After offering some quick answers to the questions "What is learning?" and "What is a quantum computer?", I'll explain how you can improve your learning with a quantum computer. We measure the efficiency of a learning algorithm by its query complexity, and in this field one tries to find upper bounds for the query complexity (by creating algorithms) as well as lower bounds (by proving the optimality of certain algorithms). Many of the first quantum algorithms are in fact learning algorithms, and we'll discuss two important ones: Grover's search algorithm and the Bernstein-Vazirani algorithm. These offer amazing speedups in query complexity over classical computers. If time permits, I'll describe recent research which introduces a huge family of learning problems with nonabelian symmetries for which little is known but the (upper and lower bounds for the) query complexity for any given problem can be easily computed on computer algebra software such as GAP or SAGE. This work is in collaboration with Orest Bucicovschi, Hanspeter Kraft, David Meyer and Jamie Pommersheim.

In this talk, I focus on factor analysis of high-dimensional time series
data, in which the dimension of data is allowed to be even larger than the
length of data. Analysis of high-dimensional data suffers from the curse
of dimensionality. Factor analysis is considered as an effective way for
dimension reduction. Factor models presume that a few common factors can
explain most of the variation/dynamics of an observed process in high
dimensions. In the models, factor loadings are introduced to reflect the
percentages of variations explained and contributions made by these common
factors. Based on real data analysis, it has been discovered that the
loadings may vary in different situations/regimes. To interpret this
observation and capture the regime-switching mechanism often encountered
in practice, we propose a threshold factor model for high-dimensional time
series data, in which a threshold variable is introduced to distinguish
different regimes. Loadings controlled by the threshold variable vary
across regimes. The theoretical properties of the procedure are
investigated.

The affine sieve is a technique first developed by Bourgain, Gamburd, and Sarnak in 2006 and later completed by Salehi Golsefidy and Sarnak in 2010 to study almost-primality in a broad class of affine linear actions. The beauty of this is that it gives us effective bounds on the saturation number for thin orbits coming from $GL_n$ - in particular, producing infinitely many $R$-almost primes for some $R$. However, in practice this value of $R$ is often far from optimal. The case of thin Pythagorean triangles has been of particular interest since the outset of the affine sieve, and I will discuss recent progress on improving bounds for the saturation numbers for their hypotenuses and areas using Archimedean sieve theory.

We give a geometric characterization of mean ergodic convergence in the Calkin algebras for certain Banach spaces. (Joint work with William B. Johnson)

Shrinking gradient Ricci solitons (shrinkers) are models for the local geometry of singular regions of solutions to the Ricci flow and their classification is critical to the understanding of singularity formation under the flow. Growing evidence suggests that the asymptotic geometry of complete noncompact shrinkers may be particularly constrained; in fact, all examples currently known which do not split locally as products are smoothly asymptotic to a regular cone at infinity. I will present some results from a joint project with Lu Wang, in which we study the uniqueness of shrinkers asymptotic to such structures as a problem of parabolic unique continuation, and discuss the applications of these results to a conjectured classification in four dimensions.

We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$
over a finite field of characteristic $p$, and consider the
$\mathbb{Z}_{p^{\ell}}$-Artin--Schreier--Witt tower defined by $\bar
f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to
\cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically
isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified
extension of $\mathbb{Z}_p$, which is abstractly isomorphic to
$(\mathbb{Z}_p)^\ell$ as a topological group.
We study the Newton slopes of zeta functions of this tower of curves.
This reduces to the study of the Newton slopes of L-functions
associated to characters of the Galois group of this tower. We prove
that, when the conductor of the character is large enough, the Newton
slopes of the L-function
asymptotically form a finite union of arithmetic progressions. As a
corollary, we prove the spectral halo property of the spectral variety
associated to the $\mathbb{Z}_{p^{\ell}}$-Artin--Schreier--Witt tower.
This extends the main result of Davis--Wan--Xiao from rank one
case $\ell=1$ to the higher rank case $\ell\geq 1$.

Motivated by Kolmogorov's theory of hydrodynamic turbulence, we considerdissipative weak solutions to the 3D incompressible Euler equations and the 2D surface quasi-geostrophic equations. We prove that up to a certain regularity threshold weak solutions are not unique. In the case of the Euler system this is the threshold determined by the Onsager conjecture.
For SQG, this answers an open problem posed by De Lellis and Szekelyhidi Jr.

Following ideas of Marian and Oprea, finite quot schemes can be
used to investigate Le Potier's strange duality conjecture for surfaces. I
will discuss recent work with Aaron Bertram and Drew Johnson in which we
prove the existence of a large class of finite quot schemes on the
projective plane. We use nice resolutions of general stable vector
bundles, which also yield an easy proof that these bundles are globally
generated whenever their Euler characteristic suggests that they should
be.