Title: Hilbert's Third Problem and the Dehn Invariant Abstract: Of Hilbert's 23 problems posed in 1900 the third was resolved first a year later by Max Dehn. It was settled by Dehn's construction of an algebraic invariant that detects when … Read more
Part 3: Spectral Asymptotics and the Heat Kernel. Abstract: In this lecture series, our goal will be to prove a result known as Weyl’s law, which tells us how the Laplace eigenvalues of a compact manifold are distributed. Since eigenvalues are inherently … Read more
Part 4: Spectral Asymptotics and the Heat Kernel. Abstract: In this lecture series, our goal will be to prove a result known as Weyl’s law, which tells us how the Laplace eigenvalues of a compact manifold are distributed. Since eigenvalues are inherently … Read more
Title: Coarse-grained biological simulations: Representing biological systems using molecular dynamics Abstract: At the intersection of biology and data science, there is a need for simulations of small-scale biological processes. Even as imaging technology continues to improve, biologists still look to … Read more
Title: Optimal Transport and Discrete/Continuum Laplacians Abstract: In the past 20 years, spectral methods have shown to be increasingly effective at a wide variety of data analytic problems. These spectral methods usually involve a graph Laplacian constructed from the data … Read more
Title: Optimal Transport and Discrete/Continuum Laplacians Abstract: In the past 20 years, spectral methods have shown to be increasingly effective at a wide variety of data analytic problems. These spectral methods usually involve a graph Laplacian constructed from the data … Read more
Title: Homology: Redux Abstract: Topological Data Analysis (TDA) is a relatively recent area of applied topology, in which tools from algebraic topology are used to understand the topology of data sets. The TDA pipeline is, in short, the following: a simplicial complex … Read more
Title: Bundle up, it's cold outside! Abstract: Vector bundles are an important part of modern math, as they provide us with valuable geometric information and allow us to name natural classes in the cohomology of the space over which they live. In … Read more
Title: Data-driven Mesh Optimization for Staggered-grid Lagrangian Hydrodynamics Abstract: The Euler equations are used in many industrial applications to model gas dynamics, but face a multitude of numerical issues. The foremost of which is the development of shock-waves, at which finite difference … Read more
Title: Emergent Clustering from Pairwise Interactions in Polymer Genome Models Abstract: The physical organization of chromosomes is known to be vital to cell function. Highly efficient compaction is achieved through various means, including histones, loop extrusion, and crosslinking. Prior work … Read more
Title: Symplectic Toric Orbifolds Abstract: Toric varieties are a special class of algebraic T-varieties whose geometry is encoded in convex polytopes. In addition to being interesting in its own right, toric geometry often serves as a source simple examples where … Read more
Title: Symmetric Functions Abstract: The most basic type of symmetric functions appears when studying roots of a polynomial: the coefficients of a polynomial are the elementary symmetric functions of its roots. Much of mathematics was motivated by trying to write down formulas … Read more
