MATH-IMS Joint Colloquium Series (Pure Mathematics)
2020-21 Term 2

Date/Time/Venue Talks
Feb 4, 2021 (Thur)
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Speaker: Fangyang Zheng (Chongqing Normal University)
Title: Kahler manifolds with positive orthogonal Ricci curvature

Abstract: The orthogonal Ricci curvature of a tangent direction in a Kahler manifold is the difference between the Ricci and the holomorphic sectional curvature of that direction. This curvature notion is closely related to Ricci and holomorphic sectional curvature. We are interested in understanding the class of compact Kahler manifolds with everywhere positive orthogonal Ricci curvature. In this talk, I will report on some recent progress on the topic.
Feb 18, 2021 (Thur)
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Speaker: Vincent Colin (University of Nantes)
Title: Reeb dynamics in dimension 3 and broken book decompositions

Abstract: The study of the dynamics of vector fields in dimension 3 was at center of Poincaré's pioneering work on the 3-body problem. Vector fields arising from mechanics often have the extra property of preserving a contact structure and fall in the family of Reeb vector fields. I will describe how to extend the notion of a Poincaré's surface of section to the more general notion of a broken book decomposition and derive general dynamical properties. Amongst those, a nondegenerate Reeb vector field has either 2 or infinitely many periodic orbits. This is joint work with Patrick Dehornoy and Ana Rechtman.
Feb 25, 2021 (Thur)
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Speaker: Gang Liu (East China Normal University)
Title: Gromov-Hausdorff convergence of Kahler manifolds

Abstract: We discuss some recent development of Gromov-Hausdorff convergence of Kahler manifolds with geometric applications.
Mar 4, 2021 (Thur)
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Speaker: Hélène Esnault (Freie Universität Berlin)
Title: Local Systems in Geometry and Arithmetic

Abstract: Galois laid the foundations of Galois theory and Galois groups, Riemann and Poincaré the ones of algebraic topology and the fundamental group. Grothendieck showed the two worlds are analog and developed the notion of étale fundamental group. Where do we find (continuous) representations of those in mathematics? Essentially the only ones we have at disposal are those of geometric nature. I’ll discuss conjectures on the density of those in the parameter space of all representations, and some results giving a small evidence for it (based on joint work with Moritz Kerz and Michael Groechenig).
Mar 11, 2021 (Thur)
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Speaker: Alberto Abbondandolo (Ruhr Universität Bochum)
Title: Systolic questions in metric and symplectic geometry

Abstract: The prototypical question in metric systolic geometry is to bound the length of a shortest closed geodesic on a closed Riemannian manifold by the volume of the manifold. This question has been extensively studied for non simply connected manifolds, but in the recent years there has been some progress also for simply connected manifolds, on which closed geodesics cannot be found simply by minimizing the length. This progress involves extending systolic questions to Reeb flows, a class of dynamical systems generalising geodesic flows. On the one hand, this extension and the use of symplectic techniques provide some answers to classical questions within metric systolic geometry. On the other hand, new questions arise from the more general setting and relate seemingly distant fields such as the study of rigidity properties of symplectomorphisms and the integral geometry of convex bodies. I will give a non-technical panoramic view of some of these recent developments.
Mar 18, 2021 (Thur)
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Speaker: Thomas Haines (University of Maryland)
Title: The geometry of affine Schubert varieties and applications

Abstract: Classical Schubert varieties are orbit-closures of a Borel subgroup acting on a partial flag variety attached to a connected reductive group. They play a central role in representation theory and combinatorics. Their geometric properties -- whether they are normal, Cohen-Macaulay, or Frobenius-split; when they are singular, and what kind of singularities arise, etc -- have been intensively studied and are now well understood. Affine Schubert varieties are similar objects but attached to a loop group rather than a group. They play a role in representation theory, mathematical physics, and in geometric approaches to automorphic forms. In the last 20 years they have been studied in large part because of their connection to certain Shimura varieties through the theory of Rapoport-Zink local models. But some key geometric properties -- including normality -- remain somewhat mysterious to this day, at least in some positive characteristic settings. This talk will survey some recent advances in the understanding of basic geometric properties of affine Schubert varieties, including the recently discovered surprising fact that ``most'' affine Schubert varieties in ``bad'' positive characteristic are not normal. We will indicate how these results are used to understand the geometry of certain Shimura varieties.
Mar 25, 2021 (Thur)
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Speaker: Ruochuan Liu (BICMR, Peking University)
Title: Topological cyclic homology for p-adic local fields

