s](F), and let K[s](F) denote the kernel of the restriction map I[s](F) → J [s](F). As before, we can also define K(F), J(F), HK(F) and HJb (F), and φt induces endomorphisms of all of these spaces, denoted by φt∗. The short exact sequences of topological differential complexes, 0 → K[s](F) → I[s](F) → J [s](F) → 0 , induces a sequence

· · · → Hb r+1J(F) → HrK(F) → Hb rI(F) → Hb rJ(F) → · · ·

We show that this sequence is exact at the terms Hb rI(F) and Hb rJ(F), and the exactness at the terms HrK(F) is still an open problem.

Distributional trace formula

Some understanding of HIb (F) can be achieved with the previous sequence. In particular, assuming its exactness and other mild conditions, a distributional trace of φt∗ on Hb rI(F) can be defined using the trace on HrK(F) and a version of Melrose b-trace on Hb rJ(F) [4]. Here, the term distributional trace refers to a “trace” whose values are distributions on R. Then we prove a trace formula for the corresponding Leftchetz distribution, involving data from the closed orbits and leaves preserved by φt, as conjectured by Christopher Deninger [3]. Some unexpected η-invariant of M 0 is also involved in the formula, which is produced by the b-trace like in the proof of the Atiyah-Bott-Patodi index formula given by Melrose [4]. The case without leaves preserved by φt was solved by the first two authors in [2], as an application of the results from [1].

11 [1] J.A. Alvarez´ Lopez´ and Y.A. Kordyukov. Long time behavior of leafwise heat flow for Riemannian foliations, Compositio Math. 125 (2001), 129–153. [2] J.A. Alvarez´ Lopez´ and Y.A. Kordyukov. Lefschetz distribution of Lie foliations. In: C∗-algebras and elliptic theory II, Trends Math., Birkh¨auser, Basel, (1–40) 2008. [3] C. Deninger. Analogies between analysis on foliated spaces and arithmetic geometry. In: Groups and analysis, London Math. Soc. Lecture Note Ser. 354, Cambridge Univ. Press, Cambridge, (174–190) 2008. [4] R.B. Melrose. The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics 4, A.K. Peters, Ltd., Wellesley, MA, 1993.

Tuesday Limit aperiodic colorings 16:25–16:50 Ram´onBarral Lij´o Universidade de Santiago de Compostela (Spain) [email protected]

A vertex-coloring of a graph is aperiodic if the colored graph has a trivial isomor- phism group. If the same condition applies for every graph contained in its closure in the Gromov space of pointed colored graphs, then we say that the coloring is limit aperiodic. In this talk we aim to present a result concerning the existence of limit aperiodic graph-colorings with an optimal number of colors, as well as some applications of this theorem to edge-colorings, tilings, and more. This is a joint work with Jes´usAntonio Alvarez´ L´opez. Let (X,E) (or simply X) be a simple1 undirected graph. Given a set of natural numbers F , a coloring φ: X → F is said to be aperiodic or distinguishing if there is no nontrivial automorphism of X preserving φ. The distinguishing number, denoted by D(X), is the smallest positive integer such that there is some aperiodic coloring φ of X by D(X) colors. This concept was introduced in [1] by Albertson and Collins, and the calculation of D(X)(or bounds thereof) for many families of graphs has been the subject of much research in recent years(e.g. [5] and [6]). A graph Y is said to be a limit of X if, for every n ∈ N and y ∈ Y , we can find an isomorphic copy of B(y, n) inside X. Analogously, we can define when a colored graph (Y, ψ) is the limit of (X, φ). A coloring φ: X → F is limit aperiodic or limit distinguishing if every limit colored graph (Y, ψ) is distinguishing, and the limit distinguishing number(denoted by DL(X)) is the least n ∈ N such that there is a limit distinguishing coloring by n colors. We also say that X (respectively, (X, φ))is repetitive if every finite pattern (resp. of (X, φ)) appears uniformly in X with respect to the graph distance. The main theorem of [2] is then the following, showing DL(X) ≤ deg X. Theorem A. Let X be an infinite connected simple graph with deg X < ∞. Then there is a limit-aperiodic coloring φ of X by deg X colors. Moreover, if X is repetitive, then there is a repetitive limit-aperiodic coloring by deg X colors. The bound DL(X) ≤ deg X is optimal, as exemplified by Z with the obvious graph structure.

1Recall that a graph is simple if for any two vertices there is at most one edge joining them.

12 The first application of this result (in fact, the question that motivated this re- search) shows that every Riemannian manifold of bounded geometry can be realized isometrically as a leaf of a compact Riemannian foliated space [2]. This follows embedding the manifold into a space that resembles the Gromov space of pointed Riemannian manifolds endowed with the smooth topology. All of these ideas are already present in [3]. Another application concerns the existence of limit aperiodic tilings. For the sake of simplicity, consider a tiling T of an n-dimensional (connected) Riemannian manifold with corners, M, by tiles meeting face to face, taken from (i.e., isometric to) a finite set of prototiles, T , consisting of compact Riemannian manifolds of dimension n with boundary (see [4] for the definition of tilings of more general spaces). The tiling isomorphisms of (M,T ) are the isometries of M that map tiles to tiles. Using such tiling isomorphisms, there are obvious versions of limit-aperiodicity and repetitivity in this setting. Also, colored tilings and face-colored tilings have an obvious meaning, as well as their limit-aperiodicity and repetitivity.

We can associate to T a graph X, with one vertex vt for each tile t ∈ T , and 0 declaring that vt is adjacent to vt0 if and only if t and t meet at some (n − 1)- dimensional face. Then X is an infinite graph of bounded degree, and every tiling isomorphism of T (a tiling preserving isometry of (M,T )) induces an isomorphism of X. Let φ: X → [deg X] := {0, 1,..., deg X − 1} be a limit-aperiodic coloring of X. This coloring induces a limit-aperiodic colored tiling T 0 by colored tiles taken from the finite set of colored prototiles T 0 := T × [deg X]. Moreover, if the tiling is repetitive, then the resulting colored tiling can be chosen to be repetitive as well. In summation, we have the following result. Theorem B. Let T be a tiling by finitely many prototiles meeting face to face, and let ∆ be the maximum number of (n − 1)-dimensional faces of the prototiles. Then there is a limit-aperiodic coloring of the tiling by ∆ colors. If T is repetitive, then the coloring can be assumed to be repetitive. Coloring the faces of the tiles instead of the tiles themselves, we can also derive the following result from Theorem B. Theorem C. Let T be a tiling by finitely many prototiles meeting face to face, and let ∆ be the maximum number of (n − 1)-dimensional faces of the prototiles. Then there is a limit-aperiodic edge coloring of the tiling by 2(∆ − 1) colors. If T is repetitive, then the edge coloring can be assumed to be repetitive. Since the coloring of the faces of a tile is a local matching rule, it can be enforced by shape. Then Theorem C has the following consequence. If a (repetitive) Rie- mannian manifold M admits a tiling T as before, with a finite set of prototiles T , then there is another finite set of prototiles T 0 such that there is a limit-aperiodic (repetitive) tiling T 0 of M using prototiles from T 0. A perhaps more direct application provides a result analogous to Theorem A about edge-colorings. The analogue of the distinguishing number and limit distin- guishing number for edge-colorings are the distinguishing index and limit distin- 0 0 guishing index, denoted by D (X) and DL(X), respectively. Using the line or dual 0 graph of X, we can obtain directly the bound DL(X) ≤ 2(deg X − 1). The authors 0 suspect that this can probably be improved to DL(X) ≤ deg X by adapting the proof of Theorem A.

13 Finally, we indicate a possible line of research. Let P be a property about foliated spaces (or graph matchbox manifolds). For example, P may be “having 0 entropy”, or “being an equicontinuous foliated space”. Then we may ask if there is a corresponding property P 0 of colored graphs, such that the closure of the canonical embedding of a colored graph satisfies P if and only if the coloring satisfies P 0. One may also want to classify for which graphs there is a coloring satisfying P 0. For example, the case where P is “being an equicontinuous foliated space” is related to the problem of whether all groups are sofic.

[1] Michael O. Albertson and Karen L. Collins . Symmetry breaking in graphs, The Electronic Journal of Combinatorics 3 (1996), no. 1, 17pp. [2] J.A. Alvarez´ Lopez´ and Ramon´ Barral Lijo´. Limit aperiodic colorings of graphs. In preparation. [3] Jesus´ Antonio Alvarez´ Lopez´ and Ramon´ Barral Lijo´. Bounded ge- ometry and leaves, Math. Nachr. 290 (2017), no. 10, 1448-1469. MR 3672890. [4] J. Block and S. Weinberger. Aperiodic tilings, positive scalar curvature and amenability of spaces, J. Amer. Math. Soc. 5 (1992), 907-918. MR 1145337. [5] Karen L. Collins and Ann N. Trenk. The distinguishing chromatic num- ber. The Elec- tronic Journal of Combinatorics 13. 2006, 19pp. [6] Poppy Immel and Paul S. Wenger. The list distinguishing number equals the distin- guishing number for interval graphs. Discussiones Mathematicae Graph Theory 37(1), 165–174, 2017.

Tuesday A conjecture on the BNS-invariant for Artin Groups 16:50–17:15 Kisnney Almeida Universidade Estadual de Feira de Santana (Brazil) [email protected]

The Bieri-Neumann-Strebel-invariant, or Σ1-invariant, is a geometric invariant of finitely generated groups that can be used to decide if a given subgroup above the commutator is also finitely generated. It is the first of a series of topological and homological Sigma-invariants that can be similarly used to study other finiteness properties of groups. Artin groups form a large class of groups, combinatorially constructed from underlying graphs, source of interesting examples in geometry and group theory. Meier, Meinert and VanWyk have obtained a partial description of Σ1 for Artin groups - given an Artin group G they establish a necessary and a sufficient condition, depending on the topology of the underlying graph, for a character of G to be on the invariant [7]. We conjecture, as stated in a joint work with Kochloukova [3], the above sufficient condition to be also necessary, resulting on a complete description for all Artin groups. Let the circuit rank of an Artin group be the free rank of the fundamental group of its underlying graph. When stated, the conjecture was already known to be true for Artin groups of circuit rank 0 [6] and for right-angled Artin groups [8]. Right now we also know the conjecture to be true for Artin groups of circuit rank 1 [3], Artin groups of circuit rank 2 [1], Artin groups of finite type [5] and other subclasses [2],[4]. We also know the class of

14 Artin groups that satisfy the conjecture is closed for finite direct products [5] and no counterexample for the conjecture has been found until now. We will talk about these ideas and results.

[1] K. Almeida. The BNS-invariant for Artin groups of circuit rank 2. Journal of Group Theory 21(2), 189–228, 2018. [2] K. Almeida. The BNS-invariant for some Artin groups of arbitrary circuit rank. Journal of Group Theory 20(4), 793–806, 2017. [3] K. Almeida & D. H. Kochloukova. The Σ1-invariant for Artin groups of circuit rank 1. Forum Mathematicum 27(5), 2901–2925, 2015. [4] K. Almeida & D. H. Kochloukova. The Σ1-invariant for some Artin groups of rank 3 presentation. Communications in Algebra 43(2), 702–718, 2015. [5] K. Almeida & F. Lima. Finite direct product closeness for a conjecture on the BNS-invariant of Artin groups. In preparation. [6] J. Meier. Geometric invariants for Artin groups. Proc. London Math. Soc. 74(1), 151–173, 1997. [7] J. Meier, H. Meinert & L. VanWyk. On the Σ-invariants of Artin Groups. Topology and its Applications 110(1), 71–81, 2001. [8] J. Meier, H. Meinert & L. VanWyk. Higher generation subgroup sets and the Σ-invariants of graph groups. Comment. Math. Helv. 110(1), 71–81, 2001.

