Let $K$ be a number field. Beginning in the 1970s, Mazur's "Program B" kicked off efforts to classify the $K$-rational points on all modular curves $X_H$, as $H$ ranges through open subgroups of $\text{GL}_2(\hat{\mathbb Z})$. Fifty years later, it remains a very active field of research in arithmetic geometry: even as late as 2017, the determination of the rational points on a single "cursed curve" was heralded a breakthrough in the subject. In this talk, we will outline a possible approach to settle Mazur's Program B completely. The inputs required are (1) a resolution to Serre's uniformity question in full generality, and (2) an algorithm to obtain rational points on any modular curve of genus at least 2. For (1), we discuss a possible approach via Borcherds products, and for (2), we discuss equationless approaches to quadratic and motivic Chabauty algorithms, following the respective recent work of Balakrishnan-Dogra-Muller-Tuitman-Vonk and Corwin.

We will give a taste of the flavors of math that constitute the study of random walks on compact groups, followed by which we will describe the author's work with Prof. Golsefidy in solving a question of Lindenstrauss and Varju. Namely, can the spectral gap of a random walk on a product of groups be related to those of the projections onto its factors.

I will talk about recent joint work with Nansen Petrosyan where we studied the behavior of $L^2$-Betti Numbers under group-theoretic Dehn filling, a quotienting process of groups motivated by 3-manifold theory. As applications, we verified the Singer Conjecture for Einstein manifolds constructed from arithmetic lattices of $SO(n, 1)$. Another application appears in my collaboration with Francesco Fournier-Facio where we constructed the first uncountable family of finitely generated torsion-free groups which are mutually non-measure equivalent.

The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function in the presence of the drift term. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains and discuss how the introduction of the transport term influences the regularity of the solutions.

The goal of quantized compressed sensing (QCS) is to recover structured signals from quantized measurements. The performance bounds of hamming distance minimization (HDM) were well established and known to be optimal in recovering sparse signals, but HDM is in general computationally infeasible. In this talk, we propose an efficient projected gradient descent (PGD) algorithm for QCS which generalizes normalized binary iterative hard thresholding (NBIHT) in one-bit compressed sensing for sparse vectors.  Under sub-Gaussian design, we identify the conditions under which PGD achieves essentially the same error rates as HDM, up to logarithmic factors. These conditions are easy to validate and include estimates of the separation probability, a small-ball probability and some moments. We specialize the general framework to several popular memoryless QCS models and show that PGD achieves the optimal rate O(k/m) in recovering sparse vectors, and the best-known rate O((k/m)^{1/3}) in recovering effectively sparse signals. This is joint work with Ming Yuan. An initial version is available in https://arxiv.org/abs/2407.04951. 

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.

My research is mostly about exploring how symmetries can be used to generate randomness or unveil structural insights. In this talk, I will focus on random walks on compact groups, and give you a glimpse of some of the tools that I use to study such a random process:

• Connection with expander graphs,
• Property (T),
• Growth within algebraic structures: sum-product and product results,
• Entropy and the Bourgain-Gamburd technique.

At the end, I will mention more recent results of Srinivas and mine on random walks on group extensions.

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.

In this talk, we will present a new software application for publishing interactive math content online. It works like Overleaf, where you type text, LaTeX, and more in your browser, but instead of a PDF it produces a live, interactive website.  This app has now been tested in several math courses at UCSD, and we hope it can support your teaching as well!