A Primer on Carnot Groups
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Locally Compact Groups: Traditions and Trends Karl Heinrich Hofmann Technische Universitat Darmstadt, [email protected]
University of Dayton eCommons Summer Conference on Topology and Its Department of Mathematics Applications 6-2017 Locally Compact Groups: Traditions and Trends Karl Heinrich Hofmann Technische Universitat Darmstadt, [email protected] Wolfgang Herfort Francesco G. Russo Follow this and additional works at: http://ecommons.udayton.edu/topology_conf Part of the Geometry and Topology Commons, and the Special Functions Commons eCommons Citation Hofmann, Karl Heinrich; Herfort, Wolfgang; and Russo, Francesco G., "Locally Compact Groups: Traditions and Trends" (2017). Summer Conference on Topology and Its Applications. 47. http://ecommons.udayton.edu/topology_conf/47 This Plenary Lecture is brought to you for free and open access by the Department of Mathematics at eCommons. It has been accepted for inclusion in Summer Conference on Topology and Its Applications by an authorized administrator of eCommons. For more information, please contact [email protected], [email protected]. Some Background Notes Some \new" tools Near abelian groups Applications Alexander Doniphan Wallace (1905{1985) Gordon Thomas Whyburn Robert Lee Moore Some Background Notes Some \new" tools Near abelian groups Applications \The best mathematics is the most mixed-up mathematics, those disciplines in which analysis, algebra and topology all play a vital role." Gordon Thomas Whyburn Robert Lee Moore Some Background Notes Some \new" tools Near abelian groups Applications \The best mathematics is the most mixed-up mathematics, those disciplines in which -
Rigidity of Quasiconformal Maps on Carnot Groups
RIGIDITY OF QUASICONFORMAL MAPS ON CARNOT GROUPS Mark Medwid A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2017 Committee: Xiangdong Xie, Advisor Alexander Tarnovsky, Graduate Faculty Representative Mihai Staic Juan Bes´ Copyright c August 2017 Mark Medwid All rights reserved iii ABSTRACT Xiangdong Xie, Advisor Quasiconformal mappings were first utilized by Grotzsch¨ in the 1920’s and then later named by Ahlfors in the 1930’s. The conformal mappings one studies in complex analysis are locally angle-preserving: they map infinitesimal balls to infinitesimal balls. Quasiconformal mappings, on the other hand, map infinitesimal balls to infinitesimal ellipsoids of a uniformly bounded ec- centricity. The theory of quasiconformal mappings is well-developed and studied. For example, quasiconformal mappings on Euclidean space are almost-everywhere differentiable. A result due to Pansu in 1989 illustrated that quasiconformal mappings on Carnot groups are almost-everywhere (Pansu) differentiable, as well. It is easy to show that a biLipschitz map is quasiconformal but the converse does not hold, in general. There are many instances, however, where globally defined quasiconformal mappings on Carnot groups are biLipschitz. In this paper we show that, under cer- tain conditions, a quasiconformal mapping defined on an open subset of a Carnot group is locally biLipschitz. This result is motivated by rigidity results in geometry (for example, the theorem by Mostow in 1968). Along the way we develop background material on geometric group theory and show its connection to quasiconformal mappings. iv ACKNOWLEDGMENTS I would like to acknowledge, first and foremost, my wife, Heather. -
Casimir Functions of Free Nilpotent Lie Groups of Steps Three and Four
Casimir functions of free nilpotent Lie groups of steps three and four∗ A. V. Podobryaev A.K. Ailamazyan Program Systems Institute of RAS [email protected] June 2, 2020 Abstract Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3, 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups we conclude that extremal controls corresponding to two-dimensional coad- joint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group. Keywords: free Carnot group, coadjoint orbits, Casimir functions, integration, geometric control theory, sub-Riemannian geometry, sub-Finsler geometry. AMS subject classification: 22E25, 17B08, 53C17, 35R03. Introduction The goal of this paper is a description of Casimir functions and coadjoint orbits for arXiv:2006.00224v1 [math.DG] 30 May 2020 some class of nilpotent Lie groups. This knowledge is important for the theory of left- invariant optimal control problems on Lie groups [1]. Casimir functions are integrals of the Hamiltonian system of Pontryagin maximum principle. -
