# Seminars & Events for 2012-2013

##### Minerva Lecture II - The virtual Haken conjecture: What is geometric group theory?

Waldhausen conjectured in 1968 that every aspherical 3-manifold has a finite-sheeted cover which is Haken (contains an embedded essential surface). Thurston conjectured that hyperbolic 3-manifolds have a

finite-sheeted cover which fibers over the circle. The first lecture will be an overview of 3-manifold topology in order to explain the meaning of Waldhausen's virtual Haken conjecture and Thurston's virtual fibering conjecture, and how they relate to other problems in 3-manifold theory. The second lecture will give some background on geometric group theory, including the topics of hyperbolic groups and CAT(0) cube complexes after Gromov, and explain how the above conjectures may be reduced to a conjecture of Dani Wise in geometric group theory. The third lecture will discuss the proof of Wise's conjecture, that cubulated hyperbolic groups are virtually special. Part of this result is joint work with Daniel Groves and Jason Manning. We will attempt to make these lectures accessible to a general mathematical audience at the level of a colloquium talk.

##### In search of a Langlands transform

Class field theory expresses Galois groups of abelian extensions of a number field F in terms of harmonic analysis on the multiplicative group of locally compact topological ring, the adèle ring, attached to F.

##### Percolation on Z^d

Percolation is a stochastic process defined on graphs which was introduced by Broadbent and Hammersley as a model for the flow of a fluid through a porous medium. I will define independent bond percolation on the lattice Z^d and explain the percolation phase transition in this case. I will then give some results which describe the subcritical and supercritical phases. Time permitting, I will end with some theorems about percolation at the critical point in two dimensions.

##### Expanders with respect to random regular graphs

The discrete Poincare inequality (PI) for a regular graph G=(V,E) is the following: For any mapping f:V-->H of the vertices into Hilbert space, the average of ||f(u)-f(v)||^2 over all pairs of vertices is at most the average of ||f(u)-f(v)||^2 over the edges (E) times 1/(1-lambda_2(G)), where lambda_2(G) is the second normalized eigenvalue of G.

##### Random lozenge tilings of polygons and their asymptotic behavior

I will discuss the model of uniformly random tilings of polygons drawn on the triangular lattice by lozenges of three types (equivalent formulations: dimer models on the honeycomb lattice, or random 3-dimensional stepped surfaces).

##### Central values of Rankin-Selberg L-functions and period relations

In his work of the early 1980s, Shimura observed that expressions of special values of automorphic L-functions in terms of period invariants could be used to obtain relations among the latter. This observation has since been applied in numerous situations by the speaker, among others. Most of these applications involve the theta correspondence. This talk will describe a different approach, b

##### Minerva Lecture III: Geometric group theory and the virtual Haken conjecture

Waldhausen conjectured in 1968 that every aspherical 3-manifold has a finite-sheeted cover which is Haken (contains an embedded essential surface). Thurston conjectured that hyperbolic 3-manifolds have a

##### Behavior of Welschinger invariants under Morse simplification

Welschinger invariants, real analogs of genus 0 Gromov-Witten invariants, provide non-trivial lower bounds in real algebraic geometry. In this talk I will explain how to get some wall-crossing formulas relating Welschinger invariants of the same (up to deformation) rational algebraic surface with different real structures. This relation is obtained via a real version of a formula by Abramovich and Bertram which computes Gromov-Witten invariants using deformations of complex structures. It can also be seen as a real version, in our special case, of Ionel and Parker's symplectic sum formula. If time permits, I will give some qualitative consequences of this study, for example the vanishing of Welschinger invariants in some cases, and will discuss some generalizations. This is joint work with Nicolas Puignau (UFRJ, Rio de Janeiro)

##### Hamiltonian S^1 Actions with Isolated Fixed Points on 6-Dimensional Symplectic Manifolds

The question of what conditions guarantee that a symplectic circle action is Hamiltonian has been studied for many years. In 1998, Sue Tolman and Jonathon Weitsman proved that if the action is semifree and has a non-empty set of isolated fixed points then the action is Hamiltonian. In 2010, Cho, Hwang, and Suh proved in the 6-dimensional case that if we have b_2^+=1 at a reduced space at a regular level \lambda of the circle valued moment map, then the action is Hamiltonian. (Please click on seminar title for complete abstract.)

##### Fourier Law and Non-Isothermal Boundary in the Boltzmann Theory

In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the steady problem for the Boltzmann equation in a general bounded domain with diffuse reflection boundary conditions corresponding to a non-isothermal temperature of the wall.

##### TBA

##### On the numerical dimension of pseudo-effective divisors in positive characteristic

Suppose that X is a smooth algebraic variety over an algebraically closed field and that D is a pseudo effective R-divisor on X. In characteristic zero, by utilizing vanishing theorems, Nakayama proved that if D is not numerically equivalent to the negative part of its Zariski decomposition, then D is pseudo-effective.

##### Analytic Number Theory in Function Fields

Function fields are a good toy model for number fields. I'll give an easy proof of the prime number theorem for the function field F_q(t), along the way proving the Riemann hypothesis, and talk a little bit about number theory in field extensions. If time permits, I'll discuss the Mertens conjecture in both number fields and function fields. No algebraic geometry will be harmed in this talk.

##### On Dettmann's 'Horizon' Conjectures

In the simplest case consider a $\mathbb{Z}^d$-periodic ($d \geq 3$) arrangement of balls of radii $< 1/2$, and select a random direction and point (outside the balls). CLICK ON THE SEMINAR TITLE FOR THE COMPLETE ABSTRACT.

##### On the Erdos-Sos conjecture

We are going to prove that if k is large enough then a graph with average degree k contains all trees on k vertices. This is joint work with Miklos Ajtai, Janos Komlos and Miklos Simonovits.

##### Combinatorial number theory arising from algebraic topology.

We will show how studying v1-periodic homotopy groups of SU(n) led to the following question. Let f(n) denote the sum of the reciprocals of the binomial coefficients (n choose i). For which p-adic integers x does the

##### The Tate conjecture for K3 surfaces over fields of odd characteristic

The classical Kuga-Satake construction, over the complex numbers, uses Hodge theory to attach to each polarized K3 surface an abelian variety in a natural way. Deligne and Andre extended this to fields of characteristic zero, and their results can be combined with Faltings's isogeny theorem to prove the Tate conjecture for K3 surfaces in characteristic zero.

##### Mom 1.5

This talk will discuss joint work in progress with Robert Haraway and Craig Hodgson. Mom Technology has had considerable success in proving facts about low-volume 1-cusped hyperbolic 3-manifolds. We are attempting to generalize Mom Technology to the case of hyperbolic 3-manifolds with totally geodesic boundary. The generalization of Mom Technology to this setting has a number of interesting

##### Behavior of Welschinger invariants under Morse simplification

Welschinger invariants, real analogs of genus 0 Gromov-Witten invariants, provide non-trivial lower bounds in real algebraic geometry. In this talk I will explain how to get some wall-crossing formulas relating Welschinger invariants of the same (up to deformation) rational algebraic surface with different real structures.

##### An arithmetic refinement of homological mirror symmetry for the 2-torus

We establish a derived equivalence of the Fukaya category of the 2- torus, relative to a basepoint, with the category of perfect complexes on the Tate curve over Z.