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Mathematics r research reports M r Boris Hasselblatt, Svetlana Katok, Michele Benzi, Dmitry Burago, Alessandra Celletti, Tobias Holck Colding, Brian Conrey, Josselin Garnier, Timothy Gowers, Robert Griess, Linus Kramer, Barry Mazur, Walter Neumann, Alexander Olshanskii, Christopher Sogge, Benjamin Sudakov, Hugh Woodin, Yuri Zarhin, Tamar Ziegler Editorial Volume 1 (2020), p. 1-3. <http://mrr.centre-mersenne.org/item/MRR_2020__1__1_0> © The journal and the authors, 2020. Some rights reserved. This article is licensed under the Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/ Mathematics Research Reports is member of the Centre Mersenne for Open Scientific Publishing www.centre-mersenne.org Mathema tics research reports Volume 1 (2020), 1–3 Editorial This is the inaugural volume of Mathematics Research Reports, a journal owned by mathematicians, and dedicated to the principles of fair open access and academic self- determination. Articles in Mathematics Research Reports are freely available for a world-wide audi- ence, with no author publication charges (diamond open access) but high production value, thanks to financial support from the Anatole Katok Center for Dynamical Sys- tems and Geometry at the Pennsylvania State University and to the infrastructure of the Centre Mersenne. The articles in MRR are research announcements of significant ad- vances in all branches of mathematics, short complete papers of original research (up to about 15 journal pages), and review articles (up to about 30 journal pages). They communicate their contents to a broad mathematical audience and should meet high standards for mathematical content and clarity. The entire Editorial Board approves the acceptance of any paper for publication, and appointments to the board are made by the board itself. -
Arxiv:1108.5351V3 [Math.AG] 26 Oct 2012 ..Rslso D-Mod( on Results Introduction the to 0.2
ON SOME FINITENESS QUESTIONS FOR ALGEBRAIC STACKS VLADIMIR DRINFELD AND DENNIS GAITSGORY Abstract. We prove that under a certain mild hypothesis, the DG category of D-modules on a quasi-compact algebraic stack is compactly generated. We also show that under the same hypothesis, the functor of global sections on the DG category of quasi-coherent sheaves is continuous. Contents Introduction 3 0.1. Introduction to the introduction 3 0.2. Results on D-mod(Y) 4 0.3. Results on QCoh(Y) 4 0.4. Ind-coherent sheaves 5 0.5. Contents of the paper 7 0.6. Conventions, notation and terminology 10 0.7. Acknowledgments 14 1. Results on QCoh(Y) 14 1.1. Assumptions on stacks 14 1.2. Quasi-coherent sheaves 15 1.3. Direct images for quasi-coherent sheaves 18 1.4. Statements of the results on QCoh(Y) 21 2. Proof of Theorems 1.4.2 and 1.4.10 23 2.1. Reducing the statement to a key lemma 23 2.2. Easy reduction steps 24 2.3. Devissage 24 2.4. Quotients of schemes by algebraic groups 26 2.5. Proof of Proposition 2.3.4 26 2.6. Proof of Theorem 1.4.10 29 arXiv:1108.5351v3 [math.AG] 26 Oct 2012 3. Implications for ind-coherent sheaves 30 3.1. The “locally almost of finite type” condition 30 3.2. The category IndCoh 32 3.3. The coherent subcategory 39 3.4. Description of compact objects of IndCoh(Y) 39 3.5. The category Coh(Y) generates IndCoh(Y) 42 3.6. -
Report for the Academic Year 1995
Institute /or ADVANCED STUDY REPORT FOR THE ACADEMIC YEAR 1994 - 95 PRINCETON NEW JERSEY Institute /or ADVANCED STUDY REPORT FOR THE ACADEMIC YEAR 1 994 - 95 OLDEN LANE PRINCETON • NEW JERSEY 08540-0631 609-734-8000 609-924-8399 (Fax) Extract from the letter addressed by the Founders to the Institute's Trustees, dated June 6, 1930. Newark, New jersey. It is fundamental in our purpose, and our express desire, that in the appointments to the staff and faculty, as well as in the admission of workers and students, no account shall be taken, directly or indirectly, of race, religion, or sex. We feel strongly that the spirit characteristic of America at its noblest, above all the pursuit of higher learning, cannot admit of any conditions as to personnel other than those designed to promote the objects for which this institution is established, and