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Weitzenböck Formula; analytic properties of Dirac operators; functional calculus; Hodge theory overview

Fourier analysis is an important tool in additive combinatorics, famously used by Roth to find 3-term arithmetic progressions in dense subsets of the integers. One of the key properties used in his proof is that a Fourier-uniform set A consisting…

An overview of more mathematical tools in harmonic analysis

We present a proof of the Gromov non-squeezing theorem following the scheme of Gromov’s original proof, with a more modern perspective on some of the techniques involved.

 Let p be a prime. I want to explain how to use the geometry of modular curves at infinite level and the Hodge–Tate period map to study regular de Rham p-adic Galois representations appearing in the p-adically completed cohomology of modular curves. We will show that these Galois representations…

We study the capacity of the range of a simple random walk in three and higher dimensions. It is known that the order of the capacity of the random walk range in n dimensions is similar to that of the volume of the random walk range in n-2 dimensions. We show that this correspondence breaks down…

Abstract

We will discuss the following version of the homological Arnold conjecture: for any closed symplectic manifold, the number of 1-periodic orbits of a non-degenerate Hamiltonian is bounded from below by a version of total Betti number over Z which takes account of torsions of all characteristics.…

Abstract

Abstract