We introduce a Swarm-Based Random Descent (SBRD) method for non-convex optimization. The swarm consists of agents, identified with positions, x, and masses, m. There are three key aspects to the SBRD dynamics: (i) persistent transition of mass from high to lower ground; (ii) marching along the gradient descent: an m-dependent random choice of marching direction…
This talk will introduce cluster algebras, with an emphasis on their combinatorics, and describe a recent joint result with Vincent Pilaud and Sibylle Schroll. At the heart of a cluster algebra is a complicated, branching recursion that defines cluster variables (certain rational functions organized into finite sets called clusters). The recursion looks bizarre at first…
In this talk we present some recent result about the long time behavior of the solutions for some diffusion processes on a metric graph. We study evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions of the heat…
Chromatin structure tightly regulates gene expression and epigenetic processes. The nuclear environment is complex, featuring tension exerted by force-generating proteins and molecular crowding modulated by different ionic concentrations. Understanding the impact of these factors on chromatin structure is crucial for elucidating the molecular mechanisms of chromatin accessibility and organization. However, it is unclear how chromatin…
It is common to assume that the data was sampled from a low-dimensional manifold in a high-dimensional space. In real life, neither the dimension of this manifold nor its geometry is known, and the data is often contaminated with noise and outliers. In this talk, we first present a method for denoising and reconstructing a…
In traditional Euclidean geometry, points serve as the foundational elements for constructing and analyzing space. In contrast, Laguerre geometry, a non-Euclidean geometry, uses oriented circles (or hyperspheres, in the context of higher dimensions) and oriented lines (or hyperplanes), as fundamental objects. Here, a “point” is simply a circle with radius zero, i.e. having no special…
Quantum cellular automata (QCA) are models of discrete-time unitary dynamics of quantum spin systems. They can be characterized algebraically as certain automorphisms of the associative algebra generated by local observables of a spin system. We will give a gentle introduction to this topic, and explain some of our recent contributions to the problem of classification…
The talk is about oscillatory integral operators on manifolds. Manifolds of constant sectional curvatures are particularly interesting, and we will see that very good estimates on these manifolds can be expected. We will also discuss Kakeya and Nikodym problems on general manifolds, in particular, manifolds satisfying Sogge’s chaotic curvatures.
In this interactive seminar, we will examine how data (qualitative and quantitative) can be leveraged to interrogate, disrupt, and enact changes in introductory math programs. In particular, I will share insights from the ACT UP Math project, which is studying the role and impact of research-practice partnerships between mathematics education experts and mathematics department faculty…
Khovanov homology is a link invariant which categorifies Jones polynomial. In this talk we present several results concerning Khovanov homology of fibered positive links; in particular, we extend the result by Stosic stating that braid positive links have vanishing Khovanov homology in homological grading 1. We also explore Khovanov homology of certain cable links and…
The formulation of Bayesian inverse problems involves choosing prior distributions; choices that seem equally reasonable may lead to significantly different conclusions. We develop a computational approach to better understand the impact of the hyperparameters defining the prior on the posterior statistics of the quantities of interest. Our approach relies on global sensitivity analysis (GSA) of…
Permutations $w$ in $S_n$ for which the (type-A) Schubert variety $\Omega_w$ is smooth are characterized by avoidance of the patterns 3412 and 4231. The smaller family of codominant permutations, those avoiding the pattern 312, seems to explain a lot about character evaluations at Kazhdan-Lusztig basis elements $C'_w(q)$ of the (type-A) Hecke algebra. In particular, for…
In this talk, I will talk about some recent well-posedness and stability results for several fluid models in 2D. More precisely, I will discuss the global well-posedness for the 2D Boussinesq equations with fractional dissipation. For the Oldroyd-B model, we show that small smooth data lead to global and stable solutions. When Navier-Stokes is coupled…
In volume-constrained capillarity problems, minimizers may have free boundaries adhering to the container. Recent work in the study of capillarity problems has utilized stability theory for the volume-constrained isoperimetric problem to classify the shape of global minimizers and (in the case without free boundary) critical points. In this talk, I will discuss joint work with…
In this talk, we discuss second-order optimality conditions for optimal control problems. This analysis is very important when we study the stability of the solution to the control problem with respect to small perturbations of the data. It is also crucial for proving superlinear or quadratic convergence of numerical algorithms for solving the problem, as…
Over the past three decades, a fruitful approach for analysis and data-driven modeling of dynamical systems has been to consider the action of (nonlinear) dynamics in state space on linear spaces of observables. These methods leverage the linearity of the associated evolution operators, namely the Koopman and transfer operators, to carry out tasks such as…