These lectures will present a board survey of recent work on new q-series invariants of 3-manifolds labeled by Spin-C structures. While the original motivation for studying these invariants is rooted in topology, they exhibit a number of unexpected properties and connections to other areas of mathematics, e.g. turn out to be characters of logarithmic vertex algebras. The integer coefficients of these q-series invariants can be understood as the answer to a certain enumerative problem, and when q tends to special values these invariants relate to other invariants of 3-manifolds labeled by Spin and Spin-C structures.
On October 14, 2021, Ms. Angela Rodel, Executive Director of the Fulbright Bulgaria Program, and Ms. Maria Kostova, Program Officer, Bulgarian Grantees, visited the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences. They presented the work of the Bulgarian-American Fulbright Educational Exchange Commission and the opportunities it offers for scientists, researchers and PhD students. After the presentation, Ms. Rodel visited the International Center for Mathematical Sciences (ICMS), where she was hosted by the director of the Center - Prof. Oleg Mushkarov, and Prof. Lyudmil Katzarkov, scientific director of ICMS, who joined the event through zoom. Opportunities for cooperation and participation in Fulbright programs were discussed at the meeting.
We will start these series by reviewing the general framework of geometric Langlands correspondence, and state the main conjectures in the de Rham and Betti settings. We will also recall V. Lafforgue's theorem about the spectral decomposition in the classical Langlands over function fields. We will then proceed to the formulation of "restricted" Langlands correspondence, which unifies the different contexts. We will state the restricted version of the geometric Langlands conjecture, and explain its relation with the classical Langlands conjecture via the operation of categorical trace.
One of the key developments in combinatorics and algebra of recent years has been the discovery of Lefschetz principles beyond Hodge structures, resolving several long-standing conjectures. I will provide an overview of recent developments, and discuss joint work with Johanna Steinmeyer, Stavros Papadakis and Vasiliki Petrotou.
During the course of the last few years a number of startling connections between quantum invariants of knots and 3-manifolds and high-level number theory have emerged. Already the rigidity theorems of 3-dimensional hyperbolic topology, which have been known for many years, had a quite non-trivial arithmetic content, with the volume of every hyperbolic 3-manifold being linked via the dilogarithm to the so-called Bloch group and algebraic K-theory, and another connection comes from the Kashaev invariant, which is linked via his famous conjecture to the hyperbolic volume but also belongs to the so-called Habiro ring, which is a beautiful number-theoretical object that is not yet well known to number theorists.
In this talk, I will briefly introduce homological mirror symmetry for certain hypersurface singularities. I will introduce basic definitions, Berglund-Hubsch duality of invertible polynomials, and some known results. Then I will discuss several examples in detail.
In a joint work with Frank Calegari and Yunqing Tang, we use methods from transcendental number theory to prove a conjecture that goes back to Atkin and Swinnerton-Dyer, in a special case, and generalized by Mason to the following form: A vector-valued modular form on SL(2,Z) whose components have q-expansions with bounded denominators are exactly the ones for which the underlying representation of SL(2,Z) has a finite image with kernel containing the congruence subgroup of matrices reducing to the identity modulo some positive integer N. In this talk, I will outline the basic ideas of the proof of the conjecture, describe the relation to mathematical physics and the representation theory of vertex algebras, and explain how our result in particular recovers a completely new proof of the so-called "congruence property" in rational conformal field theory.
Now that we have set up the Frobenius Manifold of a singularity, we can see 2 more places where the steenbrink spectrum of a singularity appears in its singularity category. Namely, the non-commutative mixed hodge structure of a singularity and the dimensional properties of the category. The former appearance being somehow natural, and the latter somewhat mysterious. In this talk we'll conduct a surface level investigation of these appearances.
Let $C$ be a smooth projective curve of genus at least 2, and let $N$ be the moduli space of semistable rank-two vector bundles of odd degree on $C$. We construct a semi-orthogonal decomposition in the derived category of $N$ conjectured by Belmans, Galkin and Mukhopadhyay and by Narasimhan. It has blocks of the form $D(C_d)$ where $C_d$ are $d$-th symmetric powers of $C$, and the semi-orthogonal complement to these blocks is conjecturally trivial. In order to prove our result, we use the moduli spaces of stable pairs over $C$. Such spaces are related to each other via GIT wall crossing, and the method of windows allows us to understand the relationship between the derived categories on either side of a given wall. This is a joint work with J. Tevelev.
On August 20, 2021, Prof. Rene Mboro gave the talk "On determinantal cubic hypersurfaces" and prof. Rodolfo Aguilar gave the talk "Quantum representations of fundamental groups of curves with infinite image."