particularly with no regard whatever to accidents of race, creed, or sex. TABLE OF CONTENTS 4 BACKGROUND AND PURPOSE 5 • FOUNDERS, TRUSTEES AND OFFICERS OF THE BOARD AND OF THE CORPORATION 8 • ADMINISTRATION 11 REPORT OF THE CHAIRMAN 15 REPORT OF THE DIRECTOR 23 • ACKNOWLEDGMENTS 27 • REPORT OF THE SCHOOL OF HISTORICAL STUDIES ACADEMIC ACTIVITIES MEMBERS, VISITORS AND RESEARCH STAFF 36 • REPORT OF THE SCHOOL OF MATHEMATICS ACADEMIC ACTIVITIES MEMBERS AND VISITORS 42 • REPORT OF THE SCHOOL OF NATURAL SCIENCES ACADEMIC ACTIVITIES MEMBERS AND VISITORS 50 • REPORT OF THE SCHOOL OF SOCIAL SCIENCE ACADEMIC ACTIVITIES MEMBERS, VISITORS AND RESEARCH STAFF 55 • REPORT OF THE INSTITUTE LIBRARIES 57 • RECORD OF INSTITUTE EVENTS IN THE ACADEMIC YEAR 1994-95 85 • INDEPENDENT AUDITORS' REPORT INSTITUTE FOR ADVANCED STUDY: BACKGROUND AND PURPOSE The Institute for Advanced Study is an independent, nonprofit institution devoted to the encouragement of learning and scholarship. -
How to Glue Perverse Sheaves”
NOTES ON BEILINSON'S \HOW TO GLUE PERVERSE SHEAVES" RYAN REICH Abstract. The titular, foundational work of Beilinson not only gives a tech- nique for gluing perverse sheaves but also implicitly contains constructions of the nearby and vanishing cycles functors of perverse sheaves. These con- structions are completely elementary and show that these functors preserve perversity and respect Verdier duality on perverse sheaves. The work also de- fines a new, \maximal extension" functor, which is left mysterious aside from its role in the gluing theorem. In these notes, we present the complete details of all of these constructions and theorems. In this paper we discuss Alexander Beilinson's \How to glue perverse sheaves" [1] with three goals. The first arose from a suggestion of Dennis Gaitsgory that the author study the construction of the unipotent nearby cycles functor R un which, as Beilinson observes in his concluding remarks, is implicit in the proof of his Key Lemma 2.1. Here, we make this construction explicit, since it is invaluable in many contexts not necessarily involving gluing. The second goal is to restructure the pre- sentation around this new perspective; in particular, we have chosen to eliminate the two-sided limit formalism in favor of the straightforward setup indicated briefly in [3, x4.2] for D-modules. We also emphasize this construction as a simple demon- stration that R un[−1] and Verdier duality D commute, and de-emphasize its role in the gluing theorem. Finally, we provide complete proofs; with the exception of the Key Lemma, [1] provides a complete program of proof which is not carried out in detail, making a technical understanding of its contents more difficult given the density of ideas. -
Professor Peter Goldreich Member of the Board of Adjudicators Chairman of the Selection Committee for the Prize in Astronomy
The Shaw Prize The Shaw Prize is an international award to honour individuals who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. The award is dedicated to furthering societal progress, enhancing quality of life, and enriching humanity’s spiritual civilization. Preference is to be given to individuals whose significant work was recently achieved and who are currently active in their respective fields. Founder's Biographical Note The Shaw Prize was established under the auspices of Mr Run Run Shaw. Mr Shaw, born in China in 1907, was a native of Ningbo County, Zhejiang Province. He joined his brother’s film company in China in the 1920s. During the 1950s he founded the film company Shaw Brothers (HK) Limited in Hong Kong. He was one of the founding members of Television Broadcasts Limited launched in Hong Kong in 1967. Mr Shaw also founded two charities, The Shaw Foundation Hong Kong and The Sir Run Run Shaw Charitable Trust, both dedicated to the promotion of education, scientific and technological research, medical and welfare services, and culture and the arts. ~ 1 ~ Message from the Chief Executive I warmly congratulate the six Shaw Laureates of 2014. Established in 2002 under the auspices of Mr Run Run Shaw, the Shaw Prize is a highly prestigious recognition of the role that scientists play in shaping the development of a modern world. Since the first award in 2004, 54 leading international scientists have been honoured for their ground-breaking discoveries which have expanded the frontiers of human knowledge and made significant contributions to humankind. -
