We introduce directed graphs and demonstrate how to generate a C*-algebra which reflects certain features of the graph. Time permitting, we will introduce two uniqueness theorems for their representations and explore a few of their consequences.

We construct the first positive examples to the Connes' Rigidity Conjecture, i.e., we construct groups $G$ with Kazhdan's property (T) such that if $H$ is a group with the same von Neumann algebra as $G$, then $H\cong G$. In this talk, I will focus on the group-theoretic side of this result and talk about how we applied geometric group theory to solve problems from von Neumann algebra. This is joint work with Ionut Chifan, Adrian Ioana, and Denis Osin.

In previous work of Akhmechet, Krushkal, and the speaker, a unification of lattice cohomology and the $\widehat{Z}$-invariant was established. Both theories are combinatorially defined invariants of plumbed 3-manifolds, but with quite different origins. Lattice cohomology, due to Némethi, is motivated by the study of normal surface singularities and is isomorphic to Heegaard Floer homology for plumbing trees. On the other hand, $\widehat{Z}$, due to Gukov-Pei-Putrov-Vafa, is a power series coming from a physical theory and is conjectured to recover quantum invariants of 3-manifolds at roots of unity. In this talk, I will discuss work in progress relating knot lattice homology and the Gukov-Manolescu 2-variable series, the knot theoretic counterparts to lattice homology and $\widehat{Z}$. This is joint work with Ross Akhmechet and Sunghyuk Park.

I will discuss a discrete non-deterministic flow-firing process for topological cell complexes. The process is a form of discrete diffusion; a flow is repeatedly diverted according to a discrete Laplacian. The process is also an instance of higher-dimensional chip-firing. I will motivate and introduce the system and then focus on two important features – whether or not the system is terminating and whether or not the system is confluent.

Polynuclear growth is one of the basic models in the Kardar-Parisi-Zhang universality class, which describes a one-dimensional crystal growth. For a particular initial state, it describes the length of the longest increasing subsequence for uniformly random permutations (the problem first studied by S. Ulam). In my joint work with J. Quastel and D. Remenik we expressed the distribution functions of the polynuclear growth in terms of the solutions of the Toda lattice, one of the classical integrable systems. A suitable rescaling of the model yields a non-trivial continuous limit of the polynuclear growth (the KPZ fixed point) and the respective equations (Kadomtsev-Petviashvili).

What Should a Good Deep Neural Network Look Like? Insights from a Layer-Peeled Model and the Law of Equi-Separation
See https://sites.google.com/view/minds-seminar/home

In favorable circumstances, topological 4-manifolds and surfaces in them can be classified. In contrast, little is known about smooth 4-manifolds and smooth surfaces. Several of the hardest problems in 4-dimensional topology (eg. the Poincare conjecture) simply ask whether the topological classification fails in the smooth setting; such failures are called exotica. In this talk, I will discuss some historic and recent progress towards detecting exotic phenomena, and outline some promising approaches.

Let $K/\bf{Q}_p$ be a local number field of absolute ramification index $e$, and let $X$ be a proper smooth $O_K$-scheme. I will discuss how one can obtain bounds on ramification of the mod $p$ Galois representations arising as the étale cohomology of (the geometric generic fiber of) $X$ in terms of $e$, the given prime $p>2$ and the cohomological degree $i$. The key tools for achieving this are the Breuil-Kisin and $A_{\rm inf}$-cohomology theories of Bhatt, Morrow and Scholze, and a series of conditions based on a criterion of Gee and Liu regarding crystallinity of the representation attached to a free Breuil-Kisin-Fargues $G_K$-module.

A quantum particle restricted to a lattice of points has been well studied in many different contexts. In the absence of disorder or environmental interaction, the particle simply undergoes ballistic transport for many suitable Hamiltonian operators. Recently, progress has been made on introducing a Lindbladian interaction term to the model, which drastically changes the dynamics in the large time limit. We prove that indeed diffusion is present in this context for an arbitrary periodic Hamiltonian. Additionally, we show that the diffusion constant is inversely proportional to the particles' coupling strength with its environment.

When Björner and Wachs introduced one of the main forms of lexicographic shellability, namely CL-shellability, they also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering. We generalize the notion of recursive atom ordering, and we prove that any such generalized recursive atom ordering may be transformed via a reordering process into a recursive atom ordering. We also prove that a finite bounded poset admits a generalized recursive atom ordering if and only if it is ``CC-shellable'' by way of a CC-labeling which is self-consistent in a certain sense. This allows us to conclude that CL-shellability is equivalent to self-consistent CC-shellability. As an application, we prove that the uncrossing orders, namely the face posets for stratified spaces of planar electrical networks, are dual CL-shellable.
During this talk, we will review plenty of background on poset topology and specifically regarding the technique of lexicographic shellability. This is joint work with Grace Stadnyk

We present results on the stability of equilibria (time-independent solutions) of the Vlasov-Maxwell equation. In particular, linear stability criteria for certain classes of equilibria are discussed. We also give a result on the nonlinear stability of an initial-boundary value problem for the Vlasov-Poisson equation.
**Note: speaker will present Virtually. Participants can join in person to view the presentation in C304, or through the Zoom link.**

If a and b are integers that satisfy a simple nonvanishing
condition, the cubic equation y^2 = x^3 + ax + b defines an elliptic
curve over the field of rational numbers. Elliptic curves have been
studied for millennia and seem to occur all over the place in
mathematics, physics and other sciences. In my talk, I'll explain how a
specific elliptic curve provides the solution to a surprisingly hard "brain
teaser" that had a big run on social media a few years ago.