Tag - Manifolds

Mohammed Abouzaid: Theory of bordisms

6th August, 2024

In this introductory lecture, which should be accessible to a general mathematical audience, I will review the classical bordism theory of manifolds, from its origin in Poincare's work, to the subsequent development by Pontryagin, Thom, Milnor, Wall, and Quillen among others.

Lecture 2: Bordism of orbifolds

An orbifold is a space with additional structure that describes it locally as the quotient of a manifold by a finite group. I will describe Pardon's recent result which reduces the study of orbifolds to the study of manifolds with Lie group actions. Then I will explain the relationship between equivariant and orbifold bordism, and formulation some structural properties of this theory.

Lecture 3: Bordism of derived orbifolds

The notion of a derived orbifold arises naturally in pseudo-holomorphic curve theory, and plays a central role in the emerging field of Floer homotopy. I will explain how it is related to the notion of "homotopical bordism" due to tom Dieck in the 1970s, and formulate some conjectures about its structure in the complex oriented case.

Amie Wilkinson: Dynamical Asymmetry Is C1-Typical

5th February, 2024

I will discuss a result with Bonatti and Crovisier from 2009 showing that the C1 generic diffeomorphism f of a closed manifold has trivial centralizer; i.e., fg = gf implies that g is a power of f. I'll discuss features of the C1 topology that enable our proof (the analogous statement is open in general in the Cr topology, for r > 1). I'll also discuss some features of the proof and some recent work, joint with Danijela Damjanovic and Disheng Xu that attempts to tackle the non-generic case.

Michael Robert Magee: Convergence of Unitary Representations and Spectral Gaps of Manifolds

26th January, 2024

Let G be an infinite discrete group. Finite dimensional unitary representations of G are usually quite hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of G alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps.

Oscar Randal-Williams: Homeomorphisms of Euclidean Space

6th October, 2023

The topological group of homeomorphisms of d-dimensional Euclidean space is a basic object in geometric topology, closely related to understanding the difference between diffeomorphisms and homeomorphisms of all d-dimensional manifolds (except when d=4). Over the last few years a great deal of progress has been made in understanding the algebraic topology of this group. I will report on some of the methods involved, and an emerging conjectural picture.

Michael Entov: Kähler-type and Tame Embeddings of Balls into Symplectic Manifolds

5th May, 2023

A symplectic embedding of a disjoint union of domains into a symplectic manifold M is said to be of Kähler type (respectively tame) if it is holomorphic with respect to some (not a priori fixed) integrable complex structure on M which is compatible with (respectively tamed by) the symplectic form. I'll discuss when Kähler-type embeddings of disjoint unions of balls into a closed symplectic manifold exist and when two such embeddings can be mapped into each other by a symplectomorphism. If time permits, I'll also discuss the existence of tame embeddings of balls, polydisks and parallelepipeds into tori and K3 surfaces.

Peter Humphries: Restricted Arithmetic Quantum Unique Ergodicity

4th May, 2023

The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds. This conjecture has been resolved by Lindenstrauss when this manifold is the modular surface assuming these eigenfunctions are additionally Hecke eigenfunctions, namely Hecke-Maass cusp forms. I will discuss a variant of this problem in this arithmetic setting concerning the mass equidistribution of Hecke-Maass cusp forms on submanifolds of the modular surface.

Thomas Koberda: First-order rigidity of homeomorphism groups of manifolds

9th December, 2022

I will discuss some aspects of the first-order theory of homeomorphism groups of connected manifolds. The main result is as follows. Let M be a compact, connected manifold. There is a sentence S(M) in the language of groups such that if N is an arbitrary manifold and the homeomorphism group of N models S(M) then N is homeomorphic to M. This resolves a conjecture of Rubin from the 1980s. I will illustrate some of the ingredients of the proof, including an interpretation of second order arithmetic in the theory of homeomorphism groups of manifolds.

Melanie Matchett Wood: A visit to 3-manifolds in the quest to understand random Galois groups

13th October, 2022

Cohen, Lenstra, and Martinet gave conjectural distributions for the class group of a random number field. Since the class group is the Galois group of the maximum abelian unramified extension, a natural generalization would be to give a conjecture for the distribution of the Galois group of the maximal unramified extension. Previous work (joint with Liu and Zurieck-Brown) produced a plausible conjecture for the part of this Galois group relatively prime to the number of roots of unity in the base field. There is a deep analogy between number fields and 3-manifolds. Thus, an analogous question would be to describe the distribution of the profinite completion of the fundamental group of a random 3-manifold. In this talk, I will explain how Will Sawin and I answered this question for a model of random 3-manifolds defined by Dunfield and Thurston, and how the techniques we used should allow us, in future work, to prove large q limit theorems in the function field analogue and give a general conjecture in the number field case, taking into account roots of unity in the base field. This is part two of a series of two talks on joint work, some in progress, with Will Sawin. Both talks should be understandable on their own.

Paul Nelson: The sup norm problem in the level aspect

29th September, 2022

The sup norm problem concerns the size of L2-normalized eigenfunctions of manifolds. In many situations, one expects to be able to improve upon the general bound following from local considerations. The pioneering result in that direction is due to Iwaniec and Sarnak, who in 1995 established an improvement upon the local bound for Hecke-Maass forms of large eigenvalue on the modular surface. Their method has since been extended and applied by many authors, notably to the 'level aspect' variant of the problem, where one varies the underlying manifold rather than the eigenvalue. Recently, Raphael Steiner introduced a new method for attacking the sup norm problem. I will describe joint work with Raphael Steiner and Ilya Khayutin in which we apply that method to improve upon the best known bounds in the level aspect.