Algebraic D-Modules and Representation Theory Of
Contemporary Mathematics Volume 154, 1993 Algebraic -modules and Representation TheoryDof Semisimple Lie Groups Dragan Miliˇci´c Abstract. This expository paper represents an introduction to some aspects of the current research in representation theory of semisimple Lie groups. In particular, we discuss the theory of “localization” of modules over the envelop- ing algebra of a semisimple Lie algebra due to Alexander Beilinson and Joseph Bernstein [1], [2], and the work of Henryk Hecht, Wilfried Schmid, Joseph A. Wolf and the author on the localization of Harish-Chandra modules [7], [8], [13], [17], [18]. These results can be viewed as a vast generalization of the classical theorem of Armand Borel and Andr´e Weil on geometric realiza- tion of irreducible finite-dimensional representations of compact semisimple Lie groups [3]. 1. Introduction Let G0 be a connected semisimple Lie group with finite center. Fix a maximal compact subgroup K0 of G0. Let g be the complexified Lie algebra of G0 and k its subalgebra which is the complexified Lie algebra of K0. Denote by σ the corresponding Cartan involution, i.e., σ is the involution of g such that k is the set of its fixed points. Let K be the complexification of K0. The group K has a natural structure of a complex reductive algebraic group. Let π be an admissible representation of G0 of finite length. Then, the submod- ule V of all K0-finite vectors in this representation is a finitely generated module over the enveloping algebra (g) of g, and also a direct sum of finite-dimensional U irreducible representations of K0. -
Shtukas for Reductive Groups and Langlands Correspondence for Functions Fields
SHTUKAS FOR REDUCTIVE GROUPS AND LANGLANDS CORRESPONDENCE FOR FUNCTIONS FIELDS VINCENT LAFFORGUE This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject. We begin with a short historical account (all notions used below are recalled in the text). The Langlands correspondence [49] is a conjecture of utmost impor- tance, concerning global fields, i.e. number fields and function fields. Many excellent surveys are available, for example [39, 14, 13, 79, 31, 5]. The Langlands correspondence belongs to a huge system of conjectures (Langlands functoriality, Grothendieck’s vision of motives, special val- ues of L-functions, Ramanujan-Petersson conjecture, generalized Rie- mann hypothesis). This system has a remarkable deepness and logical coherence and many cases of these conjectures have already been es- tablished. Moreover the Langlands correspondence over function fields admits a geometrization, the “geometric Langlands program”, which is related to conformal field theory in Theoretical Physics. Let G be a connected reductive group over a global field F . For the sake of simplicity we assume G is split. The Langlands correspondence relates two fundamental objects, of very different nature, whose definition will be recalled later, • the automorphic forms for G, • the global Langlands parameters , i.e. the conjugacy classes of morphisms from the Galois group Gal(F =F ) to the Langlands b dual group G(Q`). b For G = GL1 we have G = GL1 and this is class field theory, which describes the abelianization of Gal(F =F ) (one particular case of it for Q is the law of quadratic reciprocity, which dates back to Euler, Legendre and Gauss). -
On the Kinematic Formula in the Lives of the Saints Danny Calegari
