Polyhedral fans are geometric objects, which arise naturally in many areas of mathematics, for example in toric geometry, the theory of hyperplane arrangements and representation theory. In many cases, there…
We will discuss a special family of 2D incompressible inviscid fluid flows in the form of logarithmic spiral vortex sheets. Such flows are determined by a vorticity distribution of a curve R^2,…
Spreading (diffusion) of new products is a classical problem. Traditionally, it has been analyzed using the compartmental Bass model, which implicitly assumes that all individuals are homogeneous and connected to…
We study Merton’s expected utility maximization problem in an incomplete market, characterized by a factor process in addition to the stock price process, where all the model primitives are unknown.…
Reduced-order models (ROMs) are a critical tool for sensitivity analysis, parameter inference, and uncertainty quantification where high-fidelity models would be computationally intractable. Galerkin POD-ROMs are one particular class of ROMs…
Gene co-expression graphs are a rich source of information, revealing critical insights into cellular functions, states, and activities. Yet, extracting meaningful signals from these graphs presents a formidable challenge. This…
In this talk we discuss partial data inverse boundary problems for magnetic Schrödinger operators on bounded domains in the Euclidean space as well as some Riemannian manifolds with boundary. In…
The pop-stack sorting method takes an ordered list or permutation and reverses each descending run without changing their relative positions. In this talk we will review recent combinatorial results on…
I first give a quick introduction to front propagations, Hamilton-Jacobi equations, level-set forced mean curvature flows, and homogenization theory. I will then show the optimal rates of convergence for homogenization of…
In the past decade, deep learning has made astonishing breakthroughs in various real-world applications. It is a common belief that deep neural networks are good at learning various geometric structures…
In this talk, I will go through some old and new results concerning the rigidity and flexibility of scalar curvature.
Mean field game theory was developed to analyze Nash games with large numbers of players in the continuum limit. The master equation, which can be seen as the limit of an…
We consider a class of disordered mean-field combinatorial optimization problems, focusing on the Gibbs measure, where the inverse temperature does not vary with the size of the graph and the…
In wound healing applications, platelet-like particles (PLPs) are engineered biomaterials that aim to mimic the behavior of natural platelets. Platelets play an essential role in the successful formation of the…
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The famous knot complement theorem of Gordon and Luecke states that two knots in the three-sphere are equivalent if and only if the complements are homeomorphic. This was proved more…
In this talk, I will discuss the evolution of rigid bodies in a perfect incompressible fluid, and the limit systems that can be obtained as the bodies shrink to points. The…
Spatio-temporal dynamical systems, such as fluid flows or vibrating structures, are prevalent across various applications, from enhancing user comfort and reducing noise in HVAC systems to improving cooling efficiency in…