### Analysis and Geometry Seminar

#### Spring 2024

##### 30 April 2024

Artem Novozhilov: Systems with inheritance and their mathematical analysis (Part 2)

Abstract: My plan is to introduce a class of differential equations, often called selection systems or systems with inheritance, and discuss known properties of their solutions. These differential equations can be considered as a somewhat intermediate mathematical object between the classical ordinary differential equations and first order partial differential equations (transport equations). The analysis and applications of these equations have a long history (they were studied systematically for at least 40 years to the best of my knowledge), and yet there are quite a few interesting (in my biased opinion) open mathematical questions, which I plan to discuss. I will start with the simplest examples, and the talk will be mostly example driven, no previous knowledge about such equations will be assumed, so I encourage the graduate students to attend the talk.

##### 23 April 2024

Vincent Homlund: Totally symmetric self-complementary plane partition matrices: enumerative properties and polytopes

Abstract: Plane partitions in the totally symmetric self-complementary symmetry class (TSSCPP) are known to be equinumerous with $n \times n$ alternating sign matrices, but no explicit bijection is known. In this talk I will discuss a set of {0,1,-1}-matrices, called Magog matrices, which are in bijection with TSSCPP. I will explore their enumerative properties and compare them to those of ASMs. I will also look at them from a geometric perspective by discussing two different polytopes that are formed from TSSCPP.

##### 16 April 2024

Artem Novozhilov: Systems with inheritance and their mathematical analysis

Abstract: My plan is to introduce a class of differential equations, often called selection systems or systems with inheritance, and discuss known properties of their solutions. These differential equations can be considered as a somewhat intermediate mathematical object between the classical ordinary differential equations and first order partial differential equations (transport equations). The analysis and applications of these equations have a long history (they were studied systematically for at least 40 years to the best of my knowledge), and yet there are quite a few interesting (in my biased opinion) open mathematical questions, which I plan to discuss. I will start with the simplest examples, and the talk will be mostly example driven, no previous knowledge about such equations will be assumed, so I encourage the graduate students to attend the talk.

##### 2 April 2024

Lane Morrison:  (Part 2)

##### 26 March 2024

Lane Morrison:  Wall Stabilization

Abstract: 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 $S^2 \times S^2$. 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)

Abstract: In this second talk we will be discussing the proof of the problem posed in the previous talk. In the process we will examine a rather pleasant basis for $L^2(S^{n-1})$, and highlight the harmonic analysis techniques used in the proof.

##### 12 March 2024

Chase Reuter: A Local Result Relating Centroid and Intersection Bodies (Part 1)

Abstract: 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

##### Seminar information
• Location: Morrill 109
• Time: Tuesdays, 11:00-11:50 am
• Organizer: Maria Alfonseca-Cubero

##### 5 December 2023

Azer Akhmedov: Bi-orderability in one-relator groups (Part 3)

##### 28 November 2023

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

##### 14 November 2023

Azer Akhmedov: Bi-orderability in one-relator groups

Abstract: 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

Abstract: 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

Abstract: 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.