# Seminars & Events for 2015-2016

##### Polynomials, Representations, and Stability

I will introduce recent work of Church, Farb, and others on homological/representation stability, primarily through the elementary and explicit example of configurations of points in the plane.

##### Topological Complexity of Spaces of Polygons

The topological complexity of a topological space X is the number of rules required to specify how to move between any two points of X. If X is the space of all configurations of a robot, this can be interpreted as the number of rules required to program the robot to move from any configuration to any other. A polygon in the plane or in 3-space can be thought of as linked arms of a robot.

##### Positroids, non-crossing partitions, 1/e^2, and a conjecture of Da Silva

We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition.

##### Probabilistic global well-posedness of the energy-critical defocusing nonlinear wave equation bellow the energy space

We consider the energy-critical defocusing nonlinear wave equation (NLW) on $R^d$, $d = 3, 4, 5$. In the deterministic setting, Christ, Colliander, and Tao showed that this equation is ill-posed below the energy space $H^1× L^2$. In this talk, we take a probabilistic approach.

##### Geometric Deformations of Orthogonal and Symplectic Galois Representations

For a representation of the absolute Galois group of the rationals over a finite field of characteristic p, we would like to know if there exists a lift to characteristic zero with nice properties. In particular, we would like it to be geometric in the sense of the Fontaine-Mazur conjecture: ramified at finitely many primes and potentially semistable at p.

##### On the tangent cones in collapsed limit spaces with lower Ricci curvature bound

We will discuss open questions and some recent results regarding the tangent cones in collapsed limit spaces of manifolds with lower Ricci curvature bounds.

##### Classification of Bernoulli shifts

Bernoulli shifts over amenable groups are classified by entropy (this is due to Kolmogorov and Ornstein for $Z$ and Ornstein-Weiss in general). A fundamental property is that entropy never increases under a factor map. This property is violated for nonamenable groups.

##### Low-area Floer theory and non-displaceability

**Please note change of time: 2:30-3:30. **I will introduce a "low-area" version of Floer cohomology of a non-monotone Lagrangian submanifold and prove that a continuous family of Lagrangian tori in CP2, whose Floer cohomology in the usual sense vanishes, is Hamiltonian non-displaceable from the monotone Clifford torus. Joint work with Renato Vianna.

##### Global existence for quasilinear wave equations close to Schwarzschild

We study the quasilinear wave equation $[g] u = 0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when u is identically 0. Under a couple of extra assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.

##### When does injectivity imply surjectivity?

Any injective map from a finite set to itself is surjective. Ax's Theorem extends this to algebraic varieties and regular maps. Gromov invented sofic groups as a way to extend to this result to cellular automata and other settings. We'll re-prove his results via sofic entropy theory.

##### Deformation of quotient singularities

##### Several Nonarchimedean Variables, Isolated Periodic Points, and Zhang's Conjecture

**Please note special day, time, and location.** We study dynamical systems in several variables over a complete valued field. If x is a fixed point, we show that in many cases there exist fixed analytic subvarieties through x.

##### Geometric control: from damped waves to microfluid mixing

The damped wave equation is a prototype of a non-selfadjoint PDE, which means that the stationary problem has complex spectrum. The asymptotic distribution of the spectrum is heavily influenced by properties of the geodesic flow, and in part determines the rate of decay to equilibrium. In this talk, I will survey recent results on such decay estimates for the damped and over-damped wave equat

##### Dissipation at Maximal Rate

The lecture will present facets of the conjecture that the role of entropy is to maximize the rate of dissipation. BIO: Constantine Dafermos was born in Athens, in 1941. He received a diploma in Civil Engineering from the National Technical University, in 1964, and a Ph.D. in Mechanics from Johns Hopkins, in 1967.

##### Non-Archimedean geometry in rank >1

Recent work of Nisse-Sottile, Hrushovski-Loeser, Ducros, and Giansiracusa-Giansiracusa has demonstrated that valuation rings of rank >1 play an important role in the geometry of analytic and tropical varieties over non-Archimedean valued fields of rank 1.

##### Random matrices, differential operators and carousels

The Sine_\beta process is the bulk limit process of the Gaussian beta-ensembles. We show that this process can be obtained as the spectrum of a self-adjoint random differential operator.

##### Veering triangulations and pseudo-Anosov flows

We’ll discuss veering triangulations associated to pseudo-Anosov mapping tori, and how they arise dynamically. We’ll survey some of the results obtained regarding these triangulations. Then we’ll discuss a new construction of these triangulations associated to certain pseudo-Anosov flows, which is joint work with François Guéritaud.

##### Yang-Mills flow and entropy

We introduce the Yang-Mills functional and flow, and discuss its history, motivation and past results. We then discuss a notion of entropy and introduce a Yang-Mills soliton, sketching some joint work with Jeff Streets.

##### Characteristic polynomials for 1D band matrices from the localization side

The physical conjecture about the crossover for $N\times N$ 1D random band matrices with the band width $W$ states that we get the same behavior of eigenvalues correlation functions as for GUE for $W\gg \sqrt{N}$ (which corresponds to delocalized states), and we get another behavior, which is determined by the Poisson statistics, for $W\ll \sqrt{n}$ (and corresponds to localized st

##### Permutons

What do permutations of 1 through n, for large n, look like? For example, how can we generate a random permutation that inverts a third of its pairs? How many such permutations are there?