Georgia Institute of Technology College of Sciences

The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.

A $k \times n$ partial Latin rectangle is \textit{$C$-sparse} if the number of nonempty entries in each row and column is at most $C$ and each symbol is used at most $C$ times. We prove that the probability a uniformly random $k \times n$ Latin rectangle, where $k < (1/2 - \alpha)n$, contains a $\beta n$-sparse partial Latin rectangle with $\ell$ nonempty entries is $(\frac{1 \pm \varepsilon}{n})^\ell$ for sufficiently large $n$ and sufficiently small $\beta$. Using this result, we prove that a uniformly random order-$n$ Latin square asymptotically almost surely has no Latin subsquare of order greater than $c\sqrt{n\log n}$ for an absolute constant $c$. This is joint work with Tom Kelly, Camille Kennedy, and Jasdeep Sidhu.

How complicated can successive manifolds get in a tower of covering
spaces? Specifically, how large can the dimension of the first
cohomology get? We will begin with a tour of possible behaviors for
low-dimensional spaces, and then focus on arithmetic manifolds.
Specifically, for towers of complex-hyperbolic manifolds, I will
describe how to bound the rates of growth using known instances of
Langlands functoriality.

In her thesis, Maryam Mirzakhani counted the number of simple closed geodesics of bounded length on a (real) hyperbolic surface. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll discuss some of these connections and a qualitative strengthening of her theorem, describing what these curves, and their complements, actually (generically) look like. This is joint work with Francisco Arana-Herrera.