Seminário das Sextas, 24/11

Hidden Markov models

13h-14h | Sérgio Loiola - [hidden Markov models](

Higgs Bundles and Mirror Symmetry

14h-16h | Lucas Branco

The moduli space M(G) of Higgs bundles for a complex reductive group G on a compact Riemann surface carries a natural hyperkahler structure and it comes equipped with an algebraically completely integrable system through a flat projective morphism called the Hitchin map. Motivated by mirror symmetry, I will discuss certain complex Lagrangians (BAA-branes) in M(G) coming from real forms of G and give a proposal for the mirror (BBB-brane) in the moduli space of Higgs bundles for the Langlands dual group of G.  In this talk, I will focus on the real groups SU^*(2m), SO^*(4m) and Sp(m,m). Higgs bundles for these groups have non-reduced spectral curves and we are led to describe certain subvarieties of the moduli space of rank 1 torsion-free sheaves on ribbons. If time permits we will also discuss another class of complex Lagrangians in M(G) which can be constructed from symplectic representations of G.

Matrix factorizations, part I

 16h | Victor el Adji

(based Tobias Dyckerhoff's thesis)
We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field. We deduce a quasiequivalence between the category of matrix factorizations and the dg-derived category of an explicitly computable dg algebra. Building on this result, we employ a variant of Töen’s derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of these categories. Finally, we give interpretations of the results of this thesis in terms of noncommutative geometry based on dg categories.