Past Events
In a disordered potential, eigenfunctions of the Schrödinger operator localize in seemingly unpredictable places. Recently, Filoche and Mayboroda discovered an effective potential which magically governs the structure of eigenfunctions. In this talk, we'll examine the predictive power…
Given a homogeneous ideal I in a polynomial ring, one may apply the following combinatorial operation: for each degree d, make a list of all subsets S of the set of degree-d monomials such that S is the set of nonzero coefficients of an element of I. For each d, this set of subsets is a…
A degree d>1 self-map f of P^n is called post critically finite (PCF) if its critical hypersurface C_f is pre-periodic for f, that is, if there exist integers r ≥ 0 and k>0 such that f^{r+k}(C_f) is contained in f^{r}(C_f).
I will discuss the question: what does the locus of…
I'll describe recent work with Mikhail Karpukhin, in which we relate the problem of maximizing Laplacian eigenvalues over unit-area metrics on a given Riemann surface to natural variational constructions of harmonic maps to high-dimensional spheres. Our results give a new proof of the existence…
We will take a journey to a strange and distant land (called France) where "cohomology" is pronounced "derived pushforward of the constant sheaf to a point". We will make a bunch of definitions and, إن شاء الله, prove something.
In this talk, we will continue our discussion of Yang-Mills theory and provide some perspective on the usage of Lagrangians in solving the Yang-Mills equation. In addition, we will explore how the Yang-Mills field is a generalization of the electromagnetic-field which can be applied in other…
The Fourier transform associates a polynomial to each linear differential operator with constant coefficients, and a formal calculation shows that elements in the kernel of such a differential operator have their Fourier transforms supported on the vanishing set of that polynomial. For…
A knot K in S^3 is slice if it is the cross section of an embedded sphere in S^4, and it is doubly slice if the sphere is unknotted. Although slice knots are very well studied, doubly slice knots have been given comparatively less attention. We prove that an odd pretzel knot is doubly slice if…
We consider the random Cayley graph of a finite group $G$ formed by picking $k$ random generators uniformly at random:
- We prove universality of cutoff for the random walk, provided $k-d(G) >> 1$ where $d(G)$ is the size of the smallest generating set of $G$. As conjectured…