Publications of Members, 1930-1954
THE INSTITUTE FOR ADVANCED STUDY PUBLICATIONS OF MEMBERS 1930 • 1954 PRINCETON, NEW JERSEY . 1955 COPYRIGHT 1955, BY THE INSTITUTE FOR ADVANCED STUDY MANUFACTURED IN THE UNITED STATES OF AMERICA BY PRINCETON UNIVERSITY PRESS, PRINCETON, N.J. CONTENTS FOREWORD 3 BIBLIOGRAPHY 9 DIRECTORY OF INSTITUTE MEMBERS, 1930-1954 205 MEMBERS WITH APPOINTMENTS OF LONG TERM 265 TRUSTEES 269 buH FOREWORD FOREWORD Publication of this bibliography marks the 25th Anniversary of the foundation of the Institute for Advanced Study. The certificate of incorporation of the Institute was signed on the 20th day of May, 1930. The first academic appointments, naming Albert Einstein and Oswald Veblen as Professors at the Institute, were approved two and one- half years later, in initiation of academic work. The Institute for Advanced Study is devoted to the encouragement, support and patronage of learning—of science, in the old, broad, undifferentiated sense of the word. The Institute partakes of the character both of a university and of a research institute j but it also differs in significant ways from both. It is unlike a university, for instance, in its small size—its academic membership at any one time numbers only a little over a hundred. It is unlike a university in that it has no formal curriculum, no scheduled courses of instruction, no commitment that all branches of learning be rep- resented in its faculty and members. It is unlike a research institute in that its purposes are broader, that it supports many separate fields of study, that, with one exception, it maintains no laboratories; and above all in that it welcomes temporary members, whose intellectual development and growth are one of its principal purposes. -
Oswald Veblen
NATIONAL ACADEMY OF SCIENCES O S W A L D V E B LEN 1880—1960 A Biographical Memoir by S A U N D E R S M A C L ANE Any opinions expressed in this memoir are those of the author(s) and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 1964 NATIONAL ACADEMY OF SCIENCES WASHINGTON D.C. OSWALD VEBLEN June 24,1880—August 10, i960 BY SAUNDERS MAC LANE SWALD VEBLEN, geometer and mathematical statesman, spanned O in his career the full range of twentieth-century Mathematics in the United States; his leadership in transmitting ideas and in de- veloping young men has had a substantial effect on the present mathematical scene. At the turn of the century he studied at Chi- cago, at the period when that University was first starting the doc- toral training of young Mathematicians in this country. He then continued at Princeton University, where his own work and that of his students played a leading role in the development of an outstand- ing department of Mathematics in Fine Hall. Later, when the In- stitute for Advanced Study was founded, Veblen became one of its first professors, and had a vital part in the development of this In- stitute as a world center for mathematical research. Veblen's background was Norwegian. His grandfather, Thomas Anderson Veblen, (1818-1906) came from Odegaard, Homan Con- gregation, Vester Slidre Parish, Valdris. After work as a cabinet- maker and as a Norwegian soldier, he was anxious to come to the United States. -
Algebraic Topology - Wikipedia, the Free Encyclopedia Page 1 of 5
Algebraic topology - Wikipedia, the free encyclopedia Page 1 of 5 Algebraic topology From Wikipedia, the free encyclopedia Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Contents 1 The method of algebraic invariants 2 Setting in category theory 3 Results on homology 4 Applications of algebraic topology 5 Notable algebraic topologists 6 Important theorems in algebraic topology 7 See also 8 Notes 9 References 10 Further reading The method of algebraic invariants An older name for the subject was combinatorial topology , implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW-complex ). The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove. Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups. -