Concluding the classification of Handel-Thurston examples Wednesday 09:30–10:20 Thierry Barbot Plenary Talk Avignon University (France) [email protected]

In [5], M. Handel and W. Thurston introduced a family of Anosov flows on 3- manifolds that are topologically transitive, but not algebraic. It follows from a work by Foulon-Hasselblatt ([4]) that these flows are Reeb flows for some contact form. In summary, these flows are obtained by gluing finite coverings of pieces of geodesic flows, where the index of the covering is the same for the all the pieces. In [1], I proved that in any closed 3-manifold M equipped with a R-covered Anosov flow (for example, a contact Anosov flow), if P is one Seifert piece of the JSJ decomposition, then P is isotopic to a submanifold of M in which the restriction of the Anosov flow is orbitally equivalent to a finite covering of a geodesic flow. Therefore, a contact Anosov flow on a graphmanifold (i.e. a 3-manifold in which all the JSJ pieces are Seifert) is Thurston-Handel-like, in the sense that they are obtained by gluing finite coverings of pieces of geodesic flows, but not necessarily with the same index. This work has been improved and generalized to the case of pseudo-Anosov flows by Fenley and myself ([2]). In this talk, I will present a recent work with S.R. Fenley where we fill the missing gap ([3]): we prove that finite coverings of pieces of geodesic flows can be glued one to the other, without any condition on the index of the coverings, as long as the glued boundaries admit the same number of periodic orbits - which is an obvious

15 necessary condition. This achieves the classification of contact Anosov flows on graphmanifolds up to orbital equivalence.

[1] T. Barbot. Flots d’Anosov sur les vari´et´esgraph´eesau sens de Waldhausen. Ann. Inst. Fourier (Grenoble) 46(5), 1451–1517, 1996. [2] T. Barbot & S. Fenley. Free Seifert pieces of pseudo-Anosov flows. arXiv:1512.06341 [3] T. Barbot & S. Fenley. Work in progress. [4] P. Foulon & B. Hasselblatt. Contact Anosov flows on hyperbolic 3-manifolds. Geom. Topol. 17(2), 1225–1255, 2013. [5] M. Handel, W.P. Thurston. Anosov flows on new 3-manifolds. Invent. Math. 59, 95–103, 1980.

Wednesday Deformations of compact Hausdorff foliations 10:30–10:55 Matias L. del Hoyo Universidade Federal Fluminense (Brazil) [email protected]

In a recent work with R. Fernandes we show that a compact Hausdorff foliation over a compact connected manifold is rigid, in the sense that every one-parameter defor- mation of it is trivial. We study the foliation by means of its holonomy groupoid, a Lie groupoid integrating it, and combine some classical stability properties of folia- tions with our linearization results for Lie groupoids involving Riemannian metrics. In this talk I will overview the concept of Lie groupoid, explain the theory of metrics over them, and apply it to show the rigidity theorem for foliations, illustrating the use of Lie groupoids in geometry and topology.

Overview

A deformation of a foliation F on a manifold M parametrized by some interval 0 ∈ I ⊂ R is defined as a foliation F˜ on the cylinder M × I that is tangent to the slices M × t and restricts to F on M × 0. A deformation F˜ is trivial if, after restricting to a smaller interval, it is isomorphic to the product foliation F × 0. A foliation admiting only trivial deformations is called rigid. The foliation F on a manifold M is called compact Hausdorff if its leaves are compact and the orbit space is Hausdorff. A version of the local Reeb stability shows that this is the case if and only if every leaf admits arbitrarily small saturated opens over which F is equivalent to a flat bundle foliation [e]. If F is compact Hausdorff then its holonomy groups are finite, its leaves without holonomy L0 are all diffeomorphic, and they comprise a dense open set. In the joint work with R. Fernandes [df2] we show that a compact Hausdorff foliation F on a compact connected manifold M is rigid. We do it by combining classic results such as Thurston stability theorem [t] with the new theory of Rieman- nian metrics on Lie groupoids developed in [df1]. Our result is closely related to the

16 stability properties of actions and foliations studied by R. Palais and H. Rosenberg, among others [lr]. The language of Lie groupoids provide a unifying framework for several geome- tries, such as group actions, pseudogroups, fibrations and foliations. Any foliation F can be integrated to a global object, its holonomy groupoid Hol(F ) → M, and this is Hausdorff and s-proper if and only if F is compact Hausdorff [cmf]. In this talk I will overview the concept of Lie groupoid, explain the theory of metrics over them, apply it to show the rigidity theorem for foliations, and illustrate in this way the use of Lie groupoids in differential geometry and topology.

[cmf] M. Crainic & D. Martinez-Torres & R. Fernandes. Regular Poisson Manifolds of Compact Types (PMCT 2). arxiv preprint arXiv:1603.00064. [df1] M. del Hoyo & R. Fernandes. Riemannian metrics on Lie groupoids. Journal fr die reine und angewandte Mathematik (Crelles Journal) 735 (2018), 143– 173. [df2] M. del Hoyo & R. Fernandes. Riemannian metrics on differentiable stacks. arXiv preprint arXiv:1601.05616. [e] D. Epstein. Foliations with all leaves compact. Ann. Inst. Fourier 26 (1976), 265–282. [lr] R. Langevin & H. Rosenberg. On stability of compact leaves and fibrations. Topology 16 (1977), no. 1, 107–111. [t] W. Thurston. A generalization of the Reeb stability theorem. Topology 13 (1974), 347–352.

Taut foliations on surface bundles over S1 Wednesday 11:45–12:10 Hiraku Nozawa Ritsumeikan University (Japan) [email protected]

We construct new examples of transversely affine foliations on surface bundles over S1, which are generalizations of Meigniez’s examples called suspension foliations of pseudo-Anosov diffeomorphisms in special cases. We also show that these foli- ations have rigidity among transversely affine foliations to generalize a theorem of Nakayama. This is a joint work with Gilbert Hector.

Background

Which hyperbolic 3-manifolds admits a taut foliation? It is a fundamental and deep question on the geometry and topology of foliations and 3-manifolds. One knows that many hyperbolic 3-manifolds admit taut foliations due to constructions of Gabai, Roberts and their collaborators, while Roberts-Shareshian-Stein [RSS], Calegari-Dunfield [CD], Fenley [Fe] showed that there are infinitely many hyper- bolic 3-manifolds which do not admit taut foliations. To the best of the authors’ knowledge, no good criterion is known.

17 Our approach here is to construct and analyse simple examples. The most sim- plest taut foliations on hyperbolic 3-manifolds in foliation point of view can be Meigniez’s examples [Me] on surface bundles over S1, which are called suspension foliations of pseudo-Anosov diffemorphism in special cases. It is a generalization of the stable or unstable foliations of the suspension Anosov flow on torus bundles with hyperbolic monodromy. We generalize Meigniez’s construction to produce new examples of taut foliations (Section 2) and show that they have certain rigidity (Section 3).

New examples of taut foliations

Let us recall Meigniez’s construction of taut foliations on surface bundles on S1, which uses a closed 1-form on a surface which is an eigenvector of the monodromy map. Later we shall describe our construction with a cohomology class of degree 1 of a surface which is an eigenvector of the monodromy map. Let F be an oriented closed surface. Let M be an F -bundle over S1 whose monodromy is f ∈ Diff(F ). Meigniez’s construction is as follows: Assume that ∗ there exists a closed 1-form α on F such that ϕ α = λα for some λ ∈ R>0. Consider a 1-form t ωb± = α ± λ dt on F ×R, which induces a foliation on M = (F ×R)/hfˆi, where fˆ(x, t) = (f(x), t+1). Here we generelize the construction as follows:

Theorem 1. Assume that there exists [α] ∈ H1(F ; R) such that f ∗[α] = λ[α] for some λ ∈ R>0. Then M admits a transversely affine foliation F± with one sided-branching. Theorem 1 produces more examples than Meigniez’s example. If f is pseudo- Anosov, then there are few closed 1-forms α such that f ∗α = λα for some λ ∈ R>0. In fact, such α corresponds to a fixed point on the Thurston boundary of the Teichmuller space. On the other hand, Theorem 1 applies to any eigenvectors of f ∗ : H1(F ; R) → H1(F ; R) associated with positive real eigenvalues. The outline of the construction is as follows: Take [α] ∈ H1(F ; R) such that ∗ ˆ ˆ ϕ [α] = λ[α] for some λ ∈ R>0. For s > 0, consider fs : F × R → F × R; fs(x, t) = ∼ ˆ (f(x), t + s). Note that M = (F × R)/hfsi for any s > 0. Take the same 1-form t ˆ ωb± = α ± λ dt on N × R as Meigniez’s example. In general, fs does not preserves ker ωb±. But, by applying Moser’s trick to non-singular closed ωb, we see that there ˆ ∗ exists ϕ ∈ Diff(F × R) such that ϕ is isotopic to the identity and (ϕ ◦ fs) ωb± = ωb±. ˆ Since the quotient space (F × R)/hϕ ◦ fsi is diffeomorphic to M for s  0, we get a foliation F± on M. Here F± is independent of s up to isotopy. Meigniez’s examples have several good properties. For example, they admit transversely affine structures and they were first examples of taut foliations with one-sided branching. The examples in Theorem 1 share such good propeties with Meigniez’s foliations.

18 Rigidity

We will discuss rigid properties of taut foliations F± in Theorem 1. Ghys-Sergiescu [GS] showed that the stable or unstable foliations of the suspension Anosov flow on torus bundles over S1 with hyperbolic monodromy are unique among foliations with- out compact leaves on the manifolds. One can see that this result does not extend to higher genus surface bundles over S1, but Nakayama [Na] proved that Meigniez’s foliations have certain rigidity among transversely affine foliations. More precisely, Nakayama proved that, under certain topological conditions, any transversely affine foliations without compact leaves and with the maximal Euler class on surface bun- dles over S1 is isotopic to one of Meigniez’s foliation up to a finite covering. Here we generalize this theorem of Nakayama for the foliations F± in Theorem 1. Theorem 2. Any oriented taut transversely affine foliation F whose holonomy homomorphism and the Euler class are equal to those of F+ is isotopic to F+.A similar statement holds for F−. In the first step to prove this theorem, we isotope given F so that there exists a vector field transverse to both of F and the foliation whose leaves are surface fibers. Here, we use Roussarie’s technique [R1,R2] and Cerf theorem to cancel tangent points of the leaves of F to surface fibers. Then we isotope F to make it transverse to a 1-dimensinal horizontal foliation V, which is induced from F ×R = tx∈F {x}×R, by using Cerf type theorem due to Hatcher [Ha] in a way similar to Quˆe-Roussarie [QR]. Then, since F± is also transverse to V, Moser type argument for transversely affine foliations with fixed holonomy homomorphisms implies that F is isotopic to F±.

[CD] D. Calegari & N.M. Dunfield. Laminations and groups of homeomor- phisms of the circle. Invent. Math. 152, 149–204, 2003. [Fe] S. Fenley. Laminar free hyperbolic 3-manifolds. Comment. Math. Helv. 82, 247–321, 2007. [GS] E. Ghys & V. Sergiescu. Stabilite et conjugaison differentiable pour certains feuilletages. Topology 19, 179–197, 1980. [Ha] A. Hatcher. Homeomorphisms of sufficiently large P 2-irreducible 3-manifolds. Topology 15, 343–347, 1976. [Me] G. Meigniez. Bouts d’un groupe op´erant sur la droite. II. Applications `ala topologie des feuilletages. Tohoku Math. J. (2) 43, 473–500, 1991. [Na] H. Nakayama. Transversely affine foliations of some surface bundles over S1 of pseudo-Anosov type. Ann. Inst. Fourier (Grenoble) 41, 755–778, 1991. [QR] Ngoˆ Van Queˆ & R. Roussarie, Sur l’isotopie des formes ferm´eesen di- mension 3. Invent. Math. 64 69–87, 1981. [RSS] R. Roberts, J. Shareshian & M. Stein. Infinitely many hyperbolic 3- manifolds which contain no Reebless foliation. J. Amer. Math. Soc. 16, 639–679, 2003. [R1] R. Roussarie. Sur les feuilletages des vari´et´esde dimension trois. Ann. Inst. Fourier (Grenoble) 21, 13–82, 1971. [R2] R. Roussarie. Plongements dans les vari´et´esfeuillet´eeset classification de feuilletages sans holonomie. Inst. Hautes Etudes´ Sci. Publ. Math. 43 101–141, 1974.

