# Seminars & Events for Department Colloquium

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

##### 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.

It was conjectured by R. Schoen in the 1980s (and, independently, by Aubin) that the set of all metrics of constant scalar curvature 1 in a given conformal class is compact, except if the underlying manifold is conformally equivalent to the sphere $S^n$ equipped with its standard metric. The significance of Schoen's conjecture is that it would imply Morse inequalities for the total scalar curvature functional.

##### Incompressible Fluids: Simple Models, Complex Fluids

Complex fluids are fluids with particles suspended in them. The particles are carried by the fluid, interact among themselves, and influence the fluid's behavior. I will describe some of the basic questions of existence, uniqueness, regularity and stability of solutions of models of complex fluids, in the broader context of incompressible hydrodynamic PDE.

##### Rearrangement and convection

Rearrangement theory is about reorganizing a given function (or map) in some specific order (monotonicity, cycle monotonicity etc...). This is somewhat similar to the convection phenomenon in fluid mechanics, where fluid parcels are continuously reorganized in a stabler way (heavy fluid at bottom and light fluid at top). Convection theory is one of the most important piece of geo-sciences, related to weather forecasting, oceanography, volcanism, earthquake etc... In our talk, we make these analogies more precise by analyzing the Navier-Stokes equations with buoyancy and Coriolis forces. We will see how these approximations are related to the concept, well known in optimal transport theory, of rearrangement of maps as gradient of convex functions.

##### Metaphors in systolic geometry

The systolic inequality says that any Riemannian metric on an $n$-dimensional torus with volume 1 contains a non-contractible closed curve with length at most $C(n)$ - a constant depending only on $n$. One remarkable feature of the inequality is it holds for such a wide class of metrics. It's much more general than an inequality that holds for all metrics obeying a certain curvature condition.

The systolic inequality is a difficult theorem, and each proof is guided by a metaphor that connects the systolic inequality to a different area of geometry or topology. In this talk, I will explain three metaphors. They connect the systolic inequality to minimal surface theory, topological dimension theory, and scalar curvature.

##### Acoustical spacetime geometry and shock formation

In 2007 I published a monograph which treated the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. In this monograph I considered initial data which outside a sphere coincide with the data corresponding to a constant state. Under a suitable restriction on the size of the initial departure from the constant state, I established theorems which gave a complete description of the maximal classical development.. In particular, I showed that the boundary of the domain of the maximal classical development has a singular part where the inverse density of the wave fronts vanishes, signaling shock formation. In fact, the theorems which I established give a complete picture of shock formation in three-dimensional irrotational fluids .

##### Natural maps old and new

In 1995, G. Courtois, S. Gallot and myself constructed a family of maps with very good properties regarding volume elements between certain manifolds. We used it to give an alternative proof of Mostow's rigidity for rank one closed symmetric spaces as well as a rigidity result for their geodesic flow, conjectured by A. Katok. Various modifications of the original construction have been made since yielding new results in different settings. We shall describe the basic construction, the modifications, some applications and open questions.

##### 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. Whether it actually does is a long-standing open question. After Ledrappier's unexpected 1978 confirmation for two commuting actions the theory for general compact abelian groups has been studied, and several problems have been posed by Klaus Schmidt. In this context the speaker in 2004 proved that when multiple mixing fails it does so in an orderly fashion thanks to the so-called “non-mixing sets", and in 2006 with Harm Derksen that the size of the smallest such set could be effectively determined. Last year we gave a finiteness result for these smallest sets. It is closely related to the problem of finding all “shortest" elements of a given ideal in a polynomial ring.

##### Modular representations of $p$-adic groups

The Langlands program relates complex representations of $GL_n(Q_p)$ to $n$-dimensional Galois representations. For $n=1$ this is explained by class field theory and for $n=2$ this is closely related to the theory of modular forms. For general $n$, this is now understood by the work of Harris-Taylor and Henniart. In the last decade, a mod-p (as well as a $p$-adic) version of the Langlands program have been emerging, and they have already played an important role in some recent progress in number theory. But so far understanding has been limited to $n=1$ and $n=2$. We survey some of the known story for $n=2$, and then discuss some recent progress on the classification of mod p representations of $GL_n(Q_p)$.

