It is well known that any directed graph induces an undirected graph by forgetting the direction of the edges and keeping the underling structure. In fact, this assignment can be extended to consider graph morphisms, thus obtaining a functor from the category of simple directed graphs and directed graph morphism, to the category of undirected graphs and undirected graph morphisms. This particular functor is known as a “forgetful” functor, since it forgets the notion of direction.

In this talk, I will present a bijective functor that relates the category of simple directed graphs with a particular category of undirected graphs, whose objects we call “prime graphs”. Intuitively, prime graphs are undirected bipartite graphs endowed with a label that evokes a notion of direction. As an application, we use two undirected graph techniques to study directed graphs: spectral clustering and network alignment.

In 2015, Marcus, Spielman, and Srivastava realized that expected characteristic polynomials of sums and products of randomly rotated matrices behave like finite versions of Voiculescu's free convolution operations. In 2022, I obtained a similar result for commutators of such random matrices; one feature of this result is the special role of even polynomials, in parallel with the situation in free probability.

It turns out that a certain family of special polynomials, called hypergeometric polynomials, arises naturally in relation to convolution of even polynomials and finite free commutators. I will explain how these polynomials can be used to approach questions of real-rootedness and asymptotics for finite free commutators. Based on arXiv:2209.00523 and ongoing joint work with Rafael Morales and Daniel Perales.

This talk discusses computationally feasible machine learning methods, based on optimal transport and neural network theory, applied to measure-valued data.  We first analyze linearized optimal transport (LOT), which essentially embeds measure-valued data into an $L^2$ space, where out-of-the-box machine learning techniques are available.  We analyze the situations when LOT provides an isometric embedding with respect to the Wasserstein-2 distance and provide necessary bounds when we can achieve a pre-specified linear separation level in the LOT embedding space.  Second, we produce a computationally feasible algorithm to recover low-dimensional structures in measure-valued data by using the LOT embedding along with dimensionality reduction techniques.  Using computational methods for solving optimal transport problems such as the Sinkhorn algorithm or linear programming, we provide approximation guarantees in terms of the sampling rates.  Third, we study structured approximations of measures in Wasserstein space by a scaled Voronoi partition of $\mathbb{R}^d$ generated from a full rank lattice.  We show that these structured approximations match rates of optimal quantizers and empirical measure approximation in most instances.  We then extend these results for noncompactly supported measures that decay fast enough.  Finally, we study methods for comparing probability measures by analyzing a neural network two-sample test.  In particular, we perform time-analysis on a related neural tangent kernel (NTK) two-sample test and extend the analysis to the neural network two-sample test with a small-time training regime.  We also show the amount of time needed before the two-sample test detects a deviation $\epsilon > 0$ in the case the probability measures considered are different versus when they are the same.

We introduce repetition noise into quantum error-correcting codes with a tensor network structure. Our approach employs a quantum channel, which is a superposition of exact encodings and repetition encoding with a small probability. The boundary states of our models capture key features of conformal field theory states, particularly the power law of the two-point function and logarithmic entanglement, which are precisely obeyed in many cases. The noisy holographic quantum error-correcting codes on trees and tilings of two-dimensional hyperbolic space preserve the bulk/boundary duality in AdS/CFT, and their boundary states exhibit the features of conformal field theory accordingly.

In this talk we will define the Rel foliation for a stratum of translation surfaces with at least two singularities. We will focus on the real Rel flow in the stratum $\mathcal{H}(1,1)$. We will provide some examples of orbits, and their closures. Finally, we will describe the real Rel orbits of tremors of surfaces and provide explicit examples of trajectories that are not recurrent, but do not diverge.

The Laplace operator is the infinitesimal generator of Brownian motion with a real state space.  The Vladimirov operator, a $p$-adic analogue of the Laplace operator, similarly gives rise to Brownian motion with a $p$-adic state space.  This talk aims to introduce the concept of a $p$-adic Brownian motion and demonstrate a further similarity with its real analogue: $p$-adic Brownian motion is a scaling limit of a discrete-time random walk on a discrete group.  Attendees need not have prior knowledge of $p$-adic analysis, as the talk will provide a brief review of necessary background information.