19 Reeb graph: from the analysis through topology to the Wednesday group theory and conversly 12:10–13:00 Plenary Talk Wac law Marzantowicz Adam Mickiewicz University of Pozna´n(Poland) [email protected]

We will present classical and recent facts on the Reeb graph R(f) including a new definition of it. We concentrate our attention on the problem of realization of given graph as the Reeb graph R(f) of a C1-function f with finitely many critical points on a closed manifold. The answer to this problem has various versions depending what idetification of graphs we admit: up combinatorial isomorphism of graphs, up homeomorphism, or up to homotopy. Also a requirement on the function f leads to different answer depending whether it is a C1- function with finitely many critical points, a Morse function, or a simple Morse function respectively. Finally, we will discuss the problem of realization of a homomorphism from any finitely presented group to a free group as the homomorphism, induced by Reeb map, from the fundamental group of a manifold M into the fundamental group of Reeb graph of some function f on M. This is a joint work withLukasz Patryk Michalak.

A Combinatorial/Algebraic Topological Approach to Thursday Nonlinear Dynamics 09:30–10:20 Plenary Talk Konstantin Mischaikow Rutgers, The State University of New Jersey (USA) [email protected]

I will discuss a combinatorial/algebraic topological approach to characterizing non- linear dynamics with particular emphasis on how it can be used to identify regulatory networks from time series data.

20 Existence of common zeros for commuting vector fields on three manifolds Thursday 10:30–10:55 Bruno Santiago Universidade Federal Fluminense (Brazil) [email protected]

We study existence of fixed points for R2 actions on three manifolds. This can be viewed as the joint dynamics of two commuting vector fields, and a fixed point for the action is a common zero. The classical work of Elon Lima gives such a common zero for commuting vector fields of surfaces with non-vanishing Euler characteristic. It is natural then to search for these type of result on three manifolds. As in this case every Euler characteristic vanishes, on has has to replace it by another topological condition. Bonatti proposed the Poincar´e-Hopfindex of one of the involved vector fields as a candidate and showed that if X and Y are real analytic commuting vector fields on a manifold of dimension up to 4 then in every open set U, with compact closure, such that X does not vanishes on the boundary ∂U and has a non-zero Poincar´e-Hopfindex Ind(X,U), there exists a common zero of X and Y . Our aim is to remove analyticity from the three-dimensional version of this result, proving it in the C3 category. We succeed to do so assuming that every periodic orbit γ of Y along which X and Y are collinear (this amounts to saying that γ is a one- dimensional closed orbit of the R2 action) is partially hyperbolic. Our proof sheds new light on the problem and reveals a possible global strategy to attack to full C3 case. This a joint work with S´ebastien Alvarez´ and Christian Bonatti.

[1] E. L. Lima. Common singularities of commuting vector fields on 2-manifolds.. Commentarii Mathematici Helvetici 39.1 (1964): 97-110. [2] C. Bonatti. Champs de vecteurs analytiques commutants, en dimension 3 ou 4: existence de z´eros communs.. Boletim da Sociedade Brasileira de Matemtica- Bulletin/Brazilian Mathematical Society 22(2) (1992): 215-247. [3] S. Alvarez, C. Bonatti and B. Santiago. Existence of common ze- ros for commuting vector fields on three manifolds II. Solving global difficulties arXiv:1710.06743 (2017).

21 Almost-crystallographic groups as quotients of Artin braid Thursday groups 11:45–12:10 Oscar Ocampo Universidade Federal da Bahia (Brazil) [email protected]

Let n, k ≥ 3. We analyse the quotient group Bn/Γk(Pn) of the Artin braid group Bn by the subgroup Γk(Pn) belonging to the lower central series of the Artin pure braid group Pn. We prove that it is an almost-crystallographic group. We then focus more specifically on the case k = 3. If n ≥ 5, and if τ ∈ N is such that gcd(τ, 6) = 1, we show that Bn/Γ3(Pn) possesses torsion τ if and only if Sn does, and we prove that there is a one-to-one correspondence between the conjugacy classes of elements of order τ in Bn/Γ3(Pn) with those of elements of order τ in the symmetric group Sn. Then, we determine some 4-dimensional almost-Bieberbach groups which appears as subgroups of B3/Γ3(P3). This is a joint work with Daciberg Lima Gon¸calves and John Guaschi.

Thursday Thickness of skeletons of hyperbolic orbifolds 12:10–13:00 Plenary Talk Mikhail Belolipetsky Instituto de Matem´aticaPura e Aplicada (Brazil) [email protected]

In 2011, Gromov and Guth proved a deep theorem relating hyperbolic volume, isoperimetric constant and thickness of an embedding of a hyperbolic manifold in Euclidean space. I will first discuss this theorem and its proof. After this, I will present a generalisation of the Gromov-Guth theorem to triangulated hyperbolic orbifolds, which was obtained in a joint work with Hannah Alpert. If time permits, I will also consider some applications of our theorem.

The Borsuk-Ulam property for homotopy class in fibrations Thursday with basis S1 and fiber torus 14:30–14:55 Vinicius C. Laass Universidade Federal da Bahia (Brazil) [email protected]

Given a fibration p : M → S1 whose fiber is the 2-torus T2, suppose that there exist a free involution τ : M → M that is over S1. We say that a homomotpy class α ∈ [M,M], over S1, has the Borsuk-Ulam property (with respect to τ) if for each map f ∈ α, there exists a point x ∈ M such that f(τ(x)) = f(x). Based on [1], we will show that the problem of deciding when a given homotopy class α has the Borsuk-Ulam property is equivalent of the existence of a certain algebraic diagram involving the fundamental groups of M, the orbit space M/τ, the

22 pull-back minus the diagonal M ×S1 M − ∆ and the orbit space of this last space by the involution τ 0(x, y) = (y, x). We will provide some examples based on the list of fibrations that admit free involutions over S1 that is described in [2]. This is a joint work with Daciberg L. Gon¸calves and W. L. Silva.

[1] D. L. Gonc¸alves, J. Guaschi, V. C. Laass. The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero. Preprint (arXiv:1608.00397). [2] M. Sakuma. Involutions on Torus Bundle over S1. Osaka J. Math. 22, 1985 (163-185).

Closure of singular foliations: the proof of Molino’s conjecture Thursday 15:00–15:25 Marcos M. Alexandrino Universidade de S˜aoPaulo (Brazil) [email protected]

A singular foliation on a Riemannian manifold M is called Riemannian if their leaves are locally equidistant. A typical example of a singular Riemannian foliaton is the decomposition of M into the orbits of an isometric group action on M. Another example is the holonomy foliation, the foliation whose leaves are orbits of parallel transports on an Euclidean vector bundle with a metric connection. In this talk we review some basic concepts and examples and give an idea of the proof of Molino’s conjecture that for each singular Riemannian foliation the partition given by the closure of the leaves is again a singular Riemannian foliation. This talk is based on a joint work with Prof. Marco Radeschi [1].

[1] Marcos M. Alexandrino & Marco Radeschi Closure of singular foli- ations: the proof of Molinos conjecture. Compositio Mathematica 153(12), 2577– 2590, 2017.

Meridional rank of knots whose exterior is a graph manifold Thursday 16:25–16:50 Ederson R. F. Dutra Universidade Federal de S˜aoCarlos (Brazil) [email protected]

We prove for a large class of knots that the meridional rank coincides with the bridge number. This class contains all knots whose exterior is a graph manifold. This gives a partial answer to a question of S. Cappell and J. Shaneson (Problem 1.11 in [Kirby]). This is a joint work with Richard Weidmann, Michel Boileau and Yeonhee Jang.

23 Results

An n-bridge sphere of a knot k in the 3-sphere S3 is a 2-sphere which meets k in 2n points and cuts (S3,L) into n-string trivial tangles. An n-string trivial tangle is a pair (B3, t) of the 3-ball B3 and n arcs properly embedded in B3 parallel to the boundary of B3. It is known that every knot admits an n-bridge sphere for some positive integer n. We call a knot k an n-bridge knot if k admits an n-bridge sphere and does not admit an (n − 1)-bridge sphere. We call n the bridge number of the knot k and denote it by b(k). 3 If a knot admits an n-bridge sphere, then it is easy to see that π1(S − k) can be generated by n meridians, where a meridian is an element of the fundamental group that is represented by a curve that is freely homotopic to a meridian of k. This implies that the minimal number of meridians needed to generate the group 3 π1(S − k) is less than or equal to b(k). We denote by w(k) the minimal number of 3 meridians of π1(S − k) and call it the meridional rank of k. Thus for any knot k we always have w(k) ≥ b(k). S. Cappell and J. Shaneson [?, pb 1.11], as well as K. Murasugi, have asked whether the opposite inequality always holds, i.e. whether w(k) = b(k) for any knot k. To this day no counterexamples are known but the equality has been verified in a number of cases:

(1) For generalized Montesinos knots this is due to Boileau and Zieschang [BZ].

(2) For torus knots this is a result of Rost and Zieschang [RZ].

(3) The case of knots of meridional rank 2 (and therefore also knots with bridge number 2) is due to Boileau and Zimmermann [BoZ].

(4) For a class of knots also referred to as generalized Montesinos knots, the equal- ity is due to Lustig and Moriah [LM].

(5) For some iterated cable knots this is due to Cornwell and Hemminger [CH].

(6) For knots of meridional rank 3 whose double branched cover is a graph manifold the equality can be found in [BJW].

We prove the following

Theorem 1 Let k be a knot whose exterior C(k) = S3 − V (k) is a graph manifold. Then the meridional rank w(k) of k is equal to b(k).

[BJW] M. Boileau, Y. Jang, R. Weidmann. Meridional rank and bridge number for a class of links. Pac. J. Math. 292(1), 61–80, 2018. [BZ] M. Boileau, H. Zieschang. Nombre de ponts et g´en´erateurs m´eridiensdes entrelacs de Montesinos, Comment. Math. Helv. 60, 270–279, 1985. [BoZ] M. Boileau, B. Zimmermann. The π-orbifold group of a link, Math. Z. 200 187–208, 1989. [CH] C. R. Cornwell, D. R. Hemminger. Augmentation rank of satellites with braid pattern, Comm. Anal. Geom. 24(5), 939–967, 2016.

24 [Kirby] R. C. Kirby. Problems in low-dimensional topology. Proc. of Georgia Top. Conference, Part 2, AMS International Press, 35–473, 1995. [LM] M. Lustig, Y. Moriah. Generalized Montesinos knots, tunnels and N -torsion, Math. Ann. 295, 167–189, 1992. [RZ] M. Rost, H. Zieschang.Meridional generators and plat presentations of torus links, J. Lond. Math. Soc. 35(2), 551-562, 1987.

Quotients of the torus braid groups and crystallographic groups Thursday 16:50–17:15 Carolina de Miranda e Pereiro Universidade Federal do Esp´ıritoSanto (Brazil) [email protected]

This is a work in progress, joint with Oscar Ocampo (UFBA). Let T be the torus. We study the quotients of the torus braid groups by the lower central series of the torus pure braid groups, Bn(T)/Γk(Pn(T)), with the main goal of determining for which values of n and k these groups are crystallographic or almost-crystallographic groups. We also obtain a classification of (some) elements of finite order in Bn(T)/[Pn(T),Pn(T)].

Stable Signatures for Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence Friday 09:30–10:20 Facundo M´emoli Plenary talk Ohio State University (USA) [email protected]

When studying flocking/swarming behaviors in animals one is interested in quanti- fying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. In a similar vein, studying the dynamics of social networks leads to the problem of characterizing groups/communities as they form and disperse throughout time. Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent data. Given a finite dynamic graph (DG), we first construct a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG. Based on standard results, we then obtain a persistence diagram or barcode from this zigzag persistence module. We prove that these barcodes are stable under perturbations in the input DG under a suitable distance between DGs that we identify. More precisely, our stability theorem can be interpreted as providing a lower bound for the distance between DGs. Since it relies on barcodes, and their bottleneck distance, this lower bound can be computed in polynomial time from the DG inputs. Along the way, we propose a summarization of dynamic graphs that captures their time-dependent clustering features which we call formigrams. These set-valued

25 functions generalize the notion of dendrogram, a prevalent tool for hierarchical clus- tering. In order to elucidate the relationship between our distance between two DGs and the bottleneck distance between their associated barcodes, we exploit re- cent advances in the stability of zigzag persistence due to Botnan and Lesnick, and to Bjerkevik. This is joint work with Woojin Kim and Zane Smith.

Friday Deficient and multiple points of maps into a manifold 10:30–10:55 Tha´ısF. M. Monis Universidade Estadual Paulista (Brazil) [email protected]

Keywords multiple point, deficient point, Hopf’s absolute degree

For a map f : X → M we study the set of deficient and multiple points, where M is a manifold. In case of the set of deficient points, we estimate its dimension. For multiple points, we study its density in X, and we also provide examples where its complement is dense. This is a joint work with Daciberg L. Gon¸calves and Stanis law Spie˙z.

