The cohomology groups of tautological bundles on Grassmannians are described by the celebrated Borel-Weil-Bott theorem. Quot schemes on the projective line provide a natural generalization of Grassmannians: they parametrize rank r quotients of a vector bundle V on the projective line. In this talk, I will present formulas for Euler characteristics and for the cohomology groups of tautological bundles on these Quot schemes. Additionally, I will describe how these formulas apply to the study of the quantum K-theory of Grassmannians.

Countable Similarity Structure (CSS) groups are a class of generalized Thompson groups. I will introduce CSS$^*$ groups, a subclass, that we prove to be non-acylindrically hyperbolic, that includes the Higman-Thompson groups $V_{d,r}$, the countable R\"over-Nekrashevych groups $V_d(G)$, and the topological full groups of subshifts of finite type of Matui. I will discuss how all CSS$^*$ groups give rise to prime group von Neumann algebras, which greatly expands the class of groups satisfying a previous deformation/rigidity result. I will then discuss how CSS$^*$ groups are either C$^*$-simple with a simple commutator subgroup, or lack both properties. This extends C$^*$-simplicity results of Le Boudec and Matte Bon and recovers the simple commutator subgroup results of Bleak, Elliott, and Hyde. This is joint work with Eli Bashwinger.

This talk is devoted to combining model predictive control (MPC) and deep learning methods, specifically neural networks, to solve high-dimensional optimization and control problems. MPC is a popular method for real-life process control in various fields, but its computational requirements can often become a bottleneck. In contrast, deep learning algorithms have shown effectiveness in approximating high-dimensional systems and solving reinforcement learning problems. By leveraging the strengths of both MPC and neural networks, we aim to improve the efficiency of solving MPC problems. The talk also discusses the optimal control problem in MPC and how it can be divided into smaller time horizons to reduce computational costs. Additionally, we focus on enhancing MPC through two approaches: a machine learning–based feedback controller and a machine learning–enhanced planner, which involve implementing neural networks and iLQR. Overall, this talk provides insights into the potential of combining MPC and deep learning methods to tackle complex control problems across various fields, with applications to robotics.

The burning number of a graph is the minimal number of steps that are needed to burn all of its vertices, with the following procedure: at each step, one can choose a point to set on fire, and the fire propagates constantly at unit speed along the edges of the graph. In joint work with Alice Contat, we consider two natural random burning procedures in the discrete Euclidean torus with side-length n, in which the points that we set on fire at each step are random variables. Our main result deals with the case where at each step, the law of the new point that we set on fire conditionally on the past is the uniform distribution on the complement of the set of vertices burned by the previous points. In this case, we prove that as n tends to infinity, the corresponding random burning number (i.e, the first step at which the whole torus is burned) is asymptotic to T times n^{d/(d+1)} in probability, where T is the explosion time of a so-called generalised Blasius equation.

Consider min f(x) s.t. x>=0, where the objective function f: R→ R is smooth, and the variable is required to be nonnegative. A naive "squared variable" technique reformulates the problem to min_v f(v^2). Note that the new problem is now unconstrained, and many algorithms, e.g., gradient descent, can be applied. In this talk, we discuss the disadvantages of this approach, which have been known for decades, and the possible surprising fact of equivalence for the two problems in terms of (i) local minimizers and (ii) points satisfying the so-called second-order optimality conditions, which are keys for designing optimization algorithms. We further discuss extensions of the approach and equivalence to the vector case (where the vector variable is required to have all entries nonnegative) and the matrix case (where the matrix variable is required to be a positive semidefinite).

It is conjectured that many models of statistical mechanics have a rich, fractal-like behaviour at and near their points of phase transition, with power-law scaling governed by critical exponents that are expected to depend on the dimension but not on the small-scale details of the model such as the choice of lattice. This is now reasonably well understood in two dimensions and in high dimensions, but remains poorly understood in intermediate dimensions (e.g. d=3). I will overview the conjectures around this area and describe recent progress on related problems for models with long-range interactions.

Motivated by the Brill--Noether problems and enumerative geometry over surfaces, we study the expected behaviors of coherent sheaves. We estimate the dimension of global sections of stable sheaves. We also prove some cases of an analogue of Lange's conjecture over curves, which states that general extensions of two vector bundles are stable under some obvious conditions. These are closely related to Segre invariants of sheaves, which studies maximal subsheaves of a fixed rank. This can be understood as to determine when Grothendieck's Quot schemes are non-empty. This is based on work in progress jointly with Thomas Goller and Zhixian Zhu.