In this talk I will describe Arthur's classification of automorphic representations of symplectic and orthogonal groups using automorphic representations of $\mathrm{GL}_N$.
Given a Lagrangian submanifold invariant under a Hamiltonian loop, we partially compute the image of the loop's Seidel element under the closed-open string map into the Hochschild cohomology of the Lagrangian. This piece captures the homology class of the loop's orbits on the Lagrangian and can help to prove that the closed-open map is injective in some examples. As a corollary we prove that $\mathbb{RP}^n$ split-generates the Fukaya category of $\mathbb{CP}^n$ over a field of characteristic 2, and the same for real loci of some other toric varieties.
We consider Euler equations on a fixed Lorentzian manifold. The fluid is initially supported on a compact domain and the boundary between the fluid and the vacuum is allowed to move. Imposing the so-called physical vacuum boundary condition, we will explain how to obtain a priori estimates for this problem. In particular, our functional framework allows us to track the regularity of the free boundary. This is joint work with S. Shkoller and J. Speck.
In the talk we present a survey of recent results (see [4]-[6]) on the existence theorems for the steady-state Navier-Stokes boundary value problems in the plane and axially symmetric 3D cases for bounded and exterior domains (the so called Leray problem, inspired by the classical paper [8]). One of the main tools is the Morse-Sard Theorem for the Sobolev functions $f\in W^2_1(\mathbb R^2)$ [1] (see also [2]-[3] for the multidimensional case). This theorem guaranties that almost all level lines of such functions are $C^1$-curves besides the function $f$ itself could be not $C^1$-regular.
Also we discuss the recent Liouville type theorem for the steady-state Navier-Stokes equations for axially symmetric 3D solutions in the absence of swirl (see [1]).
References
We look at the construction of radial metrics with an isolated singularity for the constant fractional curvature equation. This is a semilinear, non-local equation involving the fractional Laplacian, and appears naturally in conformal geometry.
A permutation group is called sharply n-transitive if it acts freely and transitively on the set of ordered n-tuples of distinct points. The investigation of such permutation groups is a classical branch of group theory; it led Emile Mathieu to the discovery of the smallest finite simple sporadic groups in the 1860's. In this talk I will discuss the case where the permutation group is assumed to be a locally compact transformation group, and explain how this set-up is related to Gromov hyperbolicity and to arithmetic lattices in products of trees.
Let $G$ be a reductive group such as $SL_n$ over the field $k((t))$, where $k$ is an algebraic closure of a finite field, and let $W$ be the affine Weyl group of $G$. The associated affine Deligne-Lusztig varieties $X_x(b)$ were introduced by Rapoport. These are indexed by elements $x$ in $G$ and $b$ in $W$, and are related to many important concepts in algebraic geometry over fields of positive characteristic. Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and representation theory to address these questions in the case that $b$ is a translation. Our approach is constructive and type-free, sheds new light on the reasons for existing results and conjectures, and reveals new patterns. Since we work only in the standard apartment of the building for $G$, which is just the tessellation of Euclidean space induced by the action of the reflection group $W$, our results also hold over the p-adics. This is joint work with Elizabeth Milicevic (Haverford) and Petra Schwer (Karlsruhe).
The study of closed geodesics on a Riemannian manifold is a classical and important part of differential geometry. In 1969 Gromoll and Meyer used Morse - Bott theory to give a topological condition on the loop space of compact manifold M which ensures that any Riemannian metric on M has an infinite number of closed geodesics. This makes a very close connection between closed geodesics and the topology of loop spaces.
Nowadays it is known that there is a rich algebraic structure associated to the topology of loop spaces — this is the theory of string homology initiated by Chas and Sullivan in 1999. In recent work, in collaboration with John McCleary, we have used the ideas of string homology to give new results on the existence of an infinite number of closed geodesics. I will explain some of the key ideas in our approach to what has come to be known as the closed geodesics problem.
It is well known that there are topological obstructions to a manifold $M$ admitting a Riemannian metric of everywhere positive scalar curvature (psc): if $M$ is Spin and admits a psc metric, the Lichnerowicz–Weitzenböck formula implies that the Dirac operator of $M$ is invertible, so the vanishing of the $\hat{A}$ genus is a necessary topological condition for such a manifold to admit a psc metric. If $M$ is simply-connected as well as Spin, then deep work of Gromov--Lawson, Schoen--Yau, and Stolz implies that the vanishing of (a small refinement of) the $\hat{A}$ genus is a sufficient condition for admitting a psc metric. For non-simply-connected manifolds, sufficient conditions for a manifold to admit a psc metric are not yet understood, and are a topic of much current research.
I will discuss a related but somewhat different problem: if $M$ does admit a psc metric, what is the topology of the space $\mathcal{R}^+(M)$ of all psc metrics on it? Recent work of V. Chernysh and M. Walsh shows that this problem is unchanged when modifying $M$ by certain surgeries, and I will explain how this can be used along with work of Galatius and myself to show that the algebraic topology of $\mathcal{R}^+(M)$ for $M$ of dimension at least 6 is "as complicated as can possibly be detected by index-theory". This is joint work with Boris Botvinnik and Johannes Ebert.