Non-Minimality of Corners in Subriemannian Geometry
NON-MINIMALITY OF CORNERS IN SUBRIEMANNIAN GEOMETRY EERO HAKAVUORI AND ENRICO LE DONNE Abstract. We give a short solution to one of the main open problems in subrie- mannian geometry. Namely, we prove that length minimizers do not have corner- type singularities. With this result we solve Problem II of Agrachev's list, and provide the first general result toward the 30-year-old open problem of regularity of subriemannian geodesics. Contents 1. Introduction 1 1.1. The idea of the argument 2 1.2. Definitions 3 2. Preliminary lemmas 4 3. The main result 6 3.1. Reduction to Carnot groups 6 3.2. The inductive non-minimality argument 7 References 9 1. Introduction One of the major open problems in subriemannian geometry is the regularity of length-minimizing curves (see [Mon02, Section 10.1] and [Mon14b, Section 4]). This problem has been open since the work of Strichartz [Str86, Str89] and Hamenst¨adt [Ham90]. Contrary to Riemannian geometry, where it is well known that all length minimizers are C1-smooth, the problem in the subriemannian case is significantly more difficult. The primary reason for this difficulty is the existence of abnormal curves (see [AS04, Date: March 8, 2016. 2010 Mathematics Subject Classification. 53C17, 49K21, 28A75. Key words and phrases. Corner-type singularities, geodesics, sub-Riemannian geometry, Carnot groups, regularity of length minimizers. 1 2 EERO HAKAVUORI AND ENRICO LE DONNE ABB15]), which we know may be length minimizers since the work of Montgomery [Mon94]. Nowadays, many more abnormal length minimizers are known [BH93, LS94, LS95, GK95, Sus96]. Abnormal curves, when parametrized by arc-length, need only have Lipschitz- regularity (see [LDLMV14, Section 5]), which is why, a priori, no further regular- ity can be assumed from an arbitrary length minimizer in a subriemannian space. -
Measure Contraction Properties of Carnot Groups Luca Rizzi
Measure contraction properties of Carnot groups Luca Rizzi To cite this version: Luca Rizzi. Measure contraction properties of Carnot groups . Calculus of Variations and Partial Differential Equations, Springer Verlag, 2016, 10.1007/s00526-016-1002-y. hal-01218376v3 HAL Id: hal-01218376 https://hal.archives-ouvertes.fr/hal-01218376v3 Submitted on 21 May 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MEASURE CONTRACTION PROPERTIES OF CARNOT GROUPS RIZZI LUCA Abstract. We prove that any corank 1 Carnot group of dimension k + 1 equipped with a left-invariant measure satisfies the MCP(K, N) if and only if K ≤ 0 and N ≥ k + 3. This generalizes the well known result by Juillet for the Heisenberg group Hk+1 to a larger class of structures, which admit non-trivial abnormal minimizing curves. The number k + 3 coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent (the least N such that the MCP(0,N) is satisfied). We prove that, on a metric measure space, the curvature exponent is always larger than the geodesic dimension which, in turn, is larger than the Hausdorff one. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Optimal Control for Automotive Powertrain Applications
Universitat Politecnica` de Valencia` Departamento de Maquinas´ y Motores Termicos´ Optimal Control for Automotive Powertrain Applications PhD Dissertation Presented by Alberto Reig Advised by Dr. Benjam´ınPla Valencia, September 2017 PhD Dissertation Optimal Control for Automotive Powertrain Applications Presented by Alberto Reig Advised by Dr. Benjam´ınPla Examining committee President: Dr. Jos´eGalindo Lucas Secretary: Dr. Octavio Armas Vergel Vocal: Dr. Marcello Canova Valencia, September 2017 To the geniuses of the past, who built our future; to those names in the shades, bringing our knowledge. They not only supported this work but also the world we live in. If you can solve it, it is an