Introduction

Let Sn be the unitary sphere in Rn+1. The classical Borsuk-Ulam Theorem asserts that for every continuous map f : Sn → Rn there is x ∈ Sn such that f(x) = f(−x). A simple consequence is that there is no injective continuous map Sn → Rn. Given a continuous map f : X → Y , a point x ∈ X is called a multiple point of f if f −1(f(x)) 6= {x}. Otherwise, it is called a single point. For continuous maps from Sn into Rn, although there is no injective continuous map, one may ask “how big” could be the set of single points of a such map, also about the set of multiple points. In a general study about multiple points of continuous maps between manifolds of the same dimension, D. L. Gon¸calves gave proved the following:

Theorem 1 ([GonOre]). Let M,N be manifolds of the same dimension, with M closed. Let [f] denote the homotopy class of a continuous map f : M → N.

a) If the Hopf’s absolute degree of f is 1, then there is a map g ∈ [f] such that the set of multiple points is not dense in M.

b) If the map f has Hopf’s absolute degree different from 1 then the set of multiple points of any map g ∈ [f] is dense.

26 And S. Orevkov provided an example of a continuous map f : S2 → R2 such that the set of single points is dense, but does not contain any open set. In another direction, but also related, there is the following result of P. T. Church and J. G. Timourian. Theorem 2 ([CT]) Suppose M and N are connected n-manifolds and f : M → N is a proper map with Hopf’s absolut degree A(f) 6= 0. Let ∆f be the set of points y ∈ N for which f −1(y) has less than |A(f)| points.

(1) Then dim ∆f ≤ n − 1 and ∆f contains no closed (in N) subset of dimension n − 1.

−1 (2) If f is discrete (i.e., each f (y) is discrete), then dim ∆f ≤ n − 2.

In this work, based on the papers [Epstein] and [Olum], we extend the notion of Hopf’s absolute degree to proper maps from a connected, locally path-connected, locally compact n-dimensional space into a n-manifold. The domain also requires to satisfy a certain cohomological condition. Then, we are able to prove similar results to Theorem 1 and Theorem 2, where the domain of the map is not assumed to be a manifold.

[CT] P. T. Church, J. G. Timourian , Deficient points of maps on manifolds , Michigan Math. J. (27) (1980), no. 3, 321-338. [Epstein] D. B. A. Epstein , The degree of a map , Proc. London Math Soc. 16 (1966), 369–383. [GonOre] D. L. Gonc¸alves , The size of multiple points of maps between manifolds , with an Appendix written by S. Orevkov. Topology proceedings 48 (2016), 361– 373. [Olum] P. Olum , Mappings of manifolds and the notion of degree, Annals of Math- ematics, vol 58, no. 3, (1953) 458–480.

Limiting Actions of Infinite Surface Groups Friday 11:45–12:10 Marcel Vinhas Bertolini Universidade Federal do Par´a(Brazil) [email protected]

We consider sequences of conjugacy classes of Kleinian representations of infinite surface groups. For certain such sequences going to infinity in the topology of alge- braic convergence, we guarantee the existence of subsequences that are projectively convergent. Geometric models of the ideal limits are provided by small isometric actions of the group on non-trivial R-trees. This extends results of Morgan-Shalen / Bestvina / Paulin on finitely generated groups. Consider the space of Kleinian representations of a group, with the topology of algebraic convergence, and the respective space of conjugacy classes. For finitely generated groups, if a sequence of conjugacy classes contains no convergent sub- sequences, then it contains a subsequence that converges projectively to a small

27 isometric action of the group on a non-trivial R-tree (assuming that the group is non-elementary). This was first proved by Morgan-Shalen, and the result was later established by geometric arguments, independently, by Bestvina and Paulin (for instance, see [1,3]). Here we address to the problem of extending this geometric version to infinite surface groups. Our context will now be outlined. Let S be a topologically infinite (hyperbolic) surface, and let Γ be its fundamental group, identified as a Fuchsian group. Take on S a quasiconformal homeomorphism φ with a property that we call generating: there exist a finite set F ⊂ Γ such k that ∪k∈Zφ∗F generates Γ, where φ∗ is an automorphism of Γ representing φ. For example, the generalized pseudo-Anosov transformations built by Andr´ede Car- valho and Toby Hall [2], when properly punctured, are generating. Denoting by H3 the hyperbolic three-dimensional space, we look at two copies of S as the ends of the 3-manifold H3/Γ, to which we extend φ as a biLipschitz homeomorphism with dilatation bounded by L ≥ 1. Then, we consider the subspace of Kleinian representations ρ that are compatible with φ, in the sense that there exist a L-biLipschitz homeomorphism f of H3/ρ(Γ) such that ρ ◦ φ∗ = f∗ ◦ ρ. For example, the quasi-Fuchsian representations whose Bers’ coordinates are ([φk], [φ−k]) have this property. Under additional technical conditions, we extend the Bestvina-Paulin argument. Namely, we prove that if a sequence of conjugacy classes of compatible representations contains no convergent subsequences and is non-exceptional, then it contains a subsequence that converges projectively to a small isometric action of Γ on a non-trivial R-tree. This is a joint work with Andr´ede Carvalho.

[1] M. Bestvina. R-trees in topology, geometry, and group theory. Handbook of geometric topology, 55–91, 2001. [2] A. de Carvalho & T. Hall. Unimodal generalized pseudo-Anosov maps. Geometry & Topology 8(3), 1127–1188, 2004. [3] J.-P. Otal. Le th´eor`emed’hyperbolisation pour les vari´et´esfibr´eesde dimension 3. Ast´erisque 235, 1996.

Friday Retracts of free groups 12:10–13:00 Plenary Talk Slobodan Tanushevski Universidade Federal Fluminense (Brazil)

A subgroup R of a group G is said to be a retract of G if there is a homomorphism r : G → R that restricts to the identity on R. I will discuss the role of retracts in the study of endomorphisms of free groups and describe some recent results on the structure of retracts of free groups.

28 Topological data analysis on materials science and several problems in random topology Friday 14:30–15:20 Yasuaki Hiraoka Kyoto University (Japan) [email protected]

Topological data analysis (TDA) is an emerging concept in applied mathematics in which we characterize“shape of data” using topological methods. In particular, the persistent homology and its persistence diagrams are nowadays applied to a wide variety of scientific and engineering problems. In my talk, I will explain our recent activity of TDA on materials science, e.g. glass [1] (Figure 1), granular systems [2], iron ore sinters [3] etc. By developing several new mathematical tools based on quiver representations [4], inverse analysis [5], and machine learnings [6], we can explicitly characterize significant geometric and topological (hierarchical) features embedded in those materials, which are practically important for materials properties. I will also present several interesting mathematical problems in random topology (limit theorems of persistence diagrams [7] and higher dimensional percolations) which are motivated from those applications.

Figure 1: Persistence diagrams of silica (SiO2). Left: crystal, middle: glass, right: liquid.

[1] Y. Hiraoka, T. Nakamura, A. Hirata, E. G. Escolar, K. Matsue, and Y. Nishiura. Hierarchical structures of amorphous solids characterized by persistent homology. PNAS 113 (2016), 7035-7040. [2] M. Saadatfar, H. Takeuchi, N. Francois, V. Robins, and Y. Hiraoka.. Pore con- figuration landscape of granular crystallisation. Nature Communications. 8:15082 (2017). [3] M. Kimura, I. Obayashi, Y. Takeichi, R. Murao and Y. Hiraoka. Non-empirical identification of trigger sites in heterogeneous processes using persistent homology. Scientific Reports 8, Article number: 3553 (2018). [4] E. Escolar and Y. Hiraoka. Persistence Modules on Commutative Ladders of Finite Type. Discrete & Computational Geometry, 55 (2016), 100-157. [5] M. Gameiro, Y. Hiraoka, I. Obayashi. Continuation of Point Clouds via Persis- tence Diagrams. Physica D: Nonlinear Phenomena, 334 (2016), 118-132.

29 [6] I. Obayashi, Y. Hiraoka, M. Kimura. Persistence diagrams with linear machine learning models. J. Appl. and Comput. Topology (2018). [7] T. K. Duy, Y. Hiraoka, and T. Shirai. Limit theorems for persistence diagrams. To appear in Annals of Applied Probability.

Friday On the construction of a covering map 15:00–15:25 Samson Saneblidze A. Razmadze Mathematical Institute I.Javakhishvili Tbilisi State University (Georgia) [email protected]

Let Y = |X| be the geometric realization of a path-connected simplicial set X, and let G = π1(X) be the fundamental group. Given a subgroup H ⊂ G, let G/H be the set of cosets. Using the combinatorial model ΩX → PX → X of the path fibration ΩY → PY → Y and a canonical action µ : ΩX × G/H → G/H we construct a covering map G/H → YH → Y as the geometric realization of the associated short sequence G/H → PX ×µ G/H → X. This construction, in particular, does not use the existence of a maximal tree in X. For a 2-dimensional X and H = {1} it can be also viewed as a simplicial approximation of a Cayley 2-complex of G.

Introduction

The construction of a covering map is in fact a special case of a general one and relies on the work [1]. Given a simplicial set X, let ΩX → PX → X be a short sequence with ΩX to be a monoidal cubical set such that the 0-dimensional elements (vertices) Ω0X ⊂ ΩX form a group. (Here note that although the vertices of |ΩX| are formed by edge-loops in |X| similarly to [2], the number of the other cells of |ΩX| is much reduced.) Let F be a cubical set with an action µ : ΩX ×F → F being in particular a cubical map. One can form the associated short sequence F → PX ×µ F → X where PX ×µ F := {(PX × F )∼ :(xα, y) ∼ (x, αy), (x, y) ∈ PX × F, α ∈ ΩX}. Now take F = G/H to be a trivial cubical set, i.e., the set of n-dimensional elements is the set G/H for all n with identity face and degeneracy maps, so all non-degenerate elements are concentrated in the 0-degree. Let ΩX → π0(ΩX) = π1(X) := G be the quotient map, and G × G/H → G/H be the standard action. Then these data induce the action µ : ΩX × G/H → G/H. Thus we obtain the short sequence G/H → PX ×µ G/H → X the geometric realization of which is just a covering map G/H → YH → Y on Y = |X|. Note that YH is automatically triangulated because all elements of positive degrees are degenerate in the cubical set G/H. We mention the following difference of the above construction of a covering map from the standard combinatorial ones (cf. [3]):

• X is not necessarily 2-dimensional;

• The existence of a maximal tree in X is not used;

30 • The orientation of edges in X is fixed by the underlying simplicial structure. (Instead the two orientations of edges are just encapsulated in the definition of ΩX.)

[1] M. Rivera & S. Saneblidze. A combinatorial model for the path fibration. Preprint, math. AT/1712.02644. [2] M. Gromov. Homotopical effects of dilatation. J. Diff. Geometry 13, 303–310, 1978. [3] R.C. Lyndon & P.E. Schupp. Combinatorial group theory. Springer-Verlag, 2001.

Topology of the leaves of hyperbolic surface laminations Friday 15:50–16:15 S´ebastien Alvarez´ Universidad de la Rep´ublica(Uruguay) [email protected]

We show here how to use towers of covering maps to construct minimal laminations by hyperbolic surfaces whose leaves have prescribed topologies. In particular we construct a lamination such that every noncompact surface is realized as a leaf. In this talk we are interested in the topology of leaves of compact minimal lam- inations by hyperbolic surfaces. We know from the work of Ghys [2] and Cantwell- Conlon [3] that the topology of the generic leaf of such a lamination can be one of six types. It has one, two, or a Cantor set of ends and has no genus or all its ends have genus. On the other hand, it is proven in [1] that infinite topological type is highly contagious: if the generic leaf is not a disc then every leaf must be of infinite topological type. In fact, in all known examples every leaf have finite topological type, or every leaf have infinite topological type so we ask naturally if this is a dichotomy. We prove that it is far from being the case. In fact we will show how to use towers of finite covering maps of closed surfaces in order to construct minimal laminations by hyperbolic surfaces leaves have prescribed topologies. In particular we construct a lamination such that every noncompact surface is realized as a leaf.such that every non-compact surfaces realize as a leaf. In particular, the generic leaf of such an example must be a disc. This is joint work with J. Brum, M. Mart´ınezand R. Potrie.

