# Seminars & Events for 2010-2011

##### Gromov's knot distortion

Gromov defined the distortion of an embedding of $S^1$ into $R^3$ and asked whether every knot could be embedded with distortion less than 100. There are (many) wild embeddings of $S^1$ into $R^3$ with finite distortion, and this is one reason why bounding the distortion of a given knot class is hard. I will show how to give a nontrivial lower bound on the distortion of torus knots. I will a

##### Parahoric subgroups and supercuspidal representations of $p$-adic groups

This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is that this induced representation can (in certain situations) have finite length.

##### Metric flips with Calabi symmetry

I will discuss the metric behavior of the Kahler-Ricci flow on $\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus (m+1)})$, assuming that the initial metric satisfies the symmetry defined by Calabi. I will describe the Gromov-Hausdorff limit of the flow as time approaches the singular time and how the Kahler-Ricci flow can be continued.

##### Initial time singularities for mean curvature flow

##### Reformulation of the Covering and Quantizer Problems as Ground States of Interacting Particles

I reformulate the covering and quantizer problems, well-known problems in discrete geometry, as the determination of the ground states of interacting particles in d-dimensional Euclidean space that generally involve single-body, two-body, three-body, and higher-body interactions.

##### Concrete mathematical incompleteness

An unprovable theorem is a theorem about basic mathematical objects that can only be proved using more than the usual axioms for mathematics (ZFC = *Zermelo Frankel set theory with the Axiom of Choice*) — and that has been proved using standard extensions of ZFC generally adopted in the mathematical logic community.

##### Geometric methods for nonlinear quantum many-body systems

Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schrödinger operators. In this talk I will present a formalism which also allows to study nonlinear systems.

##### Restriction varieties and geometric branching rules

In representation theory, a branching rule describes the decomposition of the restriction of an irreducible representation to a subgroup. Let $i: F' \rightarrow F$ be the inclusion of a homogeneous variety in another homogeneous variety. The geometric analogue of the branching problem asks to calculate the induced map in cohomology in terms of the Schubert bases of $F$ and $F'$.

##### Choice numbers and coloring numbers - the infinite case

The *choice number* or *list-chromatic number* $\chi_\ell(G)$ of a graph $G=(V,E)$ is the minimum $k$ such that for every assignment of a list $s(v)$ of $k$ colors to each $v\in V$ there exists a proper coloring $c$ of $V$ that colors each $v$ by a color from $s(v)$.

##### Holomorphic Pairs of Pants in Mapping Tori

We consider invariants of mapping tori of symplectomorphisms of a symplectic surface S, such as symplectic field theory, contact homology, and periodic Floer homology, for the standard stable Hamiltonian structure on the mapping torus. These invariants involve counts of holomorphic curves in R times the mapping torus.

##### Weyl's sums for roots of quadratic congruences

It is known that the roots of congruences for a fixed irreducible quadratic polynomial are equidistributed. This statement translates to getting cancellation in the corresponding sum of Weyl's sums. In a recent work by W. Duke, J. Friedlander and H. Iwaniec we succeeded to get cancellation (so also the equidistribution) in very short sums of Weyl's sums relatively to the discriminant.

##### Filtering smooth concordance classes of topologically slice knots

Cochran, Orr, and Teichner introduced the n-solvable filtration of the knot concordance group, which has given a framework for recent advances in the study of knot concordance. However, it fails to detect anything about topologically slice knots, denoted T.

##### High-dimensional reservoir neural dynamics: rules and rewards

Neural activity recorded in behaving animals is highly variable and heterogeneous, which is especially true for neurons in the prefrontal cortex (PFC), the so called 'CEO of the brain' of central importance to many cognitive functions. In this talk, I will present a reservoir-type model of randomly connected neurons to account for the diversity of neural signals in the prefrontal cortex.

##### Multiple mixing and short polynomials

In dynamical systems the notion of multiple mixing seems to strengthen that of mixing for an action n a probability space.

##### Discrete Schrödinger operators with periodic and almost periodic potentials

The first half of the talk will be a survey; I will start from general facts about discrete Schrödinger operators with periodic potentials, and then discuss operators with almost periodic potentials, focusing on the almost Mathieu operator.

##### Phase Transitions of Achlioptas Processes

Achlioptas processes are a class of modifications of the Erdős–Rényi random graph. At each step of an Achlioptas process we add one of two randomly selected edges to the graph according to a fixed rule.

##### Bordered Floer homology and the contact category

Bordered Heegaard Floer homology is a verison of Heegaard Floer homology for 3-manifolds with boundary, developed by Lipshitz, Ozsvath, and Thurston. A key component of the theory is a DG-algebra associated to a parametrized surface $F$.

##### Impossible intersections for elliptic curves

We proved with Umberto Zannier that there are at most finitely many complex numbers $\lambda \neq {0,1}$ such that two points on the Legendre elliptic curve $y^2 = x(x-1)(x-\lambda)$ with coordinates $x=2$ and $x=3$ both have finite order. However we still do not know how to find these $\lambda$ effectively (there are probably none).

##### The logarithmic singularities of the Green functions of the conformal powers of the Laplacian

Green functions play an important role in conformal geometry. In this talk, we shall explain how to compute explicitly the logarithmic singularities of the Green kernels of the conformal powers of the Laplacian, including the Yamabe and Paneitz operators. The results are formulated in terms of explicit conformal invariants arising from the ambient metric of Fefferman-Graham.

##### A few computational and applied math problems in density functional theory related calculations

Density functional theory (DFT) has become the most widely used quantum mechanical method in material science simulations. Due to the change of computer architectures, and the corresponding change in the scope of problems amenable by the DFT method, the algorithms used in the DFT calculations are also changing.