SHORT STORIES On the Kinematic Formula in the Lives of the Saints Danny Calegari Saint Sebastian was an early Christian martyr. He served as a captain in the Praetorian Guard under Diocletian un- til his religious faith was discovered, at which point he was taken to a field, bound to a stake, and shot by archers “till he was as full of arrows as an urchin1 is full of pricks2.” Rather miraculously, he made a full recovery, but was later executed anyway for insulting the emperor. The trans- pierced saint became a popular subject for Renaissance painters, e.g., Figure 1. The arrows in Mantegna’s painting have apparently ar- rived from all directions, though they are conspicuously grouped around the legs and groin, almost completely missing the thorax. Intuitively, we should expect more ar- rows in the parts of the body that present a bigger cross section. This intuition is formalized by the claim that a subset of the surface of Saint Sebastian has an area pro- portional to its expected number of intersections with a random line (i.e., arrow). Since both area and expectation are additive, we may reduce the claim (by polygonal ap- proximation and limit) to the case of a flat triangular Saint Sebastian, in which case it is obvious. Danny Calegari is a professor of mathematics at the University of Chicago. His Figure 1. Andrea Mantegna’s painting of Saint Sebastian. email address is [email protected]. 1 hedgehog This is a 3-dimensional version of the classical Crofton 2quills formula, which says that the length of a plane curve is pro- For permission to reprint this article, please contact: portional to its expected number of intersections with a [email protected]. -
Groups of PL Homeomorphisms of Cubes
GROUPS OF PL HOMEOMORPHISMS OF CUBES DANNY CALEGARI AND DALE ROLFSEN D´edi´e`aMichel Boileau sur son soixanti`eme anniversaire. Sant´e! Abstract. We study algebraic properties of groups of PL or smooth homeo- morphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing point- wise a submanifold of codimension 1 (resp. codimension 2), and show that such groups are locally indicable (resp. circularly orderable). We also give many examples of interesting groups that can act, and discuss some other algebraic constraints that such groups must satisfy, including the fact that a group of PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no elements that are more than exponentially distorted. 1. Introduction We are concerned in this paper with algebraic properties of the group of PL homeomorphisms of a PL manifold, fixed on some PL submanifold (usually of codimension 1 or 2, for instance the boundary) and some of its subgroups (usually those preserving some structure). The most important case is the group of PL homeomorphisms of In fixed pointwise on ∂In; hence these are “groups of PL homeomorphisms of the (n-)cube”. The algebraic study of transformation groups (often in low dimension, or preserv- ing some extra structure such as a symplectic or complex structure) has recently seen a lot of activity; however, much of this activity has been confined to the smooth category. It is striking that many of these results can be transplanted to the PL category. This interest is further strengthened by the possibility of working in the PL category over (real) algebraic rings or fields (this possibility has already been exploited in dimension 1, in the groups F and T of Richard Thompson). -
Commentary on Thurston's Work on Foliations
COMMENTARY ON FOLIATIONS* Quoting Thurston's definition of foliation [F11]. \Given a large supply of some sort of fabric, what kinds of manifolds can be made from it, in a way that the patterns match up along the seams? This is a very general question, which has been studied by diverse means in differential topology and differential geometry. ... A foliation is a manifold made out of striped fabric - with infintely thin stripes, having no space between them. The complete stripes, or leaves, of the foliation are submanifolds; if the leaves have codimension k, the foliation is called a codimension k foliation. In order that a manifold admit a codimension- k foliation, it must have a plane field of dimension (n − k)." Such a foliation is called an (n − k)-dimensional foliation. The first definitive result in the subject, the so called Frobenius integrability theorem [Fr], concerns a necessary and sufficient condition for a plane field to be the tangent field of a foliation. See [Spi] Chapter 6 for a modern treatment. As Frobenius himself notes [Sa], a first proof was given by Deahna [De]. While this work was published in 1840, it took another hundred years before a geometric/topological theory of foliations was introduced. This was pioneered by Ehresmann and Reeb in a series of Comptes Rendus papers starting with [ER] that was quickly followed by Reeb's foundational 1948 thesis [Re1]. See Haefliger [Ha4] for a detailed account of developments in this period. Reeb [Re1] himself notes that the 1-dimensional theory had already undergone considerable development through the work of Poincare [P], Bendixson [Be], Kaplan [Ka] and others. -
