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…
The Fisher-KPP equation was introduced in 1937 to model the spread of an advantageous gene through a spatially distributed population. Remarkably precise information on the traveling front has been obtained…
Determining the sensitivity of model outputs to input parameters is an important precursor to developing informative parameter studies, building surrogate models, and performing rigorous uncertainty quantification. A prominent class of…
Minimal surfaces are fundamental geometric objects which have been studied intensively since the 1700's. Classes of minimal surfaces of particular interest are the complete embedded ones in Euclidean space, closed…
Determinantal systems are systems of polynomial equations which encode a rank deficiency of a given matrix with polynomial entries over the solution set to other polynomial equations. Such systems arise…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types…