Upcoming Seminars & Events
Character varieties parametrize conjugacy classes of representations of discrete groups into algebraic groups. I'll discuss some recent result on the topology of these varieties. In particular, I'll explain how studying their fundamental groups leads to information about centralizers in Lie groups. This is joint with with Biswas, Lawton, and Florentino.
An inverse cascade, energy transfer to progressively larger scales, is a salient feature of two-dimensional turbulence. If the cascade reaches the system scale, it creates a coherent flow expected to have the largest available scale and conform with the symmetries of the domain. In a square doubly periodic domain, the mean flow is expected to take the form of a vortex dipole. The velocity profile of a corresponding single vortex was recently obtained analytically and subsequently confirmed numerically. I will describe the next step in the derivation: using the mean velocity profile to predict features of the turbulent fluctuations. I will also address the mean flow in a doubly periodic (non-square) rectangle. For a rectangle, the mean flow with zero total momentum was believed to be unidirectional, with two jets along the short side. I will describe how direct numerical simulations reveal that neither the box symmetry is respected nor the largest scale is realized: the flow is never purely unidirectional since the inverse cascade produces coherent vortices, whose number and relative motion are determined by the aspect ratio. This spontaneous symmetry breaking is closely related to the hierarchy of averaging times. Long-time averaging restores translational invariance due to vortex wandering along one direction, and gives jets whose profile, however, can be deduced neither from the largest-available-scale argument, nor from the often employed maximum-entropy principle or quasi-linear approximation.
In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székelyhidi to obtain further partial results towards Onsager's conjecture. The proof of the full conjecture combines convex integration using the “Mikado flows” introduced by Daneri-Székelyhidi with a new “gluing approximation” technique. The latter technique exploits a special structure in the linearization of the incompressible Euler equations.
Please note special day (Tuesday).
I will begin with a brief overview of the existence question for conformally compact Einstein manifolds with prescribed conformal infinity. After stating the seminal result of Graham-Lee, I will discuss a non-existence result (joint with Qing Han) for certain conformal classes on the 7-dimensional sphere. I will also mention some ongoing work (with Gabor Szekelyhidi) on a version of "local existence" of Poincare-Einstein metrics.
Given a projective variety X over a field of characteristic 0, and a positive integer r, we study the rth secant variety of Veronese re-embeddings of X. In particular, I'll explain recent work which shows that the degrees of the minimal equations (and more generally, syzygies) defining these secant varieties can be bounded in terms of X and r independent of the Veronese embedding. This is based on arXiv:1510.04904 and arXiv:1608.01722.
I will describe some multiplicity result about constant Q-curvature metrics in the case of homogeneous vibrations and in the case of non compact manifolds related to a singular version of the Q-curvature problem.
The study of the Poisson geometry of the Teichmuller space and the moduli space of local systems gave rise to the discovery of the Goldman bracket of curves on a surface which in turn led Chas and Sullivan to discover string topology operations on chains on the free loop space of an oriented manifold. Their string topology operations also generalized the Turaev cobracket which did not come from a Poisson geometric origin, and the search for the geometric meaning of all string topology operations continues. I will discuss some Poisson geometry aspects of the moduli stack of Chen connections and how in the large N limit an additional relevant structure appears. This is part of a joint work in progress with Gregory Ginot and Owen Gwilliam.
We prove a structure theorem for pseudo-Anosov flows restricted to Seifert fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration so that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A non periodic Seifert fibered piece is called free. In this talk we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow is orbitally equivalent to a hyperbolic blow up of a geodesic flow piece. A geodesic flow piece is a finite cover of the geodesic flow on a compact hyperbolic surface, usually with boundary (a union of geodesics). The proof uses an associated convergence group theorem, hyperbolic blow ups and models of geodesic flows. This is joint work with Thierry Barbot.