[1] F. Alcalde Cuesta, F. Dal’bo, M. Mart´ınez, A. Verjovsky. Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology. Discrete Contin. Dyn. Syst. A. 36, 4619–4635, 2006. [2] J. Cantwell, L. Conlon. Generic leaves. Comment. Math. Helv. 73, 306–336, 1998. [3] E.´ Ghys. Topologie des feuilles g´en´eriques. Ann. of Math 141, 387–422, 1995.

31 32 Author Index

Alvarez´ Yasuaki, 29 S´ebastien,31 Alvarez´ L´opez Korinman Jes´usA., 10 Julien, 10

Alexandrino Laass Marcos, 23 Vinicius, 22 Almeida Lima Kisnney, 14 Daciberg, 4

Barbot M´emoli Thierry, 15 Facundo, 25 Barral Lij´o Marzantowicz Ram´on,12 Wac law, 20 Barreto Matucci Alexandre, 9 Francesco, 1 Belolipetsky Meni˜noCot´on Mikhail, 22 Carlos, 2 Bertolini Mio Marcel, 27 Washington, 1 Borghini Mischaikow Eugenio, 7 Konstantin, 20 Monis Caramello Tha´ıs,26 Francisco, 6 Nozawa Dekimpe Hiraku, 17 Karel, 9 del Hoyo Ocampo Matias, 16 Oscar, 22 Dutra Ederson, 23 Pereiro Carolina, 25 Geoghegan Piterman Ross, 8 Kevin, 8 Ghys Etienne,´ 2 Saneblidze Samson, 30 Hiraoka Santiago

33 Bruno, 21 Tanushevski Skipper Slobodan, 28 Rachel, 4 T¨oben Vendr´uscolo Dirk, 5 Daniel, 6

34 XXI Brazilian Topology Meeting Poster Abstracts

Niter´oi 12th–18th August 2018 ii General Info

Welcome at Niter´oi!

This booklet encloses all the abstratcs of the posters. The additional information and references given by the participants are also included.

Poster sessions will occur on tuesday and thursday, during and after the second coffe break (15:25–16:15) in the Block H, by the stilts.

iii iv Contents

General Info iii

Tuesday - Poster Session 1 1 Limited information strategies in Games (Juan Francisco Camasca Fern´andez) 1 Topologies of a pinchuk’s map via intersection homology (Nguyen Thi Bich Thuy) ...... 1 Join Product of Generators of Homotopy Groups of the Stiefel Manifolds of 5-frames and 6-frames (Nancy de Souza Cardim, M´arioOlivero Marques da Silva)...... 3 Enneper representation of minimal surfaces in the Lorentz-Minkowski 3- space (Adriana Araujo Cintra )...... 4 Heegaard Floer Homology of 3-Manifolds with Heegaard’s genus 2 (Celso Melchiades Doria) ...... 4 Some results on extension of maps and applications (Alice K. M. Libardi, Edivaldo L. dos Santos) ...... 6 The Reidemeister torsion of tetrahedral spherical space forms (Ana Paula Tremura Galves) ...... 7

Some properties of the algebraic invariant e∗(G)(Maria Gorete Carreira Andrade) ...... 7 Homological Tools in Dynamical Systems (Dahisy Lima)...... 8 Singular surfaces of revolution (Luciana de F´atimaMartins) ...... 8 Coincidences and Wecken type results (Gustavo de Lima Prado) ...... 9 Transition Matrix (Ewerton Rocha Vieira)...... 9 Optimization of Maggot Mass Rearing via Geometric Statistical Analysis (Jamil Viana Pereira, Alice Kimie Miwa Libardi) ...... 10 Good real deformations of co-rank one map germs from R3 to R3 (Taciana Oliveira Souza)...... 11

Thursday - Poster Session 2 13 Free collisions motion planning for rigid bodies: a Topological Complexity approach (C´esarIpanaque Zapata) ...... 13 Fundamental domains, a dynamical question (Gabriel Longatto Clemente) 14 Homotopy theory of finite networks (Guilherme Vituri F. Pinto) ...... 14 The hamburger theorem: a generalization of ham sandwich theorem (Le- andro Vicente Mauri) ...... 15 Geometrization of Bowen-Series like maps (Natalia A. Viana Bedoya) . . . 16

v Stiefel-Whitney Classes as Smooth Embeddings Obstrucions (Alex Melges Barbosa)...... 16 Germ of Singular Holomorphic Foliations of Codimension One Of Second Type in (C3, 0) (Allan Ramos de Souza) ...... 17 The Chomsky Hierarchy (Bianca B. Dornelas) ...... 17 Artin braid groups and some finite quotients (Caio Lima Silva) ...... 18 Equa¸c˜oesde Euler-Poisson (In´acioRabelo)...... 19 Braids Groups and the Word Problem (Raquel Magalh˜aesde Almeida de Cruz) ...... 19 A differentiable but not C1 version of the Inverse Mapping Theorem (Telmo Irineo Acosta Vellozo) ...... 20 The Nielsen fixed point theory for multi-valued maps on surfaces (Bartira Mau´es)...... 20 Persistent Homology and Optimal Cycles (Carlos Henrique Venturi Ronchi) 21

Author Index 23

vi Tuesday - Poster Session 1

Limited information strategies in Games Juan Francisco Camasca Fern´andez Universidade de S˜aoPaulo (Brazil) [email protected]

We study some results in the Gruenhag’s and Menger’s Game with limited in- formation strategies. In addition, we approach the Union Filling Game played in cardinals κ and relationships of that game with the Mengers Game. In the search for a direct relationship between the two games in question, we defined the state- ment S(κ) relating to the almost-compatibility of functions from countable subsets of κ into ω. Finally, we see some covering properties derived of Mengers Game and robustly Menger spaces.

[1] Clontz, Steven. Limited Information Strategies for Topological Games. Doc- toral Thesis. 2015. [2] Kenneth Kunen. Set theory. Volume 102 of Studies in Logic and the Founda- tions of Mathematics. North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. [3] Marion Scheepers. Concerning n-tactics in the countable-finite game. J. Symbolic Logic, 1991.

Topologies of a pinchuk’s map via intersection homology Nguyen Thi Bich Thuy Universidade Estadual Paulista (Brazil) [email protected]

Let F : Kn → Kn , where K = C or K = R, be a polynomial map and let us denote by JF the Jacobian matrix of F . In 1939, O. H. Keller stated a famous conjecture known nowadays as the Jacobian Conjecture, whose statement is the following: “A polynomial map F : Kn → Kn is nowhere vanishing jacobian, i.e. det(JF ) 6= 0, for every x ∈ K, if and only if it is a polynomial automorphism.” A polynomial map F : Kn → Kn satisfying the Jacobian condition “det(JF ) 6= 0, for every x ∈ K” is called a Keller map. The Jacobian Conjecture remains open today even for the dimension 2 in the complex case. However, in the 2-dimensional real case, the Real Jacobian Conjecture is solved by Pinchuk [P] in the year 1994. In fact, Pinchuk

1 provided a counter-example by giving a polynomial map P : R2 → R2 , called the Pinchuks map, satisfying det(JP) > 0 for every x ∈ R2 but F is not injective. Immediately, the geometry of the Pinchuks map is a desired object to study, since the Jacobian condition “det(JF ) 6= 0, for every x ∈ K” in the complex case becomes “det(JF ) is a constant polynomial” in the real case. In others words, the study of polynomial maps F : R2 → R2 with the condition det(JF ) ≡ const. 6= 0 is exacly the study of the 2-dimensional Complex Jacobian Conjecture. Then the geometry of the Pinchuks map may help us to find out a counter-example of the 2-dimensional Complex Jacobian Conjecture if there exists. Moreover, a weak Jacobian Conjecture says that if a nonzero constant Jacobian determinant polynomial map of R2 to itself is injective. This is weaker than the standard Jacobian Conjecture even though injectivity implies bijectivity, but the inverse does not always hold. The asymptotic variety of a given polynomial map F : Kn → Kn is the smallest n 1 n set SF such that the map F : K \ F (SF ) → K \ SF is proper. The Jacobian conjecture reduces to show that the asymptotic variety of a polynomial map F : Kn → Kn with nonzero constant Jacobian is empty. Therefore, the asymptotic variety SP of the Pinchuk’s map is non-empty. In [VV], Anna Valette and Guillaume Valette gave a new approach to study the Complex Jacobian Conjecture in the case of dimension 2: they constructed some real 2n-dimensional pseudomanifolds contained in Rν, where ν > 2n, associated to a given polynomial map F : Cn → Cn, such that the singular locus of these pseudomanifolds are contained in (SF × K0(F )) × {0Rν2n } where K0(F ) is the set of critical values of F . In the case of dimension 2, they prove that with a nonvanishing 2 2 jacobian polynomial map F : C → C (i.e, K0(F ) = ∅), the condition “SF = ∅” is equivalent to the condition “the intersection homology in dimension two and with any peversity of any constructed pseudomanifold is trivial” (Theorem 3.2 in [VV]). This result is generalized in the case of higher dimension in [NVV]. Singular varieties constructed by Anna and Guillaume Valette are called Valette varieties and denoted by V F . We can also construct Valette varieties associated to real polynomial maps F : Rn → Rn (see Remark 2.7 of [VV] or Proposition 3.8 of [NVV]). Two natural questions then arise: 1) How are behaviours of Valette varieties associated to the Pinchuk’s map?

2) Is there a “real version” of Anna and Guillaume Valette’s result, i.e, if F : R2 → R2 is a nowhere vanishing jacobian polynomial map then the condition 0 SF = ∅ is equivalent to the condition IH1 (VF ) = 0? (Notice that in this case, the dimension of VF is 2, then there exists only the zero perversity). We will describe in this talk a Valette variety V P associated to the Pinchuk’s map and calculate its intersection homology. The result describes topologies of singularities of the Pinchuk’s map. It is a counter-example for the “real version” of Anna and Guillaume Valette’s result in [VV].

[N] T. B. T. Nguyen Geometry of singularities at infinity of the Pinchuk’s map. ArXiv: 1710.03318 (2017). [NVV] T. B. T. Nguyen, A. Valette, G. Valette. On a singular variety as- sociated to a polynomial mapping. Journal of Singularities volume 7, 190–204, 2013.

2 [NR] N.T.B. Nguyen, M.A.S. Ruas. On singular varieties associated to a poly- nomial mapping from Cn to Cn1 , to appear in the Asia Jornal of Mathematics. [P] S. Pinchuk. A counterexample to the strong Jacobian conjecture. Math. Zeitschrift, 217, 1–4, 1994. [VV] A. Valette, G. Valette. Geometry of polynomial mappings at infinity via intersection homology. Ann. I. Fourier vol. 64-5, 2147–2163, 2014.

Join Product of Generators of Homotopy Groups of the Stiefel Manifolds of 5-frames and 6-frames Nancy de Souza Cardim, M´arioOlivero Marques da Silva Universidade Federal Fluminense (Brazil) [email protected]

Joint work with Maria Herminia Paula Leite mello. Randall and others [2] used the intrinsic join product to express the index of a k-field with finite singularities on the total space of a smooth fiber bundle F,→ M → B, for 1 < k ≤ 4. The k-field with finite singularities on the total space M arises from k-fields with finite singularities given on the fiber F and on the base B. The index of the k-field on M is the join product of indices of the k-fields with finite singularities defined on F and on B. In this work we extend the results of [2] for the cases where k = 5 and 6. We present tables for the join product of generators of πm−1(Vm,k) for k = 5 and 6, where Vm,k denotes the real Stiefel manifold of k-frames in Rm. The index of a k-field with finite singularities on a smooth, closed, oriented manifold, for k = 5 and 6, is given precisely in terms of the characteristic numbers of the manifold as coefficients of the generators of πm−1(Vm,k).

