Title: Finite approximations as a tool for studying triangulated categories.Abstract: A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll start with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.And […]
Title: Kähler-Einstein metric, K-stability and moduli of Fano varieties I. Abstract: A complex variety with a positive first Chern class is called a Fano variety. The question of whether a Fano variety has a Kähler-Einstein metric has been a major topic in complex geometry since the 1980s. The Yau-Tian-Donaldson Conjecture predicts the existence of such […]
Title and abstract TBA
Title: Kähler-Einstein metric, K-stability and moduli of Fano varieties II. Abstract: Algebraic varieties are given by solutions of a system of polynomial equations. Constructing parameter space for them has roots in many different fields, e.g. number theory, complex analysis, mathematical physics etc. It is often related to stability notions. The first framework to construct moduli […]
Title: First eigenvalue estimates on asymptotically hyperbolic manifolds and their submanifolds. Abstract: I will report on joint work with Samuel Pérez-Ayala. We derive a sharp upper bound for the first eigenvalue $lambda_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<infty$. We then prove that a particular class of conformally compact submanifolds within asymptotically hyperbolic […]
Title: Six functor formalism and the !-topology
Title: The simply typed 2λ-calculus Abstract: 2λ-calculus is a language with three levels: types, terms, and rewrites. I will explore their properties and give categorical description of the equational theory on the rewrites of the 2λ-calculus
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From Stochastic to Deterministic: SGD dynamics of non-convex models in high dimensionsStochastic gradient descent (SGD) stands as a cornerstone of optimization and modern machine learning. However, understanding why SGD performs so well remains a major challenge. In this talk, I will present a theory for SGD in high dimensions when the number of samples and […]