### Spring 2024 Information

Fridays @ 4:00 PM in Altgeld Hall 347.

Mailing list: gradgeomtopo@lists.illinois.edu

### Spring 2024 Talks

- 3 May 2024:
*PDEs, Symmetries, and Invariants (Wilmer Smilde).*

Symmetries of PDEs often arise as (Lie) Pseudogroups. Their invariants—objects preserved by the symmetries—play an essential role in determining when two solutions of a PDE are equivalent and in determining the right space of solutions to a geometric problem, or even in the construction of solutions.

In this talk, I will highlight some aspects of the theory of Lie pseudogroups. I will introduce the formalism of PDEs and Lie pseudogroups, explain what invariants are and why they are relevant, and give some examples of PDEs with symmetry and their invariants. I will also present the Lie-Tresse theorem, which states that, under mild regularity assumptions, the space of invariant functions is generated by finitely many invariants and invariant derivatives. - 26 April 2024:
*Bochner Technique and Applications (Huy Tran).*

In Riemannian Geometry, a central concern lies in understanding the

relationship between curvature and the topology of Riemannian manifolds. This is motivated by Hogde’s theorem, which asserts that every de Rham cohomology class is represented by a harmonic form. Bochner established a crucial link to geometry by showing that the first Betti number of compact manifolds with positive Ricci curvature must vanish. Berger and Meyer further contributed by establishing vanishing results for the Betti numbers of manifolds with positive curvature operators. Tachibana, utilizing Bochner’s technique, proved a rigidity result for Einstein manifolds with positive curvature operators. Recently, Petersen and Wink have extended the results of Meyer, Berger, and Tachibana by introducing more generalized curvature conditions. In this talk, I aim to provide the framework for Bochner’s technique, followed by Petersen and Wink’s new method of controlling curvature terms. Finally, I will discuss several applications of Bochner’s technique. - 4 April 2024:
*The Virtually Haken Conjecture*(Brevan Ellefsen).

After having introduced the elementary theory of Haken manifolds, we now dive into their modern theory. (A brief review will be given forthose who did not attend the first talk). This talk will focus on the history and ideas developed by Wise, Agol, and others toward the proof of the Virtually Haken Conjecture. We will also consider group theoretic applications and investigations, and will briefly discuss open and future problems as time allows. - 29 March 2024:
*Introduction to Haken Manifolds*(Brevan Ellefsen).

In dimension 2, surfaces are commonly classified by cutting and gluing along curves. Similar techniques turn out to work in dimension 3 with spheres, but there are now many more shapes to cut along. Perhaps the nicest are the*Haken*manifolds, which contain a very nice surface one can repeatedly cut along. Not every 3-manifold is Haken, but it turns out every 3-manifold*almost*is: the surface can merely be twisted a finite number of times. The proof of this was a major milestone in 21st century mathematics. In the first of two talks discussing the proof of this result, we present an introduction to the classical theory at an introductory level. Intended for a general audience with some knowledge of algebraic topology. - 22 March 2024:
*Asymptotic Dimension of Various Spaces*(Manisha Garg).

The asymptotic dimension of a metric space is a large-scale analog of topological dimension. It was introduced by Gromov in the context of geometric group theory as a quasi-isometry invariant of finitely generated groups. In this talk, we will introduce the asymptotic dimension by emphasizing on the large-scale nature of dimension and working on various examples. In particular, we will present Roe’s proof that the asymptotic dimension of Gromov-hyperbolic spaces is finite, which proves that the Novikov conjecture for these spaces is true. If time permits, we will also examine the asymptotic dimension for some groups such as free groups and one-relator groups. - 8 March 2024:
*You map, but why?*(Sam Hsu)

In this talk we will look at some ideas behind UMAP (Uniform Manifold Approximation and Projection), which (to me) seems to be a very popular manifold learning algorithm despite its unusually topological/geometric underpinnings. In addition, we will see UMAP isn’t just for dimension reduction, and we will sketch the ideas behind variants of UMAP in other parts of unsupervised learning. A portion of the time will be spent introducing the learning problems and common ways of tackling them. If there’s any time left we might try to shoehorn in a few words about a parametric UMAP, autoencoders, and umap_pytorch. This will be an introductory talk (taking longer than 1.316 seconds) focusing on the geometric and topological aspects, and in particular we will not assume prior knowledge of manifold learning or really any unsupervised learning in general. - 1 March 2024:
*Hodge theory and Cohomology*(Anthony D’Arienzo)

Hodge theory is bridge between the topological properties of a smooth manifold, its cohomology, and its analytical properties, namely harmonic forms. This bridge enables the study of spaces whose topology break down yet still have a sensible notion of a smooth function space. I will review the classical example of this bridge: showing that every cohomology class on a smooth manifold has a unique harmonic form. Afterwards I will highlight some examples where this bridge can study generalized our singular spaces. - 16 February 2024:
*Topological invariants from combinatorial data*(Jason Liu)

Given a Hamiltonian T-space, one can encode some information of the space into a polytope. In good cases, the lengths of the edges of the polytope satisfy some identities, which could be translated into topological invariants. For example, given a reflexive polygon, the sum of the affine length of its edges and that of the edges in the polar polygon is always 12. A similar result holds for 3-dimensional polytopes. Later, Gohindo-Heymann-Sabatini generalized the result to arbitrary dimension with the extra assumption that the polytope is Delzant. In this talk, I’ll introduce how one can read off some topological information (for instance, self intersection numbers, the area of invariant submanifolds, equivariant cohomology ring etc.) from the combinatorial date associated to a Hamiltonian T-space. Time permitting, I’ll sketch the proof of the result from Godinho-Heymann-Sabatini. - 9 February 2024:
*Virtually Haken Conjecture*(Brevan Ellefsen)

After Perelman’s proof of geometrization, the Virtually Haken Conjecture was perhaps the largest open question in 3-manifold theory. The proof of this was the result of a large collaborative effort. This talk will outline the tools and ideas that led to the result. Some experience with Algebraic Topology will be assumed.