In the talk, we will discuss the irreducibility and the Galois group of random polynomials over the integers. After giving motivation (coming from work of Breuillard--Varju, Eberhard, Ferber--Jain--Sah--Sawhney, and others), I will present a result, conditional on the extended Riemann hypothesis, showing that the characteristic polynomial of certain random tridiagonal matrices is irreducible, with probability tending to 1 as the size of the matrices tends to infinity. 

The proof involves random walks in direct products of \({\rm SL}_2(\mathbb{F}_p)\), where we use results of Breuillard--Gamburd and Golsefidy--Srinivas. 

Joint work with Lior Bary-Soroker and Sasha Sodin.

Some Shimura varieties are moduli spaces of Abelian varieties with extra structure.

The Tate module of a universal Abelian variety is a natural source of $\ell$-adic local systems on such Shimura varieties. Remarkably, the theory allows one to build these local systems intrinsically from the Shimura variety in an essentially tautological way, and this construction can be carried out in exactly the same way for Shimura varieties whose moduli interpretation remains conjectural.

This suggests the following program: Show that these tautological local systems "look as if" they were arising from the cohomology of geometric objects. In this talk, I will describe some recent progress. It is based on joint work with Kiran Kedlaya, Christian Klevdal, and Stefan Patrikis, as well as joint work with Yifei Zhang.

[pre-talk at 3pm]

TBA

The genealogy process is typically the most time-consuming part of -- and a limiting factor in the success of -- forensic investigative genetic genealogy, which is a new approach to solving violent crimes and identifying human remains. We formulate a stochastic dynamic program that -- given the list of matches and their genetic distances to the unknown target -- chooses the best decision at each point in time: which match to investigate (i.e., find its ancestors), which ancestors of these matches to descend from (i.e., find its descendants), or whether to terminate the investigation. The objective is to maximize the probability of finding the target minus a cost on the expected size of the final family tree. We estimate the  parameters of our model using data from 17 cases (eight solved, nine unsolved) from the DNA Doe Project. We assess the Proposed Strategy using simulated versions of the 17 DNA Doe Project cases, and compare it to a Benchmark Strategy that ranks matches by their genetic distance to the target and only descends from known common ancestors between a pair of matches. The Proposed Strategy solves cases 25-fold faster than the Benchmark Strategy, and does so by aggressively descending from a set of potential most recent common ancestors between the target and a match even when this set has a low probability of containing the correct most recent common ancestor.

This lecture is jointly sponsored by the UCSD Rady School of Management and the UCSD Mathematics Department.

The MPR2 conference room is just off Ridge Walk. It is on the same level as Ridge Walk. You will see the glass-walled MPR2 conference room on your left as you come into the Rady School area. 

FREE REGISTRATION REQUIRED: https://forms.gle/jv8nVFajV9mZ6U3v6 

The plate bending or Babuska paradox refers to the failure of convergence when a linear bending problem with simple support boundary conditions is approximated using polygonal domain approximations. We provide an explanation based on a variational viewpoint and identify sufficient conditions that avoid the paradox and which show that boundary conditions have to be suitably modified. We show that the paradox also matters in nonlinear thin-sheet folding problems and devise approximations that correctly converge to the original problem. The results are relevant for the construction of curved folding devices.

TBA

The probability that every vertex in a random orientation of the edges of a given graph has the same in-degree and out-degree is equivalent to counting Eulerian orientations, a problem that is known to be ♯P-hard in general. This count also appears under the name residual entropy in physical applications, most famously in the study of the behaviour of ice. Using a new tail bound for the cumulant expansion series, we derive an asymptotic formula for graphs of sufficient density. The formula contains the inverse square root of the number of spanning trees, for which we do not have a heuristic explanation. We will also show a strong experimental correlation between the number of spanning trees and the number of Eulerian orientations even for graphs of bounded degree. This leads us to propose a new heuristic for the number of Eulerian orientations which performs much better than previous heuristics for graphs of chemical interest. The talk is based on two recent papers arXiv:2309.15473 and arXiv:2409.04989 joint with B.D.McKay and R.-R. Zhang.

We survey some recent observations and ongoing work motivated by a hope to better understand p-adic K-theory. More specifically, we discuss arithmetic problems—and potential approaches—related to syntomic cohomology in positive and mixed characteristics. At the level of the structure sheaf, syntomic cohomology is an 'intelligent version' of p-adic étale Tate twists at the characteristic and (among other things) provides a motivic filtration on p-adic étale K-theory via the theory of trace invariants.

Solving linear systems is a crucial subroutine and challenge in the large-scale data setting. In this presentation, we introduce an iterative method for approximating the solution of large-scale multi-linear systems, represented in the form A*X=B under the tensor t-product. Unlike previously proposed randomized iterative strategies, such as the tensor randomized Kaczmarz method (row slice sketching) or the tensor Gauss-Seidel method (column slice sketching), which are natural extensions of their matrix counterparts, our approach delves into a distinct scenario utilizing frontal slice sketching. In particular, we explore a context where frontal slices, such as video frames, arrive sequentially over time, and access to only one frontal slice at any given moment is available. This talk will present our novel approach, shedding light on its applicability and potential benefits in approximating solutions to large-scale multi-linear systems.
 

Starting  with a transient irreducible diffusion process $X^0$ on a locally compact separable metric space $(D, d)$ (for example, absorbing Brownian motion in a snowflake domain), one can construct a canonical symmetric reflected diffusion process $\bar X$ on a completion $D^*$ of $(D, d)$ through the theory of  reflected Dirichlet spaces. The boundary trace process $\check X$ of $X$ on the boundary $\partial D:=D^*\setminus D$ is the reflected diffusion process $\bar X$ time-changed by a smooth measure $\nu$ having full quasi-support on $\partial D$. The Dirichlet form of the trace process $\check X$ is called the trace Dirichlet form. In this talk, I will address the following two fundamental questions:

1) How to characterize the boundary trace Dirichlet space in a concrete way?

2) How does the boundary trace process behave? 

Based on a joint work with Shiping Cao.

Professor David Hirshleifer
Robert G. Kirby Chair in Behavioral Finance
Professor of Finance and Business Economics
University of Southern California

Wednesday, May 7, 2025, 3.10-4.10 p.m.
Kavli Auditorium, Tata Hall, UC San Diego

Social Transmission Effects in Economics and Finance

Please register at 

https://docs.google.com/forms/d/e/1FAIpQLSdDNCEUSFHbiXxgJtNeWj8px-D8ftKXaiUGrW0cyRcsUTuc3A/viewform