Abstract: Topological cyclic homology, which was introduced by Bokstedt-Hsiang-Madsen in early 90s, is an important tool to compute algebraic K-groups via the cyclotomic trace map. It is also closely related to p-adic cohomology theories as revealed by the recent work of Bhatt-Morrow-Scholze. In this talk, we will give a brief introduction to topological cyclic homology and present a new approach to computing it in the case of p-adic local fields. Joint work with Guozhen Wang.
Mar 31, 2021 (Wed)
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Speaker: Rachel Roberts (Washington University in Saint Louis)
Title: Using foliations to understand manifolds

Abstract: One approach to understanding a complicated object involves breaking the object into simpler pieces that fit back together in constrained ways. In this talk, the complicated object is an n-manifold, and the simpler pieces are obtained from foliations, in various ways, including foliation charts, leaves of a foliation, branched surfaces, and sutured manifolds. Work of Thurston in the 1970s highlighted the power of codimension one. More recently, the Floer homology theories have led to new insights. After introducing the definition of foliation and describing some basic foliation results, I will specialize to the special case of codimension one foliations in 3-manifolds. Included in the discussion will be results of Gabai, Thurston, Eliashberg-Thurston, Bowden, Kazez-R, and Delman-R.
Apr 8, 2021 (Thur)
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Speaker: Patrice Le Calvez (Institut de Mathématiques de Jussieu - Paris Rive Gauche)
Title: About periodic points of conservative surface diffeomorphisms

Abstract: As explained by Poincare, periodic points are of high importance in the study of the dynamics of volume preserving diffeomorphisms on a manifold M. There is still a lot of open questions concerning periodic points , even in the case where M is two dimensional. We will report on some recent progress about periodic points of area preserving diffeomorphisms of surface and more generally of homeomorphisms with no wandering points.
Apr 15, 2021 (Thur)
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Speaker: Yiming Long (Chern Institute of Mathematics, Nankai University)
Title: Rabinowitz minimal periodic solution conjecture

Abstract: In 1978, Professor Paul Rabinowitz proved the existence of T-periodic solutions for autonomous Hamiltonian systems whose Hamiltonian function is super-quadratic at infinity and zero for any given T>0 by using a minimax variational method. Because the minimal period of this solution could be T/k for some positive integer k, he asked whether such a system possesses always a T-periodic solution with T as its minimal period. This is the famous Rabinowitz minimal periodic solution conjecture. In the last more than 40 years, many mathematicians have studied this conjecture and got many interesting results. But the conjecture is still open under the original conditions of Rabinowitz. In this lecture, I shall give a brief survey on the main results obtained and methods used in these studies so far, and hope to lead to more interests on this conjecture.
Apr 21, 2021 (Wed)
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Speaker: Emily Riehl (Johns Hopkins University)
Title: Elements of ∞-Category Theory

Abstract: Confusingly for the uninitiated, experts in weak infinite-dimensional category theory make use of different definitions of an ∞-category, and theorems in the ∞-categorical literature are often proven "analytically", in reference to the combinatorial specifications of a particular model. In this talk, we present a new point of view on the foundations of ∞-category theory, which allows us to develop the basic theory of ∞-categories --- adjunctions, limits and colimits, co/cartesian fibrations, and pointwise Kan extensions --- "synthetically" starting from axioms that describe an ∞-cosmos, the infinite-dimensional category in which ∞-categories live as objects. We demonstrate that the theorems proven in this manner are "model-independent", i.e., invariant under change of model. Moreover, there is a formal language with the feature that any statement about ∞-categories that is expressible in that language is also invariant under change of model, regardless of whether it is proven through synthetic or analytic techniques. This is joint work with Dominic Verity.
Apr 29, 2021 (Thur)
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Speaker: Shun-Jen Cheng (Academia Sinica)
Title: Representation Theory of Lie superalgebras in the BGG category

Abstract: Substantial progress on the solution of the irreducible character problem for the complex simple Lie superalgebras in the so-called BGG category has been made over the last decade or so. In this talk we shall first give a brief description of these simple Lie superalgebras and then to give an outline of some of these results on their irreducible characters.
May 6, 2021 (Thur)
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Speaker: Peter Constantin (Princeton University)
Title: Remarks on Euler and Navier-Stokes equations

Abstract: I'll discuss results concerning incompressible fluid equations at high Reynolds numbers: construction of multiscale solutions, inviscid limits, and conditions for the absence of finite time singularities.

© Martin Li, Department of Mathematics, The Chinese University of Hong Kong