# Seminars & Events for 2010-2011

##### The Conquest of Topology by Algebra, or, Topological Spaces as Relational Beta-Modules

We explore the relationship between topological spaces and modules of the ultrafilter monad, Beta. Several basic properties of topological spaces X are rephrased as properties of "convergence" relations $BX--->X$.

##### Hidden symmetries and decay for fields outside a Kerr black hole

The Kerr solutions to Einstein's equations describe rotating black holes. For the wave equation in flat-space and outside the non-rotating, Schwarzschild black holes, one method for proving decay is the vector-field method, which uses the energy-momentum tensor and vector-fields.

##### Narcissistic Skyrmions

The Skyrme model is a nonlinear sigma-model whose topologically non-trivial target space allows the existence of so-called topological solitons. Such solitons were proposed by Skyrme to model nuclear matter. In this talk I will review the history of the model and present recent work with Gary Gibbons and Willie Wong ruling out the existence of solitons with a planar reflection symmetry.

##### Unfriendly partition of graphs without an infinite subdivided clique

In this talk I prove that every graph with less than $\aleph_\omega$ vertices, which does not contain a subdivision of an infinite clique as a subgraph, must have a partition of its vertices to two sets, so that no vertex has more neighbors in its own set than in the other set.

##### Dynamics of 2D point vortics and its generalizations

##### Optimal Error Estimates in Stochastic Homogenization

Joint Analysis/PACM Colloquium

##### Optimal bounds on the Kuramoto-Sivashinsky equation

##### Asymptotic extremal graph theory is non-trivial

Many fundamental theorems in extremal graph theory can be expressed as linear inequalities between homomorphism densities. Lovasz and, in a slightly different formulation, Razborov asked whether it is true that every such inequality follows from a finite number of applications of the Cauchy-Schwarz inequality. In this talk we will show that the answer to this question is negative.

##### Gradient Ricci Solitons

We present some recent development in the study of gradient shrinking Ricci solitons. We address some questions about their classification and their geometric and topological structure.

##### Still running! Recent work on the neuromechanics of insect locomotion

I will describe several models for running insects, from an energy-conserving biped with passively-sprung legs to a muscle-actuated hexapod driven by a neural central pattern generator(CPG). Phase reduction and averaging theory collapses some 300 differential equations that describe this neuromechanical model to 24 one-dimensional oscillators that track motoneuron phases.

##### Derived Equivalence and the Picard Variety

I will explain a result, joint with Mihnea Popa, saying that if two smooth projective varieties have equivalent derived categories of coherent sheaves, then their Picard varieties are isogeneous; in particular the number of independent holomorphic one-forms is a derived invariant. A consequence of this is that derived equivalent threefolds have the same Hodge numbers.

##### Dynamical stability in the planar surface tension problem for the Gates-Penrose-Lebowitz free energy function and Kawasaki dynamics

The planar surface tension problem for the Gates-Penrose-Lebowitz free energy function concerns the minimization of this functional for profiles $m(x,y)$ on a cylinder in $R\times C\in R^d$ with cubic cross section $C$ and periodic boundary conditions.

##### Bounding chromatic number for graphs in Forb*(bull)

Given a graph $H, Forb(H)$ is the class of all graphs that do not contain $H$ as an induced subgraph, and $Forb^*(H)$ is the class of all graphs that do not contain any subdivision of $H$ as an induced subgraph.

##### Vertex-disjoint paths in tournaments

The question of linking pairs of terminals by disjoint paths is a standard and well-studied question in graph theory. The setup is: given vertices$ s1,\ldots,sk$ and $t1,\ldots,tk$, is there a set of disjoint path $P1,\ldots,Pk$ such that $Pi$ is a path from $si$ to $ti$?

##### Gradient ideals

Zero schemes of exact 1-forms have received more attention recently as moduli spaces associated to Calabi-Yau threefolds; they are called gradient schemes or critical schemes.

##### Phase transition in kinetically constrained models

Kinetically constrained models are simple lattice models of glasses with a dynamical frustration: a move can be performed only if some local constraints are satisfied, for example if the local density is low enough. These models have been introduced to explain on a purely dynamical ground the glass forming phenomenology.

##### Blow-up phenomena for the Yamabe equation

The Yamabe problem asserts that any Riemannian metric on a compact manifold can be conformally deformed to one of constant scalar curvature. However, this metric is not, in general, unique, and there are examples of manifolds that admit many metrics of constant scalar curvature in a given conformal class.

##### Shortest Curves Associated to a Degenerate Jacobi Metric on the two Torus

Let $P$ be a potential on the two torus that takes its minimum value at a unique point m. Set $E_0 := P(m)$. For a real number $E$, let $g_E$ be the Jacobi metric associated to $P$ and $E$. For $E \gt E_0, g_E$ is a Riemannian metric. An ancient theorem of Morse and Hedlund says that a $g_E$-shortest curve in an indivisible homology class is simple.

##### Random Graphs and the Parity Quantifier

The classical zero-one law for first-order logic on random graphs says that for any first-order sentence $F$ in the theory of graphs, the probability that the random graph $G(n, p)$ satisfies $F$ approaches either 0 or 1 as $n$ grows.

##### Homology of the curve complex and the Steinberg module of the mapping class group

The homology of the curve complex is of fundamental importance for the homology of the mapping class group. It was previously known to be an infinitely generated free abelian group, but to date, its structure as a mapping class group module has gone unexplored.