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."
This is a survey course on the relation of the invariants of 3 manifolds and singularity theory. Using the theory of differential equations we relate classical 3 dimensional theory with modern category theory. Applications to Birational geometry and uniformization will be discussed.
The International Center for Mathematical Sciences – Sofia (ICMS-Sofia) in collaboration with the Institute of the Mathematical Sciences of the Americas at the University of Miami (IMSA) offers three fellowships for WIMSA 2021 - 2022 Programme.
The goal of the meeting is to gather researchers interested in metric geometry of singularities and Lipschitz geometry. There will be two, three short courses and eight talks. The event will be held at the University of Chicago Center in Paris with some limited in person participation. All talks will be broadcasted through Zoom.
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