Kinetoplast DNA, often described as molecular chainmail, is found in the mitochondria of trypanosome parasites and consists of thousands of topologically interlocked circular molecules. In addition to its biological role in gene editing, it has been explored recently as a model system for materials science, due to its unique topological connectivity and its two-dimensional structure. In this talk, I will discuss some mathematical investigations that have emerged out of materials-based research of kinetoplast DNA, including the relationship between the link topology of the network and the Gaussian curvature of chainmail membranes, as well as methods to detect Borromean linking within densely linked networks.

In recent years, tensor networks have emerged as a powerful low-rank approximation framework for addressing exponentially large data science problems without requiring exponential computational resources. In this talk, we demonstrate how tensor networks, when combined with accelerations from randomized numerical linear algebra (rNLA), can enable the efficient representation and manipulation of large-scale, complex datasets originating from quantum physics, high-dimensional function approximation, and neural network compression. We will start by describing how to construct a tensor network directly from input data. Building on this foundation, we then describe a new randomized algorithm called Successive Randomized Compression (SRC) that asymptotically accelerates the tensor network analog of matrix-vector multiplication using the randomized singular value decomposition. As a demonstration, we present examples showing how tensor network based simulations of quantum dynamics in 2^100 dimensions can be performed on a personal laptop.

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.

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 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.

Nonlinearly Partitioned Runge--Kutta (NPRK) methods are a newly proposed class of time integration schemes which target differential equations in which different scales, stiffnesses or physics are coupled in a nonlinear way. In this talk, I will provide a broad overview of this new class of methods. First, I will motivate these methods as a nonlinear generalization of classical Runge--Kutta (RK) and Additive Runge--Kutta (ARK) methods. Subsequently, I will discuss order conditions for NPRK methods; we obtain the complete order conditions using an edge-colored rooted tree framework. Interestingly, NPRK methods have nonlinear order conditions which have no classical additive counterpart. We will show how these nonlinear order conditions can be used to obtain embedded estimates of state-dependent nonlinear coupling strength and present a numerical example to demonstrate these embedded estimates. I will then discuss how these methods yield efficient semi-implicit time integration of numerical partial differential equations; numerical examples from radiation hydrodynamics will be presented. Finally, I will discuss our recent work on multirate NPRK methods, which target problems with nonlinearly coupled processes occurring on different timescales. We will discuss properties of these multirate methods such as timescale coupling, stability and efficiency, and conclude with several numerical examples, such as a fast-reaction viscous Burgers’ equation and the thermal radiation diffusion equations.

TBA

I will discuss some recent results about relative entropies in QFT, with particular emphasis on the singular limits of such entropies.

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.

In his seminal paper, Ozawa demonstrated the solidity property for ${\rm II}_1$ factor arising from Biexact groups. In this talk, I will discuss a relative version of the solidity property for biexact groups in the setting of measure equivalence and its applications to measure equivalence rigidity. This is a joint work with Daniel Drimbe.

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.

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