We'll discuss the incompressible Navier-Stokes equations.

Extracting useful information from high-dimensional data is the focus of today's statistical research and practice. After broad success of statistical machine learning on prediction through regularization, interpretability is gaining attention and sparsity has been used now as its proxy. With the virtues of both regularization and sparsity, Lasso (L1 penalized L2 minimization) has been very popular recently. In this talk, I would like to discuss the theory and pratcice of sparse modeling. First, I will give an overview of recent research on model selection consistency property of l1 penalized minimization including Lasso and explain what useful insights have been learned. Second I will present collaborative research on building nonparametric sparse hierarchical models that describe fMRI responses in primary visual cortex area V1 to natural images.

By Northcott's Theorem one knows that any subset of a number field of bounded Weil height is finite. In this talk we present similar results for certain infinite field extensions of the rationals. We highlight connections between height bounds Northcott type theorems and finiteness results in discrete dynamics and Diophantine geometry.

We will discuss the cone of effective divisors of a variety, and then move on to the specific case of .

In this talk we will summarize several recent developments related to Landau-Ginzburg models. We will begin by describing how one A-twists a Landau-Ginzburg model in physics, and how a physical process known as "renormalization group flow" sometimes identifies correlation functions in such A-twisted Landau-Ginzburg models with ordinary Gromov-Witten invariants. Then, after briefly reviewing how one associates a CFT to a gerbe and associated technical issues, we will outline some applications of CFT's of gerbes, including Landau-Ginzburg (Toda) mirrors to gerbes on projective spaces, and a physical realization of Kuznetsov's homological projective duality, which provides examples of LG's and GLSM's describing non-birational spaces on the same GLSM Kahler moduli space, as well as examples in which the geometry arises via nonperturbative effects, rather than the usual complete intersection story.

One classical question in geometric group theory asks which groups act isometrically without global fixed points on certain graphs; of particular interest are those groups which act without global fixed points on CAT(0) graphs, or trees.  One generalization of this question, formulated by Benson Farb, asks whether a given Coxeter group acts isometrically without global fixed points on an n-dimensional CAT(0) cell complex.  The Coxeter FA_n conjecture states that the answer is yes if and only if has an infinite special subgroup of rank n + 1.  I'll discuss some of the progress that has been made towards a proof of this conjecture, including the work that I have done in the case n = 3.

Some of the mathematics and properties behind one of the most intricate and interesting shapes in Complex Analytic Dynamics, The Mandelbrot Set, will be explored in this presentation. Background topics presented will include the Quadratic Map, Orbit Analysis, and properties of The Julia Set in addition to Mandelbrot Set properties. All topics will be introduced from the standpoint of discrete deterministic Chaos. Finally, computer graphics will be used to visually illustrate some of the mathematics discussed including how to “count” and “add” on the Mandelbrot Set.

A subset of the algebraic numbers has the property (N) if each of its subsets of bounded Weil height is finite. In 1950 D.G. Northcott proved that sets of bounded degree have property (N) and that this in turn yields finiteness results for the number of preperiodic points under polynomial mappings. Northcott's Theorem has been used extensively in Diophantine geometry. Therefore it would be interesting to establish property (N) for other sets than sets of bounded degree; e.g. certain infinite field extensions. In 2001 Bombieri and Zannier addressed the following question: do fields generated by algebraic numbers of degree at most d have property (N)? Bombieri and Zannier showed that this is the case for d=2; more generally they established the property (N) for the fields gotten by adjoining a d-th root of all positive integers to the rationals. In this talk we show one of many possible ways to further generalize the latter result. We introduce a simple criterion for the property (N) that provides many new examples of infinite extensions with property (N).

Abstract: Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z_2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of K-homology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the K-theoretic classification is stable to interactions, but a counterexample is also given.