exercise; otherwise it's a research problem. | Richard Bellman e Resumen El Control Optimo´ (CO) es esencialmente un problema matem´aticode b´us- queda de extremos, consistente en la definici´onde un criterio a minimizar (o maximizar), restricciones que deben satisfacerse y condiciones de contorno que afectan al sistema. La teor´ıade CO ofrece m´etodos para derivar una trayectoria de control que minimiza (o maximiza) ese criterio. Esta Tesis trata la aplicaci´ondel CO en automoci´on,y especialmente en el motor de combusti´oninterna. Las herramientas necesarias son un m´etodo de optimizaci´ony una representaci´onmatem´aticade la planta motriz. Para ello, se realiza un an´alisiscuantitativo de las ventajas e inconvenientes de los tres m´etodos de optimizaci´onexistentes en la literatura: programaci´ondin´amica, principio m´ınimode Pontryagin y m´etodos directos. Se desarrollan y describen los algoritmos para implementar estos m´etodos as´ıcomo un modelo de planta motriz, validado experimentalmente, que incluye la din´amicalongitudinal del veh´ıculo,modelos para el motor el´ectricoy las bater´ıas,y un modelo de motor de combusti´onde valores medios. -
On Logarithmic Sobolev Inequalities for the Heat Kernel on The
ON LOGARITHMIC SOBOLEV INEQUALITIES FOR THE HEAT KERNEL ON THE HEISENBERG GROUP MICHEL BONNEFONT, DJALIL CHAFAÏ, AND RONAN HERRY Abstract. In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub- elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the more recent inequality of Fabrice Baudoin and Nicola Garofalo obtained using a generalized curvature criterion. Finally, we extend this inequality to the case of homogeneous Carnot groups of rank two. Résumé. Dans cette note, nous obtenons une inégalité de Sobolev logarithmique nouvelle pour le noyau de la chaleur sur le groupe de Heisenberg. La preuve est inspirée de la méthode historique de Leonard Gross à base de théorème limite central pour une marche aléatoire. Ici la nature non commutative des incréments produit un nouveau gradient qui fait intervenir naturellement un pont brownien sur le groupe de Heisenberg. Cette nouvelle inégalité contient l’inégalité de Sobolev logarithmique optimale pour la mesure gaussienne en deux dimensions. Nous comparons cette nouvelle inégalité avec l’inégalité sous-elliptique de Hong-Quan Li et avec les inégalités plus récentes de Fabrice Beaudoin et Nicola Garofalo obtenues avec un critère de courbure généralisé. -
ON the ACTION of SO(3) on Sn
Pacific Journal of Mathematics ON THE ACTION OF SO.3/ ON Sn DEANE MONTGOMERY AND HANS SAMELSON Vol. 12, No. 2 February 1962 ON THE ACTION OF SO(3) ON Sn DEANE MONTGOMERY AND HANS SAMELSON 1. Introduction* This paper contains some facts about the possible actions of the rotation group SO(3) on the ^-sphere Sn. For some of the results the action is required to be differentiate, but for others this is not necessary. We recall for an action of a compact Lie group, that a principal isotropy group is an isotropy group of the lowest possible dimension and that among these it is one with the fewest possible components. An orbit with such an isotropy group is called a principal orbit. For a compact Lie group acting on a cohomology manifold over Z, principal orbits form an open connected everywhere dense set. In fact the comple- ment is a closed set of dimension at most n — 2 [1, Chapter IX]. Two of the results to be proved are the following, where B is the set of points on orbits of dimension less than the highest dimension of any orbit: If SO(3) acts differentiably on Sn with three-dimensional principal orbits and if dim B < n — 2 then the principal isotropy group is the identity; if SO(3) acts differentially on S7, then some orbit has dimension less than 3. Part of the motivation for our work was the attempt to discover whether the latter result is true for all n. If SO(3) does act differ- entiably on Sn with all orbits three-dimensional then, as far as rational coefficients are concerned, the sheaf generated by the orbits is constant and relative to these coefficients we obtain similar results to those for a fibering by S3 (see [1, 3]); hence n — 4/b — 1.