[1] M. F. Atiyah, J. L. Dupont. Vector fields with finite singularities. Acta Math. 128 (1972), 1–40. [2] N. S. Cardim, M. H. P. L. Mello, D. Randall, M. O. M. da Silva. Join products and indices of k-fields over differentiable fiber bundles. Topology and its Applications 136 (2004), 275–291. [3] I. M. James. The topology of Stiefel manifolds. Cambridge University Press, Volume 24 (1976). [4] U. Koschorke. Vector fields and other vector bundles morphisms - A singu- larity approach. Lecture Notes in Math. Volume 847 - Springer-Verlag (1981). [5] Y. Nomura. Some homotopy groups of real Stiefel manifolds in the metastable range I. Sci. Rep., Col. Gen. Educ., Osaka Univ., Volume 27, No. 1 (1978). [6] D. Randall. Tangent Frame Fields on Spin Manifolds. Pacific Joutnal of Mathematics Vol. 76, No. 1, (1978). [7] D. Randall. On Indices of Tangent Fields with Finite Singularities. Lecture Notes in Math. Volume 1350, (1988), 213–240. [8] E. Thomas, Vector fields on manifolds. Bull. Amer. Soc. 75 (1969), 643–683. [9] H. Toda. Composition Methods in Homotopy Groups of Spheres. Ann. Math. Studies, No. 49 Princeton U. Press (1962).

3 Enneper representation of minimal surfaces in the Lorentz-Minkowski 3-space Adriana Araujo Cintra Universidade Federal de Goi´as(Brazil) [email protected]

The Weierstrass representation formula for minimal surfaces in R3 has been a fundamental tool for producing examples and proving general properties of such surfaces, since the surfaces can be parametrized by holomorphic data. In [MMP] the authors describe a general Weierstrass representation formula for simply connected minimal surfaces in an arbitrary Riemannian manifold. In [a], Andrade introduces a new method to obtain minimal surfaces in the Euclidean 3-space which is equivalent to the classical Weierstrass representation and, also, he proves that any immersed minimal surfaces in R3 can be obtained using it. This method has the advantage of computational simplicity, with respect to the Weierstrass representation formula, and allows to construct a conformal minimal immersion ψ :Ω ⊂ C → C × R, from a harmonic function h :Ω → R, provided that we choose holomorphic complex valued 2 functions L, P on the simply connected domain Ω such that Lz Pz = (hz) . The immersion results in ψ(z) = (L(z) − P (z), h(z)) and it is called Enneper immersion associated to h. Besides, the image ψ(Ω) is called an Enneper graph of h. This aim of this work is to discuss an Enneper type representation for minimal surfaces in the Lorentz-Minkowski space L3. This is a joint work with Irene Ignazia Onnis.

[a] P. Andrade. Enneper immersions, J. D’analyse Math´ematique 75 (1998), 121– 134. [Cintra] A.A. Cintra, Irene I. Onnis. Enneper representation of minimal sur- faces in the three-dimensional Lorentz–Minkowski space. Annali di Matematica Pura ed Applicata. 197 (2018), 21–39. [MMP] F. Mercuri, S. Montaldo, P. Piu. Weierstrass representation formula 2 of minimal surfaces in H3 and H × R. Acta Math. Sinica 22 (2006), 1603–1612.

Heegaard Floer Homology of 3-Manifolds with Heegaard’s genus 2 Celso Melchiades Doria Universidade Federal de Santa Catarina (Brazil) [email protected]

Let (M 3, s) be a closed 3-manifold admitting a genus 2 Heegaard decomposition c (HD) M = H1 ∪h H2 endowed with a spin -structure s. We claim the Heegaard Floer Homology Groups HFH\(M, s) introduced in [3] can be defined on the Jacobian manifold JΣ associated to the Riemann Surface Σg instead of on the Symmetric g Space Sym (Σg). Let f : M → R be a Morse function whose critical point set is

4 Cr(f) = {pm, p1, . . . , pg, q1, . . . , qg, pM } whose Morse indexes are λ(pm) = 0, λ(pi) = 1, λ(qj) = 2, 1 ≤ i, j ≤ g and λ(pM ) = −1 −1 −1 3. Assume pm = f (0), pi = f (1/3), qj = f (2/3), for all i, j = 1, . . . , g, −1 −1 pM = f (1) and Σg = f (1/2). Given a diffeomorphism h :Σg → Σg, the HD i 3−i induced by the handle decomposition is (hi = D × D )

[ p1 pg [ q1 qg [ M = h0 (h1 ∪ ... ∪ hg ) (h1 ∪ ... ∪ hg ) h3 == H1 ∪h H2, | {z } | {z } H1 H2 where Σg = ∂H1 = ∂H2 is the boundary of the handlebodies. On Σg, we consider 2 the curves {αi, βj}, i, j = 1, . . . , g, given by the images αi = f({0} × ∂D ) of pi 2 the 1-handle h1 co-core’s boundary and βj = f(∂D × {0}) the image of the 2- qj handles h2 core’s boundary. M can be assembled by the data encoded in the sets α = {α1, . . . , αg} and β = {β1, . . . , βg}. A Heegaard decomposition on M c 2 induces a spin -structure s ∈ H (M, Z) on M by fixing a point z0 ∈ Σg − (α ∪ β). On the product P (Σg, g) = Σg × ... × Σg, the symmetric group Sg action (g permutations) Sg × P (Σg, g) → P (Σg, g), (τ, (x1, . . . , xg)) → (xτ(1), . . . , xτ(g)). leads g to the quotient space Sym (Σg) = P (Σg, g)/Sg, which is a K¨ahlermanifold of real dimension 2g. The K¨ahlerform is induced from the K¨ahlerform on Σg. The sets α g and β define embedded lagrangean g-tori Tα = π(α) and Tβ = π(β) in Sym (Σg) whose intersection is a a finite set of points Tα ∩ Tβ = {ζ1 . . . , ζn}. 2 Consider D = [0, 1] × iR,Ωα(x, y) = {γα : [0, 1] → Tα | γα(0) = x, γα(1) = y} 1,2 and Ωβ(x, y), similarly defined, to be the spaces of arcs with the L topology. A g 2 g Whitney disk in Sym (Σg) is a map u : D → Sym (Σg) such that 2 (i) u({0} × iR) ⊂ Tα, u({1} × iR) ⊂ Tβ. (ii) u(int(D )) ∩ Tα ∩ Tβ = ∅. (iii) limt→−∞ u(s + it) = x and limt→∞ u(s + it) = y ∈ Ωα(x, y), A Whitney disk is pseudo-holomorphic if satisfies the equation

du du + J(s) = 0, ds dt

g where J is the cplex structure on Sym (Σg). The Heegaard Floer Homology groups HFH\(M, s) are defined by the Symplectic Floer Homology theory applied to these pseudo-holomorphic disks. The whole point of the theory is to count the number of Whitney pseudo-holomorphic disks connecting 2 points (x, y) ∈ Tα ∩ Tβ. g The relationship between the spaces Sym (Σg) and the Jacobi Torus JΣ = H1(Σ,O g H1(Σ,Z ) = C /L,(L lattice), is well known from [1]. Let {ω1, . . . , ωg} be a ba- 0 1,0 R z sis of H (Σ, O ) and Wi = ωi. The Jacobi Map J :Σ → JΣ, J (z) = z0 (W1(z),...,Wg(z)) modL, extends to a Sg-invariant map Js :Σ × ... × Σ → JΣ, P Js(z1, . . . , zg) = i J (zi). The case g = 2 has interesting properties: 2 4 2 (1) Sym (Σ2) = T #CP . (2) The Jacobi Map defines J : T 4#CP 2 → T 4. 2 (3) A Whitney holomorphic disk define a holomorphic disku ˆ : D → JΣ by applying

5 the Sacks-Uhlenbeck Theorem [5] on harmonic maps; Therefore, the HFH groups can be defined on J . There are many examples of 3-manifolds admitting genus g = 2 Heegaard decomposition [6].

[1] Arthur Mattuck. Symmetric Products and Jacobians, American Journal of Mathematics, Vol. 83, No. 1, (Jan., 1961), pp. 189-206. [2] I. G. MacDonald.- Symmetric Product of an Algebraic Curve - Topology. Vol. 1, pp. 319-343. Pergamon Press, 1962. Printed in Great Britain. [3] Peter Ozsvath´ and Zoltan´ Szabo´. Holomorphic Disks and Topological Invariants for closed Three-manifolds, Annals of Mathematics 2, 159(3), 1127-1258, 2004. [4] Peter Ozsvath´ and Zoltan´ Szabo´.Holomorphic Disks and Three-manifolds Invariants: properties and applications. Annals of Mathematics 2, 159(3), 1159- 1245, 2004. [5] J. Sacks and K. Uhlenbeck. The Existence of Minimal Immersions of 2- Spheres, Annals of Mathematics, 2nd series, vol. 113, 1 (Jan., 1981), pp. 1-24. [6] Hiroshi Goda & Chuichiro Hayaashi.Genus Two Heegaard Splittings of Exterior of 1-genus 1-Bridge Knots II, Saitama Math. J. Vol. 29 (2012), 25-53.

Some results on extension of maps and applications Alice K. M. Libardi1, Edivaldo L. dos Santos2 1Universidade Estadual Paulista, Universidade Federal de S˜aoCarlos (Brazil) [email protected]

This work concerns extension of maps using obstruction theory under a non classical viewpoint. It is given a classification of homotopy classes of maps and as an application it is presented a simple proof of a theorem by Adachi about equivalence of vector bundles. Also it is proved that, under certain conditions, two embeddings are homotopic up to surgery if and only if the respective normal bundles are SO-equivalent. This is joint work with Carlos Biasi and Thiago de Melo.

[1] M. Adachi, M.. A remark on submersions and immersions with codimension one or two. J. Math. Kyoto Univ. 9, (1969) 393–404. [2] M.A. Kervaire; J.W. Milnor. Groups of homotopy spheres. I. Ann. of Math 77(2), (1963) 504–537.

6 The Reidemeister torsion of tetrahedral spherical space forms Ana Paula Tremura Galves Universidade Federal de Uberlˆandia [email protected]

Given a free isometric action of a binary tetrahedral group

2 2 2 −1 −1 −1 3 4 P24 = hx, y, z|x = (xy) = y , zxz = xy, xyx = y , z = x = 1i on odd dimensional spheres, we obtain an explicit finite celular decomposition of the tetrahedral spherical space forms, using the concept of fundamental domain. The celular structure gives an explicit description of the associated cellular chain complex over the group P24. The 4-periodic resolution for the binary tetrahedral groups was defined in [1]. The main purpose of this presentation is use the chain complex to compute the Reidemeister torsion of these spaces using a representation of the fundamental domain that defines a complex that is acyclic which has been defined in [2]. The Reidemeister torsion was the first invariant of a variety that is not invariant by homotopy type. With this, it can be understood that the Reidemeister torsion can identify the structure of interactions between the fundamental domain and the simplicial structure. This is joint work with L´ıgiaLa´ısFˆemina.

[1] Fmina, L.L.; Galves, A.P.T.; Neto, O.M.; Spreafico, M.F.. Fundamen- tal domain and cellular decomposition of tetrahedral spherical space forms. Commu- nications in Algebra, v.44, p.768–786, 2016. [2] Galves, A.P.T. Decomposio celular e toro de Reidemeister para formas espa- ciais esfricas tetraedrais. Tese (Doutorado em Matem´atica)-Instituto de Ciˆencias Matem´aticase Computa¸c˜ao,USP, S˜aoCarlos, 2013.

Some properties of the algebraic invariant e∗(G) Maria Gorete Carreira Andrade Universidade Estadual Paulista [email protected]

In this work we present some properties of an invariant defined for finitely gener- ated groups, denoted by e∗(G), which is dual to the classical end e(G) defined by Freudenthal, Hopf and Specker. We also present some applications of this invariant to the topology and algebra.

7 Homological Tools in Dynamical Systems Dahisy Lima Universidade Estadual de Campinas (Brazil) [email protected]

Our goal is to present a topological context fruitful in obtaining information on the behaviour of a wide range of dynamical systems. The overarching idea is to define an appropriate filtered chain complex which captures connections between the invariant sets of the system. We consider as our major algebraic apparatus a spectral sequence of the given complex. The unfolding of the spectral sequence exhibits a rich algebraic procedure and provides much insight into dynamical properties of a continuation of the dynamical systems being studied, such as bifurcation phenomena due the cancellation of singularities.

