Consider a semi-simple Lie group, $G$ and a discrete torsion-free subgroup $\Gamma < G$. When $G$ has rank one, there is an important, well-understood connection between the growth rate of elements of $\Gamma$-orbits, bounds on the spectrum, and temperedness, or more generally properties of the unitary $L^2$ representation (due, in part, to work of Elstrodt and Patterson). In this talk I will present an extension of this connection to higher rank.  This is joint work with Tobias Weich and Lasse Wolf. 

I will give a short review of the physics-inspired approach to Donaldson theory based on topologically-twisted four-dimensional N=2 supersymmetric quantum field theory (SQFT) as propounded by Witten in 1988, and then describe some of my recent and ongoing work. One theme in particular is a generalization of Donaldson-Witten theory to describe invariants of smooth families of smooth, closed, oriented Riemannian 4-manifolds X using methods of SQFT and supergravity, and leading to physical derivations of relevant models of equivariant cohomology. Family Donaldson invariants are cohomology classes of the classifying space of orientation-preserving diffeomorphisms, i.e., elements of H*(BDiff^+(X)), and we propose path integral formulations of their cocycle representatives. Several interesting open questions arise concerning the observables and interpretation of the invariants, calling for a renewed math-physics dialog. I hope to (re?)ignite some interest in these issues through this talk. Time permitting, I will touch upon the second theme, a new perspective of topological twisting using the notion of ‘transfer of structure group’ associated with a continuous homomorphism between topological groups. This approach accounts for the global topology of the structure group of a quantum field theory, rather than just its simply connected cover, and leads to a proposal for twisting more general four-dimensional N=2 SQFTs. The talk will be based on collaborations with G. W. Moore, M. Roček, and R. K. Singh. 

Please join us for pre-Colloquium Tea

We use the Selberg zeta function to study the limit behavior of resonances of a degenerating family of Kleinian Schottky groups. We prove that, after a suitable rescaling, the Selberg zeta functions converge to the Ihara zeta function of a limiting finite graph associated with the relevant non-Archimedean Schottky group acting on the Berkovich projective line. Moreover, our techniques can be used to obtain effective statements. One key idea is to introduce an intermediate zeta function that captures Archimedean and non-Archimedean information (while the Selberg resp. Ihara zeta function concerns only Archimedean resp. non-Archimedean properties). This is a joint work with Jialun Li, Carlos Matheus, and Zhongkai Tao.

In this talk, I will explain how paradifferential calculus can be applied to construct quasiperiodic-in-time solution for PDEs (“KAM for PDEs”). Due to the loss of regularity caused by “small divisors”, these problems are traditionally resolved using Nash-Moser/KAM type iterative schemes. One step of the Nash-Moser scheme is to reduce a non-autonomous linear operator that involves “small divisors” into constant coefficient form, for which a Nash-Moser/KAM reducibility argument is necessary, yielding a complicated “Nash-Moser within Nash-Moser” formalism. However, it is discovered that paradifferential calculus can be used to completely avoid such formalism, yielding “fixed point” style proof. In particular, it is possible to reduce the nonlinear equation itself into constant coefficient form (modulo smoothing remainder), not just its linearization. This is because paradifferential operators share all the algebraic structures of (pseudo)differential operators while gain back regularities due to J.-M. Bony’s paralinearization process. I will use the existence problem for quasiperiodic-in-time solution of fully nonlinear hyperbolic systems with one spatial variable as illustrative example. This talk is based on joint works with Thomas Alazard.