# Seminars & Events for 2009-2010

##### Transversality and noncommutative geometry

Birationally commutative graded algebras solve the moduli problem for "point modules" over a graded ring. They have been a fruitful source of counterexamples, examples, and intuition in noncommutative ring theory. We investigate when a large subclass of birationally commutative algebras is noetherian. Formally, these are idealizer subrings of twisted homogeneous coordinate rings.

##### CMC foliations on alomost Fuchsian 3-manifolds

An alomost Fuchsian 3-manifold is a quasi-Fuchsian 3-manifold that contains an incompressible surface with principal curvatures in (-1,1). In this talk, we will show that any alomost Fuchsian 3-manifold admits a CMC foliation.

##### Nonlinear Waves

Given a nonlinear wave equation whose nonlinearity contains derivatives of the unknown function: $-\partial_t^2\phi+\sum_{i=1}^ 3 \partial_{x_i}^2\phi=N(\partial\phi)$ in $3+1$ dimensions, what can we say about the global existence of solutions? What if the initial conditions are small and compactly supported? If time permits, I will discuss the case of a wave map.

##### Counting flags in digraph

Many results in asymptotic extremal combinatorics are obtained using just a handful of instruments, such as induction and Cauchy-Schwarz inequality. The ingenuity lies in combining these tools in just the right way.

##### A parabolic flow of Hermitian metrics

I will introduce a parabolic flow of Hermitian metrics which is a generalization of Kähler-Ricci flow. This flow preserves the pluriclosed condition, and its existence and convergence properties are closely related to the underlying topology of the given complex manifold. I will discuss a stability result for the flow near Kähler-Einstein metrics.

##### Differential equations arising from group actions on manifolds

We consider ODE's arising from the action of Lie groups on manifolds and discuss how solvability of corresponding lie algebra helps to find explicit solution

##### The decay of Fourier modes in solutions of Navier-Stokes systems

##### Topologically minimal surfaces in 3-manifolds

Topologically minimal surfaces are the topological analogue of geometrically minimal surfaces. Such surfaces generalize well known classes, such as incompressible, strongly irreducible (or weakly incompressible), and critical surfaces. Applications include problems dealing with stabilization, amalgamation, and isotopy of Heegaard splittings and bridge spheres for knots.

##### Claw-free graphs with strongly perfect complements. Fractional and integral version

Strongly perfect graphs have been studied by several authors (e.g. Berge, Duchet, Ravindra, Wang). This paper deals with a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement.

##### Persistence of Essential Surfaces after Dehn filling

We show that the set of closed, essential, 2-sided surfaces (considered up to isotopy) in a 3-manifold with a torus boundary component survives unchanged in all suitably generic Dehn fillings. Furthermore, for all but finitely many non-generic fillings, we show that two essential surfaces can only become isotopic in a very constrained way.

##### Even Galois Representations and the Fontaine-Mazur Conjecture

We prove, under mild hypotheses, there are no irreducible two-dimensional ordinary even Galois representations of the Galois group of Q with distinct Hodge-Tate weights, in accordance with the Fontaine-Mazur conjecture. We also show how this method can be applied to a related circle of problems.

##### Hamiltonian Stationary Tori in Kahler Manifolds

A solution of the Hamitonian stationary variational problem is a Lagrangian submanifold (of a symplectic manifold with compatible Riemannian metric) whose volume is stationary under Hamiltonian variations.

##### Self-organized selectivity in Calcium and Sodium Channels: important biology ready for mathematical analysis

Ion channels are irresistible objects for biological study because they are the [nano] valves of life controlling an enormous range of biological function, much as transistors control computers. Ion channels are appealing objects for physical investigation because conformation changes are not involved in channel function, once the channel is open.

##### Why should you care about graph theory?

The Four Color problem has long been a poster child for graph theory and has inspired much of its development over the years. I will mention a few, perhaps surprising, equivalent formulations of the problem, outline the main ideas of its proof and discuss some of the related open questions.

##### Homotopy Theoretic methods on Chow varieties

The homotopy theoretic method has been applied to the algebraic cycle theory for a long period of time. In particular, it can be applied to compute topological invariants of Chow varieties. In this talk I will discuss this method in calculating the Euler Characteristic of Chow varieties.

##### Strong multiplicity one and $l$-adic Galois representations

The strong multiplicity one theorem and its refinements amount to a local-to-global principle in the theory of automorphic representations of $GL(N)$. I will discuss a Galois-theoretic analogue with a surprisingly elementary proof, along with some related questions about the images of $l$-adic Galois representations.

##### Nearest neighbor distances for several rotations

We will discuss results of Marklof on distributions of nearest neighbor distances. We will look at the Poisson scaling and at CLT scaling. Another point of view is to look at the number of distinct gap lengths in this scenario. Here we will explain unpublished results of Boshernitzan and Dyson.

##### Graph norms and Erdos-Simonovits-Sidorenko's conjecture

I will prove some results in the direction of answering a question of Lovasz about the norms defined by certain combinatorial structures. Inspired by the similarity of the definitions of $L_p$ norms, trace norms, and Gowers norms, we introduce and study a wide class of norms containing these, as well as many other norms.

##### Projective product spaces

A projective product space is a space obtained from a product of spheres by modding out by the antipodal action in all factors. We discuss cohomology, splittings, span, parallelizability, and immersion dimension of these spaces.

##### An arithmetic fundamental lemma for unitary group of three variable

In this talk I'll present a relative trace formula approach to the Gross-Zagier formula and its high dimensional generalization (a derivative version of the global Gross-Prasad conjecture) for unitary group. In particular, an arithmetic fundamental lemma (AFL) is proposed. Some results proved recently will be presented, including the AFL for unitary group of three variable.