[1] Lima, D.V.S.; de Rezende, K.A.; Silveira, M.R.S.; Manzoli, O.. Can- cellations for Circle-valued Morse Functions via Spectral Sequences. Topological Methods in Nonlinear Analysis, v. 51 (1), p. 259–311, 2018. [2] Lima, D.V.S.; de Rezende, K.A.; Silveira, M.R.S.; Mello, M.P.; Bertolim, M.A.. A global two-dimensional version of Smale s cancellation theo- rem via spectral sequences. Ergodic Theory and Dynamical Systems, v. 36(6), p. 1795–1838, 2016.

Singular surfaces of revolution Luciana de F´atimaMartins Universidade Estadual Paulista (Brazil) [email protected]

We give an explicit formula for singular surfaces of revolution with prescribed un- bounded mean curvature. Using it, we give conditions for singularities of that sur- faces. Periodicity of that surface is also discussed. This is joint work with Kentaro Saji, Samuel P. dos Santos and Keisuke Ter- amoto.

8 Coincidences and Wecken type results Gustavo de Lima Prado Universidade Federal de Uberlˆandia(Brazil) [email protected]

n+1 n Let f1, f2 : X → Y be maps, where X,Y are smooth manifolds, connected, being X closed (compact and without boundary) and Y without boundary. Then the minimum number of components of the coincidence set is greater than or equal ˜ to the Nielsen number, that is, MCC(f1, f2) > N(f1, f2) for all pairs of maps. A Wecken type result is a result which asserts that, for some X,Y and some conditions, ˜ these numbers are equal, that is, MCC(f1, f2) = N(f1, f2) for all pairs of maps. In [K1] and [K2], U. Koschorke obtains Wecken type results for n = 1 and for n > 4. In [KP], when X = Sn+1, we obtain a Wecken type result for n = 2, 3. Hence, in the codimension one case, when the domain is the sphere, there exists a Wecken type result for every dimension. In [P], we also obtain Wecken type results when X is a spherical space form and Y is a non positive euler characteristic surface, and when X = RP (3) and Y = S2. This work is part of the author’s thesis under the advisory of Daciberg Lima Gon¸calves and Ulrich Koschorke.

[K1] U. Koschorke, Selfcoincidences in higher codimensions, J. Reine Angew. Math. 576 (2004), 1–10. [K2] U. Koschorke, Nielsen coincidence theory in arbitrary codimensions, J. Reine Angew. Math. 598 (2006), 211–236. [KP] U. Koschorke and G. L. Prado, Coincidences in codimension one, J. Fixed Point Theory Appl. 20 (2018). [P] G. L. Prado, Coincidˆenciasem codimens˜aoum e bordismo, Doctoral thesis - USP (2015).

Transition Matrix Ewerton Rocha Vieira Universidade Federal de Goi´as(Brazil) [email protected]

The Conley index theory has been a valuable topological technique for detecting global bifurcations in dynamical systems [1], [2], [3], [4], [5], [6] and [7]. This index is a standard tool in the analysis of invariant sets in dynamical systems, and its significance owes partly to the fact that it is invariant under local perturbation of a flow (the continuation property). In this setting, we present a new definition and applications of transition matrix as a Conley-index based algebraic transformation that tracks changes in index information and thereby identifies global bifurcations that could occur. Furthermore, we do not require that there exists a continuation for a Morse Decomposition related to the transition matrix, see [8], [9] and [10].

[1] Conley, C.. Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence,

9 R.I., 1978. iii+89 pp. [2] Franzosa, R.. The connection matrix theory for Morse decompositions. Trans. Amer. Math. Soc. 311 (1989), no. 2, 561–592. [3] Franzosa, R.; Mischaikow K.. Algebraic Transition Matrices in the Conley Index Theory. Trans. Amer. Math. Soc. 350 (1998), no. 3, 889–912. [4] Kokubu, H.; Mischaikow, K.; Oka. Directional Transition Matrix. Banach Center Publications, 47, 1999. [5] McCord, C.; Mischaikow, K. Connected simple systems, transition matrices, and heteroclinic bifurcations. Trans. Amer. Math. Soc. 333 (1992), no. 1, 397–422. [6] Mischaikow, K.; Mrozek, M.. Conley index. Handbook of dynamical sys- tems, Vol. 2, 393–460, North-Holland, Amsterdam, 2002. [7] Reineck, J.F.. Connecting orbits in one-parameter families of flows. Ergodic Theory Dynam. Systems 8 (1988), Charles Conley Memorial Issue, 359–374. [8] Franzosa, R.; de Rezende, K. A.; Vieira, E.R.. Generalized Topological Transition Matrix. Topol. Methods Nonlinear Anal. 48 (2016), no. 1, 183–212. [9] Franzosa, R.; Vieira, E.R.. Transition matrix theory. Trans. Amer. Math. Soc. 369 (2017), 7737–7764. [10] Vieira, E.R. Transition matrix theory II. To appear.

Optimization of Maggot Mass Rearing via Geometric Statistical Analysis Jamil Viana Pereira, Alice Kimie Miwa Libardi Universidade Estadual Paulista (Brazil) [email protected]

In this work we investigate rearing of C. megacephala using geometric and statistical methods which allow detection and visualization of fine contrasts in mass rearing results under different conditions. The approach involves comparison of outcomes of larval rearing under different conditions based on survival rates and rather fine prop- erties of the distributions of pupal weight and size, beyond their means. The results can be easily interpreted and visualized, and the method are effective and practical for optimization of mass rearing and identification of cost-effective approaches. This work was supported by FAPESP grant 2016/24707-4, Projeto Tem´atico: Topologia Alg´ebrica,Geom´etricae Diferencial. This is joint work with Thiago de Melo, Wasghington Mio and Claudio Jos Von Zuben.

[1] Fawcett,T., An Introduction to ROC Analysis. Pattern Recognition Let- ter,27(8): 861-874, 2006. [2] Hunter, J.D.. Matplotlib: A 2d graphics environment computing Sciences & Engineering, 9(3): 90-95, 2007. [3] Johnson, A. P. & Wallman, J. F.. Effect of massing on larval growth rate. Forensic Sci. Int. 241: 141149, 2014.

10 Good real deformations of co-rank one map germs from R3 to R3 Taciana Oliveira Souza Universidade Federal de Uberlˆandia(Brazil) [email protected]

Joint work with Aldicio J. Miranda and Marcelo J. Saia. The discriminat set ∆(f) given by the image of the critical points set of a map germ f :(C3, 0) → (C3, 0) is homotopically equivalent to a wedge of 2-spheres and the number of these spheres is called the discriminant Milnor number, µ∆(f). A stable deformation of a finitely determined real map-germ from R3 to R3 is a good th real deformation if the real image has 2 homology of rank µ∆(f). We study the good real deformations of co-rank one map germs from R3 to R3. First we describe all simple co-rank one map germs that have such a real good deformation and then we study the simplest non simple co-rank one map germ in these dimensions and show that it does not have a real good deformation. To obtain these results we give a full description of the topology of the discriminant of all real stable deformations of the germ.

11 12 Thursday - Poster Session 2

Free collisions motion planning for rigid bodies: a Topological Complexity approach C´esarIpanaque Zapata Universidade de S˜aoPaulo (Brazil) [email protected]

Consider the following problem in robotics. Assume, for instance, that we have k robots (rigid bodies) moving in a topological space X with no collisions and avoid- ing the obstacles whose geometry is prescribed in advance. The motion planning problem consists in constructing an algorithm, which produces continuous paths in X to transport the robots from an initial configuration to final configuration such that in the process of motion there occur no collisions between the objects and such that the objects do not touch the obstacles in the process of motion. Michael Farber [1] gave the topological approach to the robot motion planning problem when the configuration space of the system is known in advance. He divided the whole con- figuration space of the system into pieces (local domains) and prescribed continuous motion over each of the local domains. Thus, Michael Farber defined a homotopy invariant TC(X), the topological complexity of the space X, as the minimal number of such local domains. In this work we calculate the topological complexity of the configuration space of k robots without collisions in Rn, for n = 2, 3. We will use these results to describe movements of k rigid bodies without collisions in the ndimensional space Rn. This work is a part of my PhDs thesis under the supervision of professor Denise de Mattos and it is supported by FAPESP 2016/18714-8.

[1] Farber, Michael.Topological complexity of motion planning.. Discrete and Computational Geometry 29 (2), 211–221, 2003.

13 Fundamental domains, a dynamical question Gabriel Longatto Clemente Universidade Federal de S˜aoCarlos (Brazil) [email protected]

A real function f is structurally stable if it is conjugated by a real homeomor- phism to any function belonging to a suitable neighbourhood U(f). In this work we show that a special class of real diffeomorphisms, called Morse-Smale, is structurally stable, relating the notion of structural stability to the existence of fundamental do- mains. In particular, we need to consider the space of real C1-maps, endowed with a convenient metric, carefully analysing the definition of fundamental domain and its main properties. Under the supervision of Prof. Dr. F´abioFerrari Ruffino.

Homotopy theory of finite networks Guilherme Vituri F. Pinto Universidade Estadual Paulista (Brazil) vituri [email protected]

In this work we study some constructions with networks (a finite set X with a weight function φ : X × X → R) that behave, in (path) homology, like some well known constructions: the suspension, the cone and the cartesian product. We extend the definition of network to the case when not necessarily all points x, x0 of the network X have a weight associated to them, ie, when φ is defined in a subset of X × X, and explore the concepts of distance and isomorphism in this new setting.

[1] S. Chowdhury, F. Memoli´ . Persistent Path Homology of Directed Networks. Annals of Math. Studies 88. Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 2018. [2] S. Chowdhury, F. Memoli´ . Persistent homology of asymmetric networks: An approach based on Dowker filtrations. arXiv preprint arXiv:1608.05432, 2016. [3] A. Grigoryan, Y. Lin, Y. Muranov, S.-T. Yau. Homologies of path com- plexes and digraphs. arXiv preprint arXiv:1207.2834, 2012. [4] A. Grigoryan, Y. Lin, Y. Muranov, S.-T. Yau. Homotopy theory for digraphs. Pure and Applied Mathematics Quarterly, 10(4), 2014.

14 The hamburger theorem: a generalization of ham sandwich theorem Leandro Vicente Mauri Universidade de S˜aoPaulo (Brazil) [email protected]

One the most famous consequences of the Borsuk- Ulam Theorem is the ham sand- wich Theorem. Recently, Mikio Kano and Jan Kynl demonstrated a general version of ham sandwich theorem, which was called the hamburger theorem [1]. The purpose of the poster is to show this generalization. Theorem 1 (Borsuk-Ulam Theorem, [2], 2.1.1 Theorem). Let f : Sn → Rn be a continuous mapping. If f(u) = f(u) for all u ∈ Sn, then there exists a a point v ∈ Sn such that f(v) = 0 = (0, 0, ..., 0).

Theorem 2 (The ham sandwich Theorem, [2], 3.1.1 Theorem). Let µ1, µ2, ..., µd be d absolutely continuous finite Borel measures on Rd . Then there exists a hyper- 1 d plane h such that each open halfspace H defined by h satisfies µi(H) = 2 µi(R ) for every i ∈ [d].

Definition 3 (Balanced measures in a subset,[1]). Let r ≥ d and let µ1, µ2, ..., µr d be finite Borel measures on R . We say that µ1, µ2, ..., µr are balanced in a subset d 1 r X ⊂ R if for every i ∈ [r], we have µi(X) ≤ d Σj=1µj(X). Theorem 4 (The hamburger theorem, [1]). Let d ≥ 2 be a integer. Let d µ1, µ2, . . . , µd+1 be absolutely continuous finite Borel measures on R . Let ωi = d d+1 µi(R ) for i ∈ [d + 1] and ω = min{ωi; i ∈ [d + 1]}. Assume that Σj=1ωj = 1 and d that 1, 2, . . . , µd+1 are balanced in R . Then there exists a hyperplane h such that for each open halfspace H defined by h, the measures µ1, µ2, . . . , µd+1 are balanced d+1 1 1 in H and Σj=1µj(H) ≥ min{ 2 , 1 − dω} ≥ d+1 . 1 1 Moreover, setting t = min{ 2d , d − ω} and assuming that ωd+1 = ω, the vector (µ1(H), µ2(H), ..., µd+1(H)) is a convex combination of the vectors (t, t, ..., t, 0) and (ω1 − t, ω2 − t, ..., ωd − t, ωd+1). Under the supervision of Prof. Denise de Mattos.

[1] Mikio Kano, Jan Kynclˇ . The hamburger theorem. Computational Geometry, 68:167–173, 2018. [2] Jiri Matousek. Using the Borsuk-Ulam theorem: lectures on topological meth- ods in combinatorics and geometry. Springer Science & Business Media, 2008.

