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We consider the binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory, and probability theory communities. We show that at low constraint density (m=n^{1-epsilon}), the model exhibits a strong freezing…

Fintushel-Stern's knot surgery construction on elliptic surfaces has been a central source of exotic, smooth four-manifolds since its introduction in the 1990's. The construction associates a homotopy elliptic surface to a classical knot. These homotopy elliptic surfaces are non-diffeomorphic if…

This seminar will go into some detail into a proof of the central limit theorem for the values of the Estermann function D(x) at rationals x ordered by denominators, which we worked out with Sandro Bettin (Genova) in 2018. Here D(x) is the values at s=1/2 of the analytic continuation of the…

In this talk, we will investigate the emergence of geometric patterns in well-trained deep learning models by making use of a layer-peeled model and the law of equi-separation. The former is a nonconvex optimization program that models the last-layer features and weights. We use the model to…

In this talk, we consider the strict convexity of energy functions

of harmonic maps at its critical points from…

The "Sierpinski triangle graph" (based on a fractal that Mandelbrot liked to call the "Sierpinski gasket") and the analogous "Sierpinski tetrahedron graph" are well known. They have a natural generalization to simplexes of any dimension. Several elementary properties are easily proved, and…

A meandric system of size $n$ is a non-intersecting collection of closed loops in the plane crossing the real line in exactly $2n$ points (up to continuous deformation). Connected meandric systems are called meanders, and their enumeration is a notoriously hard problem in enumerative…