Motivic degree 0 Donaldson-Thomas theory on a variety \(X\) is a point counting theory on the Hilbert scheme of points on \(X\) parametrizing zero-dimensionally supported quotient sheaves of \(\mathcal{O}_X\). On the other hand, the high rank DT theory is about the so called punctual Quot scheme parametrizing zero-dimensional quotient sheaves of the vector bundle \(\mathcal{O}_X^{\oplus r}\), while the unframed DT theory is about the stack of zero-dimensional coherent sheaves on \(X\). I will talk about some recent progresses on explicit computations of these theories for singular curves \(X\). For example, we found the exact count of \(n\times n\) matrix solutions \(AB=BA, A^2=B^3\) over a finite field (a problem corresponding to the motivic unframed DT theory for the curve \(y^2=x^3\)), and its generating function is a series appearing in the Rogers-Ramanujan identities. In other families of examples, it turns out that such computations discover new Rogers-Ramanujan type identities.

The theory of countable Borel equivalence relations analyzes the actions of countable groups on Polish spaces. The main question studied is how much information is encoded by the corresponding orbit space. The amount of encoded information reflects the extent to which the action is rigid.

In this talk we will discuss rigidity results in the theory of countable Borel equivalence relations. While the first rigidity results by Adams and Kechris use Zimmer's work, more recent results are based on newer cocycle superrigidity theorems, hinting at a deeper interplay than what we currently know. We will also discuss open questions and new directions in set theoretic rigidity.

Direct policy search has achieved great empirical success in reinforcement learning. Many recent studies have revisited its theoretical foundation for continuous control, which reveals elegant nonconvex geometry in various benchmark problems. In this talk, we introduce a new and unified Extended Convex Lifting (ECL) framework to reveal hidden convexity in classical optimal and robust control problems from a modern optimization perspective. Our ECL offers a bridge between nonconvex policy optimization and convex reformulations, enabling convex analysis for nonconvex problems. Despite non-convexity and non-smoothness, the existence of an ECL not only reveals that minimizing the original function is equivalent to a convex problem but also certifies a class of first-order non-degenerate stationary points to be globally optimal. Therefore, no spurious stationarity exists in the set of non-degenerate policies. We believe that the new ECL framework may be of independent interest for analyzing nonconvex problems beyond control. This talk is based on our recent work: arxiv.org/abs/2312.15332, and arxiv.org/abs/2406.04001.

Yang Zheng is an assistant professor in the ECE department at UC San Diego. Yang Zheng received his DPhil (Ph.D.) degree in Engineering Science from the University of Oxford in 2019. He received the B.E. and M.S. degrees from Tsinghua University in 2013 and 2015, respectively.  He was a research associate at Imperial College London and was a postdoctoral scholar in SEAS and CGBC at Harvard University. His research interests include learning, optimization, and control of network systems, and their applications to autonomous vehicles and traffic systems. Dr. Zheng received the 2019 European Ph.D. Award on Control for Complex and Heterogeneous Systems, and the 2022 Best Paper Award in the IEEE Transactions on Control of Network Systems. He was also a recipient of the National Scholarship, Outstanding Graduate at Tsinghua University, and the Clarendon Scholarship at the University of Oxford. Dr. Zheng also won an NSF CAREER Award in 2024, and the 2023 Best Graduate Teacher Award from the ECE department at UC San Diego.

TBA

Let $p$ be a prime number and $ Q_p$  the field  of $p$-adic numbers.

The representations of  a cousin of the Galois group of an algebraic closure of $ Q_p$ are related (the {\bf Langlands's bridge}) to the representations of reductive $p$-adic groups, for instance $SL_2(Q_p),  GL_n(Q_p) $.   The irreducible representations $\pi$ of reductive $p$-adic groups are  easier  to study than those of the Galois groups but they are rarely finite dimensional. Their classification is very involved but their behaviour  around the identity, that we call the ``asymptotics'' of $\pi$, are expected to be more uniform. We shall survey what is known  (joint work with Guy Henniart), and what it suggests.

In this talk I will describe some properties of Meinhardt's model of sidebranching. This is a four-species reaction-diffusion model dating from 1976 describing the interaction of four fields, the concentrations of an activator, an inhibitor, the substrate, and a marker for differentiation. The model exhibits rich dynamics that are absent from simpler RD systems. I will describe two of these: propagation failure of differentiation fronts and, in a different parameter regime, the presence of intermittent spiking. The former is traced to the presence of so-called T-points in parameter space. The latter is characterized by a Poisson probability distribution function of interspike intervals, indicating that the spiking process is memoryless. The role of a (subcritical) Turing instability in generating (unstable) spikes will be emphasized.

This is joint work with Arik Yochelis, Ben-Gurion University, Be'er Sheva, Israel.

Gaussian elimination (GE) remains one of the most used dense linear solvers. In the error analysis of GE with selected pivoting strategies on well-conditioned systems, the analysis can be reduced to studying the behavior of the growth factor, which represents the largest entry encountered at each step of the elimination process. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided very recently by Huang and Tikhomirov’s average-case analysis of GEPP, which showed GEPP growth stays at most polynomial with very high probability when using Gaussian matrices. Research on GE with complete pivoting (GECP) has also seen a lot of recent interest, with improvements to lower and upper bounds on worst-case GECP growth provided by Bisain, Edelman, and Urschel in the last year. I am interested in studying how GEPP and GECP behave on the same linear systems, with a focus on large growth systems and orthogonal matrices. One direction will explore when GECP is less stable than GEPP, which will lead to new empirical lower bounds on how much worse GECP can behave compared to GEPP in terms of growth. Another direction will include an empirical study on a family of exponential GEPP growth matrices whose polynomial behavior in small neighborhoods limits to the initial GECP growth factor.

I will give a combination math-history-show-and-tell talk in which I explain the theory and background of subtractive synthesis, an approach to making sounds via electronics. Although not technically the first type of synthesis implemented successfully in an electronic instrument, subtractive synthesis is by far the most popular approach to the electronic creation of sounds for music. We will go over the rudiments of subtractive synthesis, briefly cover the history of the original Moog synthesizer, and along the way I will showcase these ideas using the Moog Mavis, a modern, smaller instrument from the same company based on the same principles.

In this talk, I will describe how to evaluate the virtual count of maps from a fixed-domain smooth curve to a hypersurface in a Grassmannian. We use the Quot scheme to compactify the space of maps and perform a virtual intersection-theoretic calculation. I will also discuss the conditions under which the virtual count is enumerative. This talk is based on joint work with Alina Marian.