15 Geometrization of Bowen-Series like maps Natalia A. Viana Bedoya Universidade federal de S˜aoCarlos (Brazil) [email protected]

In [1], R. Bowen and C. Series show, under some differentiability conditions, that from the action on S1 of a specific fuchsian group Γ, it is possible to define a 1 1 Markov map fΓ : S → S orbit equivalent with the Γ-action. In [2], J. Los shows the existence of a similar map fΓ, a Bowen-Series-like map, for the action of a hyperbolic co-compact surface group G on ∂G ∼= S1, for a geometric presentation Γ of G. In this work we study the following realization problem: given a piecewise homeomorphism Φ : S1 → S1 , which geometrical and dynamical conditions on Φ are sufficient to realize it as a Bowen-Series-like map of a surface group? This is a joint work with J´erˆomeLos.

[1] R. Bowen, C. Series. Markov maps associated with fuchsian groups. Publi- cations Math´ematiques de L’I.H..S. Tome 50 (1979), 153–170. [2] J. Los. Volume entropy for surface groups via Bowen-Series-like maps. Journal of Topology 7 (2014), 120–154.

Stiefel-Whitney Classes as Smooth Embeddings Obstrucions Alex Melges Barbosa Universidade Federal de S˜aoCarlos (Brazil) [email protected]

Initially, we will axiomatically describe the Stiefel-Whitney classes of a vector bundle and, considering such classes as cohomology ring elements of the base space of the bundle, we will give a recursive formula to calculate the inverse of these classes, by means of the cup product.

After that, we will enunciate Whitney’s Duality Theorem and see how it allows us to consider the Stiefel-Whitney classes of a smooth manifold as obstructions of smooth embedding of that same manifold into some Euclidean space. In this work, we will show how the Stiefel-Whitney classes of tangent and normal bundles of a smooth manifold are related to smooth embeddings of these manifolds into some Euclidian space.

To do so, we will assume the concepts of vector bundles and, only for the pur- pose of fixing notations, we will give an axiomatic description of the Stiefel-Whitney classes of vector bundles and the formula that provides the inverse of these classes by means of the cup product. Thus, by Whitney’s Duality Theorem, the Stiefel- Whitney classes of tangent and normal bundles can be seen as inverse of each other.

After that, we can relate the Stiefel-Whitney classes of the normal bundle of a smooth manifold to obstructions of embedding of this manifold into some Euclidean

16 space. For example, we will see that the projective space RP 9 can not be embedded into R14, that is, if RP 9 can be embedded into R9+m, then m ≥ 6.

As other examples, we will see under what conditions an n−dimensional smooth manifold can not be embedded into Rn+1 and for which dimensions of RP n, it can be embedded into Rn+1.

[1] A.M. Barbosa. Classes de Stiefel-Whitney e de Euler. Master Dissertation - Universidade Estadual Paulista ”J´uliode Mesquita Filho”, 2017. [2] J.W. Milnor & J.D. Stasheff. Characteristic Classes. 1. ed. New Jersey: Princeton University Press and University of Tokyo Press, 1974.

Germ of Singular Holomorphic Foliations of Codimension One Of Second Type in (C3, 0) Allan Ramos de Souza [email protected]

Let F be a germ of singular holomorphic foliation of codimension one in (C2, 0) and

2 π :(X,˜ D)(C , 0) your process of reduction of singularities and F˜ = π ∗ (F) your strict transform. We say that a singularity saddle-node q ∈ Sing(F˜) is a tangent saddle-node if its weak separatrix is contained in the exceptional saddle-node q ∈ Sing(F˜) divisor. A germ of singular holomorphic foliation of codimension one in (C2, 0) is said to be second type if there are no tangent saddle-node in its reduction process. Our goal in this poster is show advances about the germs of singular holomorphic foliations of codimension one and of the second type in (C3, 0) and relate them to a certain invariant.

The Chomsky Hierarchy Bianca B. Dornelas Universidade Estadual de Campinas (Brazil) [email protected]

The Chomsky Hierarchy classify languages in four types (0 to 3): recursively enu- merable, context sensitive, context free and regular ones. In order to present the Chomsky Hierarchy, involved concepts such as graphs, languages, grammars and au- tomata are essential and will be approached. We will pass in more detail through the definitions of languages of type 2 and 3, together with their generating automata, and see some of their properties. Briefly, a graph is a pair {V,E} where V is a set of vertices and E is a set of edges. To each vertex or edge, one can associate a label (letter). One can then walk through paths in the graph, traveling a series of edges and vertices. The

17 corresponding sequence of labels will be a word and all the words that can be formed using paths in the graph make a language [1]. Automata are special graphs that can be understood as a machine that produces certain special languages, for example regular and context free languages, according to the properties one put on the automaton, which is actually a graph. The Chomsky Hierarchy attempts to classify finitely generated groups with respect to certain associated languages and, in turn, automata [2]. Having all the definitions in hand, one can see some properties of each type of language, the importance of this to our study being the properties that can be found on associated groups. Every group has associated graphs, and can also act on some other graphs. There are also those groups which are constructed from graphs, or, specifically, automata. The properties proven for the automata can then be preserved as some other properties in the generated group. For instance, we have the (finitely generated) groups which word problem language is regular being the finite groups (Anisimovs Theorem) [3]. Under the advisory of Prof. Francesco Matucci.

[1] J. Meier. Groups, graphs and trees: An introduction to the geometry of infinite groups. Cambridge University Press, 2008. [2] C. Bleak. Topics in Groups - MT5824 Course Notes. Course Notes, 2014. [3] I. Chiswell. A course in formal languages, automata and groups. Universitext, Springer- Verlag London, London, 2009.

Artin braid groups and some finite quotients Caio Lima Silva Universidade Federal da Bahia (Brazil) kl [email protected]

In this work we define the Artin braid group Bn (resp. the pure Artin braid group Pn) with n strings and exhibit a presentation of these groups. Then, we study some k quotients Bn(k) = Bn/hσ1 i of the braid group. We shall see that when k = 2 then Bn(2) is the symmetric group on n letters Sn. For other values of k determine whether Bn(k) is a finite group is not a simple task. Following [Chapter 5, 2] we illustrate Coxeter’s solution: Bn(k) is a finite group if and only if (n, k) corresponds to the type of one of the five Platonic solids.

[1] K. Murasugi and B. Kurpita. A study of braids. Mathematics and its Applications 484. Kluwer Academic Publishers, Dordrecht, 1999.

18 Equa¸c˜oesde Euler-Poisson In´acioRabelo Universidade Federal de Minas Gerais (Brazil) [email protected]

From the time of Newton, Euler, Lagrange, Laplace, de Poisson, Jacobi, Hamilton e Poincar´ethe symmetry have been aplied in mechanics. The geometric conception of the mechanics is a powerfull tool which connect many areas. The goal of this project was understanding the concept of symmetry and reduction in mechanics of a geometric point view. The focus was in cotangent bundle and Lie-Poisson reduction. The project had three parts: learning about differentiable manifolds and Lie groups, learning about sympletic geometry and Poisson Geometry and the theorems about reduction in context of poisson manifolds. To illustrate the Lie-Poisson reduction we looked at a example where G = S1 acts itself, and extract qualitative informations about the system. Under the advisory of Hassan Najafi Alishah.

Braids Groups and the Word Problem Raquel Magalh˜aesde Almeida de Cruz Universidade Federal da Bahia (Brazil) raquel [email protected]

In this poster, we define the braid groups Bn and the pure braid groups Pn of the plane. We will show the classical presentation of both in terms of the Artin generators. Also, we will show that in both groups the word problem has solution. We will give a geometric algorithm that decides when two given braids are equal, namely “braid combing”. This work is part of the undergraduate research project of R. M. A. Cruz advised by Prof. V. C. Laass.

[1] Kunio Murasugi e Bohdan I. Kurpita, A Study of Braids. Mathematics and Its Applications, 484. Netherlands: Kluwer Academic Publishers, 1999.

19 A differentiable but not C1 version of the Inverse Mapping Theorem Telmo Irineo Acosta Vellozo Universidade Federal de Uberlˆandia(Brazil) [email protected]

With an approach which involves homological and differential techniques, we present a version of the Inverse Mapping Theorem for differential maps which are not nec- essarily of C1 class. As a consequence, we improve the differential version of the Implicit Mapping Theorem proposed by Biasi, Gutierrez and dos Santos in 2008. Furthermore, we present an existence and uniqueness theorem for certain differential equations, which is also a version for an existence theorem proved by the mentioned authors. This a joint work with Marcio Colombo Fenille.

[1] A. P. Barreto, M. C. Fenille and L. Hartmann. Inverse mapping theo- rem and local forms of continuous mappings. Topology Appl. 197, 10–20, 2016. [2] C. Biasi, C. Gutierrez and E. L. dos Santos. The Implicit Function The- orem for Continuous Functions. Topol. Methods Nonlinear Anal. 32(1), 177–186, 2008.

The Nielsen fixed point theory for multi-valued maps on surfaces Bartira Mau´es Universidade de S˜aoPaulo (Brazil) [email protected]

A central question in fixed point theory is if a space has the Wecken property, i.e. if it possible to deform any self-mapping f homotopically to a function with N(f) fixed points, where N(f) is the Nielsen number of f, which is a lower bound for the number of fixed points of the homotopy class [f]. This problem is already solved, and answered positively for closed, connected manifolds of dimension higher or equal than three. For the case of surfaces only the sphere, torus, the projective plane and the Klein bottle have the Wecken property. It is possiple to formulate the Wecken property for n-valued multifunction and similarly the Wecken property holds for compact triangulable manifolds of dimension at least three [5]. For compact surfaces this question is still not entirely resolved and for the Klein bottle this problem is still open. In this context the following basic and relevant questions arise: • Classify the homotopy classes [φ] of n-valued maps on the Klein bottle. • Calculate the Nielsen number N(φ) of every homotopy class. • Does every homotopy class of n-valued maps on the Klein bottle have an representative, which number of fixed points is equal to its Nielsen number?

20 To start answering parts of this questions we will transform these geometrical ques- tion into algebraic ones, using braid theory.

[1] R.F. Brown, D.L. Gonc¸alves. On the topology of n-valued maps. Advances in Fixed Point Theory, 8(2):205220, 2018. [2] D.L. Gonc¸alves, J. Guaschi. Fixed points of n-valued maps on surfaces and the wecken property - a configuration space approach. Preprint, 2017. [3] D.L. Gonc¸alves, J. Guaschi. Fixed points of n-valued maps, the fixed point property and the case of surfacesa braid approach. Preprint, 2017. [4] H. Schirmer. Fix-finite approximation of n-valued multifunctions. Fund. Math. 121(1), 73–80, 1984. [5] Helga Schirmer. A minimum theorem for n-valued multifunctions. Fund. Math., 126(1), 83–92, 1985.

Persistent Homology and Optimal Cycles Carlos Henrique Venturi Ronchi Universidade de S˜aoPaulo (Brazil) [email protected]

Persistent homology has undergone significant progress in recent years. Lately many algorithms have been developed to increase performance and to better understand the shape of data. One way to achieve the former is to consider the representatives (cycles) of homology groups and persistent homology filtration. This review aims to understand and show how to use linear programming in the task of finding optimal cycles in persistent homology.

21 22 Author Index

Acosta Vellozo Raquel, 19 Telmo, 20 Martins Alice Luciana, 8 Libardi, 6 Mau´es Andrade Bartira, 20 Maria, 7 Mauri Leandro, 15 Barbosa Alex, 16 Nguyen Bedoya Thuy, 1 Natalia, 16 Oliveira Souza Camasca Taciana, 11 Juan Francisco, 1 Cardim Pereira Nancy, 3 Jamil, 10 Cintra Pinto Adriana, 4 Guilherme, 14 Clemente Prado Gabriel, 14 Gustavo, 9 da Silva Rabelo M´ario,3 In´acio,19 Doria Ramos de Souza Celso, 4 Allan, 17 Dornelas Ronchi Bianca, 17 Carlos, 21 dos Santos Edivaldo, 6 Silva Galves Caio, 18 Ana Paula, 7 Vieira Lima Ewerton, 9 Dahisy, 8 Zapata Magalhes C´esar,13

23