Events of the Week

"Red or blue pill?" Ioana asked the graduate student.
"What happens if I take the blue one?"
"Nothing," Alan replied. "The story ends. We will know that you are not curious enough, or that you think that finding the energy levels for charged atoms is not an important quest for physics."
"Or that you do not believe in the power of probability theory, only in the crude Riemann-Hilbert method," Ioana pressed.
The student was a bit perplexed: he hated RHP, but at the same time, their quest looked impossible without it. "So if I take the red one? What is going to happen?"
"Well," said Ioana, smiling, "you will start an amazing journey. You will uncover a world of beauty and possibility. We start with the simplest possible situation: we consider a symmetric random matrix with Gaussian entries and we will compute the joint probability density function of the eigenvalues explicitly. During this journey, we will learn how to tridiagonalize a matrix, how to use recurrence relations to express the Vandermonde determinant, and much more."
Alan stood up. "And the best of all? It is going to be a symphony where all the players play their part flawlessly."
The grad took the red pill and ate it. "Well, let's get started!"

Reference: "Matrix Models for Beta Ensembles" Ioana Dumitriu, Alan Edelman https://arxiv.org/abs/math-ph/0206043

In analytic number theory, summation formulas are often useful for understanding the properties of sequences which grow erratically. We explore Riesz type sums involving class number of real quadratic field. In particular, we extend recent work of Beckwith, Diamantis, Gupta, Rolen, and Thalagoda from harmonic Maass forms to sesquiharmonic Maass forms of weight $1/2$. Our approach adapts a method of Chandrasekharan and Narasimhan, which we apply to a sesquiharmonic Maass form first introduced by Duke, Imamo{\u g}lu, and T\'{o}th. (This is ongoing joint work with Professor Olivia Beckwith)

We develop a unified algebraic and valuative theory of Lojasiewicz exponents for pairs of graded families of ideals. Within this framework, analytic local Lojasiewicz exponents, gradient exponents, and exponents at infinity are all realized as asymptotic containment thresholds between appropriate filtrations.

The main theme is a finite-max principle: under verifiable algebraic hypothesis, the a priori infinite valuative supremum describing the Lojasiewicz exponent reduces to a finite maximum and attained by divisorial valuations. We identify two complementary mechanisms leading to this phenomenon: finite testing arising from normalized blowups and Rees algebra constructions, and attainment via compactness of normalized valuation spaces under linear boundedness assumptions. This finite-max principle yields strong structural consequences, including rigidity, stratification, and stability results. We also explain classical results/problems in toric and Newton polyhedral settings.

We will continue our journey into the realm of persistence modules.

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This learning seminar introduces the foundations of persistent homology, a key tool in topological data analysis for capturing multiscale topological features of general metric spaces. We will study persistence modules, barcodes, and distances such as the bottleneck and interleaving metrics, with a focus on their geometric meaning.

A central goal of the seminar is to understand and prove stability theorems for persistent homology, including bounds relating bottleneck distance to the Gromov–Hausdorff distance. Core examples will come from Morse theory and Vietoris–Rips filtrations of metric spaces.

The seminar is structured as a guided, collaborative reading course and is aimed at graduate students and faculty generally interested in this aspect of topology or geometry.

One form of the prime number theorem asserts that
\[\sum_{n\le x} \Lambda(n) = x(1 + o(1)),\]
where $\Lambda(n)$ is the von Mangoldt function. By the triangle inequality, this also gives
\[\sum_{x < n\le x+y} \Lambda(n) = y(1 + o(1)),\]
in the ``long interval'' setting $y\sim x$. It is expected that the prime number theorem holds for much shorter intervals, namely for $y\sim x^{\theta}$ for any fixed $\theta\in (0,1]$. From the recent zero density estimates of Guth and Maynard, this result is known for all $x$ when $\theta > \frac{17}{30} $ and for almost all $x$ when $\theta > \frac{2}{15}$. In this talk, we will discuss the connections between zero density estimates, the prime number theorem in short intervals, and the distribution of prime numbers. Further, we will present some quantitative upper bounds on the size of the exceptional set where the prime number theorem in short intervals fails. We give an explicit relation between zero density estimates and exceptional set bounds, allowing for the most recent zero density estimates to be directly applied to give upper bounds on the exceptional set via a small amount of computer assistance. This talk is based on joint work with Terence Tao.

By cutting a 3D convex polyhedron by a plane, we obtain a convex cap. By cutting on a boundary rooted spanning forest of the edge graph, we can unfold the cap into the plane. Nearly flat caps have unfoldings which are very close to their orthogonal projections. We take advantage of this to construct a necessary and sufficient condition for detecting overlap in the unfolding based on the orthogonal projection of the cap's edge graph. This is a recent result which is a joint work with Tyson Trauger. We also discuss two possible applications of this condition: a positive resolution to a special case of Durer's problem, and a necessary and sufficient condition for detecting overlap in infinitesimal edge unfoldings of arbitrary convex caps.

Title and abstract to be announced

Given a local ring R we can ask when saturation of ideals in R commutes with other operations on ideals (such as extension to a ring containing R). We show that the condition that extension of ideals along a ring map R \to S commutes with saturation controls inherent properties of the rings R & S, such as Cohen-Macaulayness and unramifiedness.

Title and abstract to be announced

Title and abstract to be announced

Title and abstract to be announced

We will share a story that brings together geometry, dynamics, and number theory in interesting and novel ways, and also creates some very compelling imagery- the talk will have lots of pictures, and all relevant background notions will be introduced and explained. Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat 3-manifolds which we call translation prisms. Using ideas of Furstenberg, and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot-Vijayraghavan expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular n-gon for n = 7, 9, 14, 16, 18, 20, 24, 30. This is joint work with Nicolas Bedaride, Pat Hooper, and Pascal Hubert.

Title and abstract to be announced

Title and abstract to be announced