In this talk I will discuss results obtained with Tatyana Barron. We suggest a way to quantize, using Berezin-Toeplitz quantization (also refered to as Kahler quantization), a compact hyperkahler manifold (equipped with a natural 3-plectic form), or a compact integral Kahler manifold of complex dimension n regarded as a (2n-1)-plectic manifold. In each of these cases the n-plectic structure is derived from the symplectic structure, so it is intuitively clear that we should be able to obtain a quantization by way of the usual Kahler quantization of the symplectic form. We show that the quantization has reasonable semiclassical properties. I will begin by giving an overview of the idea of quantization and in particular I will discuss the necessary background from Berezin-Toeplitz quantization. I will then review the main results of Berezin-Toeplitz quantization (due to Martin Bordemann, Eckhard Meinrenken, Martin Schlichenmaier, which can be found in their paper “Toeplitz Quantization of Kahler Manifolds and gl(N), N--> \infty Limits”) that we have generalized. Finally I will review our results and indicate the proofs.

Ben Arous and A.Guionnet gave (1995) the first large deviation result for the Gaussian Unitary ensemble.This was subsequently extended to general Unitary matrix ensembles. I will discuss a method for obtaining such results using potential theory and, time permitting,its extension to other mathematical models. This is joint work with N.Levenberg and F.Wielonsky. No prior knowledge of random matrices will be assumed.

I will present a proof of the Freudenthal suspension theorem in HoTT. Unlike many other proofs in HoTT, this one is not merely an adaptation of a classical argument, but involves genuinely new techniques.

Modular forms are complex-valued functions on the upper-half plane $H$ that have certain properties related to the action of $SL(2,Z)$ on $H$ (or another discrete subgroup of $G=SL(2,R))$. Theory of modular forms has always included modular forms with values in a complex vector space, too. Dimension of spaces of vector-valued automorphic forms has been computed, in certain arithmetic situations, not only for $G=SL(2,R)$ but also for other linear groups (e.g. $G=SU(n,1)$, $Sp(n,1))$. This will be a survey-style talk, with the intention to to have a glimpse at a certain part of the theory of vector-valued automorphic forms, including some classical results and some very recent results.