Abstract:Mirror symmetry is a phenomenon predicted by string theory in physics. It allows one to translate questions in symplectic geometry to questions in complex geometry, and vice versa. The homological mirror symmetry program interprets mirror symmetry within the unifying categorical framework of derived noncommutative geometry. After introducing these ideas, I will describe an approach to a theory of Kähler metrics in derived noncommutative geometry. We will see how this leads to (i) a non-Archimedean categorical analogue of the Donaldson-Uhlenbeck-Yau theorem, inspired by symplectic geometry, and (ii) the discovery of a refinement of the Harder-Narasimhan filtration which controls the asymptotic behavior of certain geometric flows. This talk is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.

Abstract: In 2009, Eisenbud and Schreyer prove that extremal rays of Betti cone over a polynomial ring are spanned by Betti diagrams of pure Cohen-Macaulay S-modules, where S={\sf k}[X_1,\dots, X_n]. In this talk, we discuss Boij-S\ddot{\text{o}}derberg theory for standard graded {\sf k}-algebras. We note the obstacles in using their techniques in the general situation and identify classes of rings where we can prove some of these results.

Abstract: We shall consider a dynamic stochastic block model with single change point. An easily implementable algorithm based on maximum pseudo-likelihood method and spectral clustering will be proposed for estimating the change point. We shall also estimate the edge-probability matrices and community structures before and after the change point. The convergence rate and asymptotic distribution for these estimators will be discussed and compared with other existing works in the literature. This is joint work with Moulinath Banerjee and George Michailidis.

Abstract: Let p be an odd prime number and K/F be a quadratic extension of p-adic fields. Say that an irreducible representation of GL(n,K) is distinguished by GL(n,F) if its vector space carries a GL(n,F)-invariant nonzero linear form. Any distinguished representation is isomorphic to the contragredient of its Gal(K/F)-conjugate, but the converse is not true. We will explain how to canonically associate to any Gal(K/F)-selfcontragredient supercuspidal representation of GL(n,K) a finite tamely ramified extension T of F and a character of the multiplicative group of T, by using Bushnell-Kutzko’s theory of types, and how to get a necessary and sufficient condition on this character for this supercuspidal representation to be distinguished.