The Geometry and Geometric Analysis Seminar at Purdue in Fall 2022 is usually on Mondays 3:30-4:30pm Eastern Time in MATH 731 if we meet in person. Some talks will be on zoom and the link will be included in the email announcement.
Lvzhou Chen and Sai-Kee Yeung are organizing this seminar in Fall 2022. If you have any questions or would to suggest speakers, please contact one of us. If you would like to be on the mailing list, please email Lvzhou.
Title: Some hyperbolic 3-manifold groups completely determined by their finite quotients
Abstract: We use ideas of Bridson-McReynolds-Reid-Spitler to give new examples of hyperbolic 3-manifold groups completely determined (among all residually finite groups) by their finite quotients.
CANCELED and will be rescheduled.
Title: Revisit the theory of laminar groups
Abstract: I will give a brief introduction to laminar groups which are groups of orientation-preserving homeomorphisms of the circle admitting invariant laminations. The term was coined by Calegari and the study of laminar groups was motivated by work of Thurston and Calegari-Dunfield. We present old and new results on laminar groups which tell us when a given laminar group is either Fuchsian or Kleinian.
Title: Generic Mean Curvature Flow with Spherical and Cylindrical Singularities
Abstract: We study the local and global dynamics of mean curvature flow with spherical and cylindrical singularities. We find the most generic dynamic behavior of such singularities, and show that the singularities with the most generic dynamic behavior are robust. We also show that the most generic singularities are isolated and type-I. Among applications, we prove that the singular set structure of the generic mean convex mean curvature flow has certain patterns, and the level set flow starting from a generic mean convex hypersurface has low regularity. This is joint work with Jinxin Xue (Tsinghua University).
Title: Equilibrium measures on metric graphs
Abstract: For a finite metric graph, the signed measure that maximizes a potential defined via pairwise effective resistance, which is the canonical, or Arakelov measure, have been studied extensively and had many applications to combinatorics and tropical geometry. We studied the (non negative) measure that maximizes such potential as well as the signed measure with given support that maximizes such potential, and found many applications of them on combinatorics as well. This is a joint work with Farbod Shokrieh and Harry Richman.
Title: Holomorphic submersions and special Kähler metrics
Abstract: Proper holomorphic submersions of Kähler manifolds can be thought of as both a generalisation of holomorphic vector bundles and as a way of studying the behaviour of Kähler manifolds in families. We will consider fibrations whose fibres are K-semistable varieties that admit a degeneration to Kähler manifolds with constant scalar curvature, in a way compatible with the fibration structure. On such fibrations, we will describe a condition, called optimal symplectic connection condition, which gives a canonical choice of a relatively Kähler metric and a generalisation of the Hermite-Einstein condition on vector bundles.
Title: Transverse measures to infinite type laminations
Abstract: Geodesic laminations are crucial tools for studying mapping class groups and three-manifolds. However, little is understood about laminations on infinite type surfaces. In this talk, we consider the cone of transverse measures to a geodesic lamination on an infinite type surface. We show that this cone admits an explicit description as an inverse limit of finite-dimensional cones. We use this explicit description to illustrate exotic new behaviors exhibited by laminations on infinite type surfaces. This talk is joint work with Mladen Bestvina.
Title: The stochastic heat equation on Heisenberg groups
Abstract: In this talk I will start by giving a brief overview of some standard results concerning the stochastic heat equation in Euclidian spaces. To this aim, I will first describe the kind of Gaussian noise usually considered by the probability community. Then I will review some standard (though recent) existence and uniqueness results for the stochastic heat equation, established for a large class of Gaussian noises.
Our long term goal is to observe how exponents related to the stochastic heat equation are affected by geometric contexts. With this objective in mind, our first example of concern are Heisenberg groups. I will recall basic facts about those objects, in particular notions of Fourier analysis which are crucial for the stochastic analysis of our equation. Next I will detail the construction of a reasonable class of noises on the Heisenberg group. Eventually I will describe some recent advances aiming at a proper definition of noisy heat equations. I will focus on the so-called Itô setting, where an explicit chaos decomposition of the solution is available. A good control of the chaos expansion is then achieved thanks to heavy use of Fourier type estimates. If possible I will show the main steps of this analytic estimate.
Notice that I’m not assuming any knowledge of stochastic analysis in the audience. I will try to introduce the objects I’m manipulating in a self-contained way.
Title: Aut-invariant quasimorphisms
Abstract: For every group G, there is a natural action of Aut(G) on the space of homogeneous quasimorphisms of G. This action is very poorly understood, in particular it is hard to produce fixpoints, i.e. Aut-invariant quasimorphisms, which can be used to estimate Aut-invariant norms on groups.
I will report on joint work with Ric Wade (Oxford) where we construct Aut-invariant quasimorphisms on all Gromov-hyperbolic groups, and more.
Title: Mapping spaces and holomorphic functions
Abstract: Consider a complex manifold X and a compact Hausdorff space S. Continuous maps from S to X form an infinite dimensional complex manifold. The talk will discuss two theorems concerning holomorphic functions on such mapping spaces, one reminiscent of the Monodromy theorem, the other of Liouville’s theorem.
Title: Homotopic rotation sets for higher genus surfaces
Abstract: The rotation number of a homeomorphism of the circle leads to several generalizations in the case of surfaces. These generalizations aim to describe the speed and the directions of the orbits of the iterates of the homeomorphism. Among the existing generalizations, there is a well studied notion of rotation set for homeomorphisms of the annulus or of the 2-torus which are isotopic to the identity and a notion of homological rotation sets for homeomorphisms of higher genus surfaces which are isotopic to the identity. However, the latter rotation sets does not detect orbits which turn around a homologically trivial loop, so the idea to define a homotopical rotation set is natural.
In this talk, after reviewing the definitions and some properties of the already existing rotation sets, I will introduce the results of a work in collaboration with Pierre-Antoine Guihéneuf where we define such a homotopical rotation set and we prove some of its properties, some of which are related to the existence of periodic orbits.
Title: On the finiteness of the classifying space of diffeomorphisms of reducible three-manifolds
Abstract: It is known that the classifying space BDiff(S, rel boundary) for a surface S with a nontrivial boundary is homotopy equivalent to a finite CW complex e.g. the corresponding moduli space of Riemann surfaces. Similarly, in dimension 3, there is a conjecture that on Kirby's list it is attributed to Kontsevich which says that the classifying space BDiff(M, rel boundary) for any 3-manifold with a non-empty boundary has a finite-dimensional model. When M is irreducible, this conjecture was solved by Hatcher-McCullough. In this talk, we discuss the solution to the homological version of this problem.