In this talk we will present some analysis aspects of gauge theory in high dimension. First, we will study the completion of the space of arbitrary smooth bundles and connections under L^2-control of their curvature. We will start from the classical theory in critical dimension and then move to the supercritical dimension, making a digression about the so called “abelian” case and thus showing an analogy between p-Yang-Mills fields on abelian bundles and a special class of singular vector fields. In the last part, we will show how the previous analysis can be used in order to build a Schoen-Uhlenbeck type regularity theory for Yang-Mills fields in supercritical dimension.

I will present several applications of Shlyakhtenko’s operator-valued semicircular systems, including characterization of Property (T) for II$_1$ factors in terms of spectral gap in inclusions, and on weak containment of bimodules.

I will discuss recent developments and ongoing work for algebraic and p-adic aspects of L-functions. Interest in p-adic properties of values of L-functions originated with Kummer’s study of congruences between values of the Riemann zeta function at negative odd integers, as part of his attempt to understand class numbers of cyclotomic extensions. After presenting an approach to studying analogous congruences for more general classes of L-functions, I will conclude by introducing ongoing joint work of G. Rosso, S. Shah, and myself (concerning Spin L-functions for GSp_6). I will explain how this work fits into the context of earlier developments, while also indicating where new technical challenges arise. All who are curious about this topic are welcome at this talk, even without prior experience with p-adic L-functions or Spin L-functions.
 

In my experience, successes often arise from circumstances that appear to be less than ideal, or even hopeless.  In the AWM Colloquium, I will discuss some key developments along my career path.

The target audience is graduate students and postdocs.  Audience engagement is encouraged.  In particular, I will allow ample time for questions.
 

In this talk, we will demonstrate that the well-known singularity Hausdorff dimension estimates for isoperimetric regions are sharp by constructing singular examples in dimensions 8 and higher. Then, to explore the isoperimetric regions under generic Riemannian metrics, we will discuss the twisted Jacobi field of singular constant mean curvature hypersurfaces under certain regularity assumptions.

For $C$ a smooth curve and $\xi$ a line bundle on $C$, the moduli space $U_C(2,\xi)$ of semistable vector bundles of rank two and determinant $\xi$ is a Fano variety. We show that $U_C(2,\xi)$ is K-stable for a general curve $C \in \overline{M}_g$. As a consequence, there are irreducible components of the moduli space of K-stable Fano varieties that are birational to $\overline{M}_g$. In particular these components are of general type for $g\geq 22$.

This study delves into the study of complex networks within a hyperbolic latent space model, presenting theoretical analysis of popular link prediction indices on hyperbolic random graphs. We investigate how different degrees of nodes influence link prediction heuristics. By modifying indices like the common neighbor and shortest path index, the study demonstrates theoretical and empirical improvements in both simulated and real-world networks. Additionally, we also explore embedding methods to recover hyperbolic geometry, introducing a modified hyperbolic ordinal embedding method. 

Come find out how to (randomly) diagonalize any $n \times n$ matrix pencil in fewer than $O(n^3)$ operations!

You use universal properties everyday. But what are they exactly? We'll give an answer via something called a universal element. It is the nice concrete thing which defines a Yoneda representation. We'll give several examples of the universal elements for common representations.

Prerequisite: Watch this video https://www.youtube.com/watch?v=mLRgKPwyg4Y

Often natural moduli problems, e.g. foliated surfaces, come without an ample line bundle, so the algebraisability of formal deformations, and the very existence of a moduli space requires a study of the mermorphic functions on the aforesaid deformations, and the flattening (by blowing up) of the resulting meromorphic maps. In such a context the flattening theorem of Raynaud & Gruson, and derivatives thereof, is close to irrelevant since it systematically uses schemeness to globally extend local centres of blowing up. This was already well understood by Hironaka in his proof of holomorphic flattening, and his ideas are the right ones. Nevertheless, the said ideas can be better organised with a more systematic use of Grothendieck's universal flatifier, and, doing so, leads to a fully functorial, and radically simpler, proof provided the sheaf of nilpotent functions is coherent-which is true for excellent formal schemes, but, unlike schemes or complex spaces, is false in general.