D-Modules on Infinite Dimensional Varieties
D-MODULES ON INFINITE DIMENSIONAL VARIETIES SAM RASKIN Contents 1. Introduction 1 2. D-modules on prestacks 4 3. D-modules on schemes 7 4. Placidity 19 5. Holonomic D-modules 30 6. D-modules on indschemes 33 References 44 1. Introduction 1.1. The goal of this foundational note is to develop the D-module formalism on indschemes of ind-infinite type. 1.2. The basic feature that we struggle against is that there are two types of infinite dimensionality at play: pro-infinite dimensionality and ind-infinite dimensionality. That is, we could have an infinite dimensional variety S that is the union S “YiSi “ colimiSi of finite dimensional varieties, or T that is the projective limit T “ limj Tj of finite dimensional varieties, e.g., a scheme of infinite type. Any reasonable theory of D-modules will produce produce some kinds of de Rham homology and cohomology groups. We postulate as a basic principle that these groups should take values in discrete vector spaces, that is, we wish to avoid projective limits. Then, in the ind-infinite dimensional case, the natural theory is the homology of S: H˚pSq :“ colim H˚pSiq i while in the pro-infinite dimensional case, the natural theory is the cohomology of T : ˚ ˚ H pT q :“ colim H pTjq: j For indschemes that are infinite dimensional in both the ind and the pro directions, one requires a semi-infinite homology theory that is homology in the ind direction and cohomology in the pro direction. Remark 1.2.1. Of course, such a theory requires some extra choices, as is immediately seen by considering the finite dimensional case. -
IMU-Net 88: March 2018 a Bimonthly Email Newsletter from the International Mathematical Union Editor: Martin Raussen, Aalborg University, Denmark
IMU-Net 88: March 2018 A Bimonthly Email Newsletter from the International Mathematical Union Editor: Martin Raussen, Aalborg University, Denmark CONTENTS 1. Editorial: ICM 2018 2. IMU General Assembly meeting in São Paulo 3. CDC Panel Discussion and Poster Session during ICM 2018 4. Fields medalist Alan Baker passed away 5. CWM: Faces of women in mathematics 6. Report on the ISC general assembly 7. ICIAM 2019 congress 8. Abel Prize 2018 to Robert Langlands 9. 2018 Wolf prizes to Beilinson and Drinfeld 10. MSC 2020: Revision of Mathematics Subject Classification 11. Subscribing to IMU-Net ------------------------------------------------------------------------------------------------------- 1. EDITORIAL: ICM 2018 Dear Colleagues, After so many months of work and expectation, the year of the Congress has finally arrived! Preparations are well under way: the bulk of the scientific program has been defined, the proceedings are being finalized (the papers by plenary and invited speakers will be distributed to all the participants at the Congress), travel grants have been awarded to participants from the developing world, communications and posters are being selected as I write, and registration of participants is also taking place right now. Keep in mind that the deadline for early advance registration, with reduced registration fee, is April 27. Much of our effort over the last couple of years has been geared towards advertising the Congress, domestically and abroad, as we take the occasion as a historic opportunity to popularize mathematics in our society and, especially, amongst the Brazilian youth. I believe we are being very successful. The Biennium of Mathematics 2017-2018, formally proclaimed by the Brazilian national parliament, encompasses a wide range of outreach initiatives throughout the country, and raised the profile of mathematics in mainstream media to totally unprecedented levels.