The classical nonabelian Hodge correspondence establishes an equivalence between certain categories of flat bundles and Higgs bundles on smooth projective varieties. I will describe an extension of this result to twisted vector bundles.

This is a quick survey of Feynman's asymptotic formula for m-point functions as a sum over graphs.

I will present formal proofs of selected results discussed in preceding seminar talks using the Coq library UniMath.

In this partially expository talk we discuss some recent progress and open problems concerning real analytic Levi flat hypersurfaces with singularities. We prove that the Levi foliation of such a hypersurface extends as a holomorphic web to a full neighborhood of singularity. This is a joint work with R. Shafikov (Comment. Math. Helvet. '15).

For a smooth manifold M with a Lie group G action, we can define equivariant cohomology based on differential forms on M. That is Weil model. This construction is analogous to Borel construction on the level of differential forms. The Cartan model is then derived from the Weil model. The Cartan model provides an explicit way to compute equivariant cohomology. We will introduce both models and prove that they are equivalent.

This is the second part of this talk.

In this talk, I will introduce the structure of finite real spectral triple first, and then I will focus on how to encode those data of a finite real spectral triple inside the so-called "Krajewski diagram".

This talk is continuation of last week's talk with the same title.

A model example of a $q$-complete (in the sense of Grauert) manifold is the complement of a projective variety of codimension $q$. In particular, 1-complete manifolds are Stein. Our main results states that the top non-zero integer cohomology class of such a manifold can be represented by analytic cycles. This is a joint work with F. Forstneric and J. Smrekar (Geometry & Topology '16).

I am interested in approximation theory, in particular, on a closed subset of an open Riemann surface R. During the talk I will focus on Arakelyan's theorem and its topological conditions and explain that, despite the fact that approximation is not always possible for a closed subset E of R, one can construct a large closed subset of E on which approximation is possible in a strong sense. Also, I will talk about pole-free approximation by meromorphic functions with respect to spherical distance. Finally, I will present an extension of the Schwarz reflection principle for open bordered Riemann surfaces.

I will talk about mod $p$ cohomology of metacyclic groups. A metacyclic group is an extension of a cyclic group by another cyclic group. Mod $p$ cohomology rings of metacyclic groups are computed by Huebschmann using homological perturbation theory. Homological perturbation theory allows one to explicitly construct projective resolutions in an inductive fashion. I will describe this method and possibly relate it to other methods existing in the literature.