Analysis and Geometry Seminar

Link to Previous Semesters

Spring 2024

Tuesdays, 11:00 am
South Engineering 120
2 April 2024

Lane Morrison: Wall Stabilization (Part 2)
26 March 2024

Lane Morrison: Wall Stabilization

In 1964 C. T. C. Wall proved the following result: Two smooth simply connected 4-manifolds that are h-cobordant are diffeomorphic after enough connected sums with the sphere bundle S2×S2. The proof itself is done using some basic yet important results from handlebody theory. This theorem provides a notion of what a "stable" exotic structure of a manifold might be. My goal in this talk is to discuss the handlebody properties necessary to prove this theorem and then give a justification of it. Afterward, I would like to discuss some of the research that has emerged which utilizes Wall's theorem as a jumping off point.

19 March 2024

Chase Reuter: A Local Result Relating Centroid and Intersection Bodies (Part 2)
12 March 2024

Chase Reuter: A Local Result Relating Centroid and Intersection Bodies

This will be the first of two talks discussing a local version of an open question in geometry. We will discuss the relevant background to state the problem. The question at the heart of the talk will be a conjecture on whether the fixed point of some non-linear operator on convex bodies is the Euclidean ball. In the process we will also explore one route to applying analysis techniques to geometric questions.


Fall 2023

Tuesdays, 11:00 am
Morrill 109
5 December 2023

Azer Akhmedov: Bi-orderability in one-relator groups
28 November 2023

Azer Akhmedov: Bi-orderability in one-relator groups (Part 2)
14 November 2023

Azer Akhmedov: Bi-orderability in one-relator groups

A left order on a group is a linear order which is invariant under left-translations; a bi-order is a linear order which is invariant under both left and right translations. I'll mention several examples from topology about the use of left orders and bi-orders. Then I'll concentrate on one-relator groups and describe an example of a non-bi-orderable one-relator group which has no generalized torsion. This example is from a recent joint work with Jimmy Thorne.

31 October 2023

Doğan Çömez: Recurrence and universally good weights defined by a class of superadditive processes (Part 3)
24 October 2023

Doğan Çömez: Recurrence and universally good weights defined by a class of superadditive processes (Part 2)
17 October 2023

Doğan Çömez: Recurrence and universally good weights defined by a class of superadditive processes

In ergodic theory recurrence plays an important role in both characterizing and constructing sequences that are good for convergence of ergodic averages.  A large class of superadditive processes have recurrence property; hence, sequences defined by such processes can be considered as weights for ergodic or superadditive averages. In this two talk series, first we will investigate some recurrence theorems for both additive and superadditive processes.  It turns out that these recurrence properties pave way to Hartman almost periodic sequences. In particular, we will obtain conditions under which such sequences are good weights for the a.e. convergence of ergodic averages.
10 October 2023

Morgan O'Brien: An excursion into noncommutative ergodic theory (Part 4)
3 October 2023

Morgan O'Brien: An excursion into noncommutative ergodic theory (Part 3)
26 September 2023

Morgan O'Brien: An excursion into noncommutative ergodic theory (Part 2)
19 September 2023

Morgan O'Brien: An excursion into noncommutative ergodic theory

Motivated by problems in quantum physics, quantum information theory, and spectral theory, noncommutative analysis studies what happens one replaces spaces of continuous and measurable functions in analysis with spaces of linear operators on a Hilbert space (in particular, C*-algebras or von Neumann algebras).  In the first two talks, we will discuss some of these motivations for going from classical analysis to noncommutative analysis in more detail, and along the way we will also introduce definitions and some basic results in the theory of von Neumann algebras and their affiliated spaces. The third talk will focus on noncommutative weighted ergodic theory, where we will discuss some recent results that generalize known theorems from classical ergodic theory to the von Neumann algebra setting, and in doing so we will mention some of the intricacies that arise when this generalization is made.