Talk

In this talk we consider lifts on the Siegel three fold. Motivated from physics, string theory, it is an interesting question to study these multiplicative (Borcherds lifts) and additive lifts (Saito-Kurokawa lifts) and their coincidence.

In this talk, I will first describe a reformulation of a construction by Gorbounov, Malikov and Schechtman in terms of global geometric data. This will then be applied to define vertex algebraic versions of Dolbeault complexes and obtain a geometric description of the Witten and elliptic genera of complex manifolds.

An important (and still open) question in algebraic geometry is to to find a geometric compactification for the moduli of polarized K3 surfaces. In this talk, I will survey some recent approaches to this problem. My focus will be on explaining the interplay between arithmetic, combinatorics, and geometry in the study of degenerations of K3 surfaces.

Noncommutative tori can be viewed as limits of elliptic curves for which the period lattice degenerates to a pseudo-lattice (a rank-2 free subgroup of the real line). A noncommutative torus whose period pseudo-lattice correspond to an order in a real quadratic field is called a real multiplication noncommutative torus. Based on strong analogies with the case of elliptic curves with complex multiplication Y.

Gromov introduced the notion of macroscopic dimension to study closed manifolds with positive scalar curvature (PSC). In the talk we will discuss two his conjectures on the subject: I. If a closed $n$-manifold $M$ admits a PSC metric, then the macroscopic dimension of its universal cover is less than $n-1$. II. If the universal cover of a closed $n$-manifold $M$ has the macroscopic dimension less than $n$, then the image of the fundamental class under a map classifying the universal cover is trivial, $f_*([M])=0$, in the rational homology of the classifying space $H_*(B\pi)$.

Let $f$ and $g$ be two cusp forms of weights $k$ and $2$ respectively. Let $\rho_f$ (resp. $\rho_g$) be the $p$-adic Galois representations attached to $f$ (resp. $g$). We will present two theorems (one of them work in progress with M. Agarwal) towards the Bloch-Kato conjecture for the motives $ad^0 \rho_f(-1)$ and $\rho_f \otimes \rho_g(-k/2-1)$. The method of the proof involves constructing congruences among either modular forms on the symplectic group of genus 2 or modular forms on the unitary group $U(2,2)$.