##### New theory of hypergeometric functions

The lecture will be devoted to the new vintage in the theory of special functions, a unification of the Bessel, hypergeometric, spherical and Whittaker functions, their p-adic and difference counterparts, and of course the theta-functions (associated with root systems) in one definition. The latter was suggested;13 years ago, but a reasonably complete analytic theory of such global spherical functions was created only recently, including the Harish-Chandra asymptotic formula and many more. These global functions generalize the classical q-hypergeometric(basic) series introduce by Heine in 1846, but the new approach is very different even for one variable. Algebraically, the global functions are actually similar to the Bessel functions.

##### The spectrum of non-normal random matrices

We will discuss the asymptotics of the spectrum of non-normal random matrices with size going to infinity, and in particular the single ring phenomenon observed for unitary invariant models.

##### Topological expansion for random matrices

Department Colloquium (joint with ORFE)

##### Rational points on algebraic varieties

I will discuss several geometric techniques and constructions that emerged in the study of rational points on higher-dimensional algebraic varieties over global fields.

##### Endoscopic transfer of the Bernstein center

The Langlands-Shelstad theory of endoscopy plays a central role in the study of Shimura varieties and the Arthur-Selberg trace formula. The fundamental lemma and a deep consequence, endoscopic transfer, have now been established in works of Ngo, Waldspurger, and Hales. Both of these statements are identities involving orbital integrals of a function $f$ on a $p$-adic group $G$ and those of certain "transfer" functions $f^H$ on related groups $H$, called endoscopic. This talk will give background on the fundamental lemma, some of its variants, and the roles they played. Then I will describe a conjectural construction of a large class of matching functions in the Bernstein centers of $G$ and $H$. This conjecture has been verified in some encouraging cases.

##### Eigenfunctions and nodal sets

Nodal sets are the zero sets of eigenfunctions of the Laplacian on a Riemannian manifold (M, g). If $\Delta \phi = \lambda^{2 \phi}$, then $\phi$ is somewhat analogous to a polynomial of degree $\lambda$ and its nodal set is somewhat analogous to a real algebraic variety of this degree. The analogy is closest if (M, g) is real analytic. Donnelly-Fefferman then showed that the hypersurface volume is of order $\lambda$. The physicists make the bold conjecture that, if the geodesic flow is chaotic, then the nodal sets become equidistributed with respect to the volume form. This is far beyond current technology, but I will show that if one complexifies everything--the manifold, the eigenfunctions and the nodal sets--then one can obtain equidistribution results in the ergodic case.

##### Limit Theorems for Translation Flows

The talk is devoted to limit theorems for translation flows on flat surfaces. Consider a compact oriented surface of genus at least two endowed with a holomorphic one-form. The real and the imaginary parts of the one-form define two foliations on the surface, and each foliation defines an area-preserving translation flow. By a Theorem of H.Masur and W.Veech, for a generic surface these flows are uniquely ergodic. The first result of the talk, which extends earlier work of A.Zorich and G.Forni, is an asymptotic formula for time integrals of Lipschitz functions. One of the main objects of the talk is the space of finitely-additive Hoelder transverse invariant measures for our foliations. These measures are classified and related to G. Forni's invariant distributions of Sobolev regularity -1 for translation flows.

##### Higher order Fourier analysis

In a famous paper Timothy Gowers introduced a sequence of norms $U(k)$ defined for functions on abelian groups. He used these norms to give quantitative bounds for Szemeredi's theorem on arithmetic progressions. The behavior of the $U(2)$ norm is closely tied to Fourier analysis. In this talk we present a generalization of Fourier analysis (called k-th order Fourier analysis) that is related in a similar way to the $U(k+1)$ norm. Ordinary Fourier analysis deals with homomorphisms of abelian groups into the circle group. We view k-th order Fourier analysis as a theory which deals with morphisms of abelian groups into algebraic structures that we call "k-step nilspaces". These structeres are variants of structures introduced by Host and Kra (called parallelepiped structures) and they are close relatives of nil-manifolds.