AGNT Ical Atom

Topological recursion is a remarkable universal recursive procedure that has been found in many enumerative geometry problems, from combinatorics of maps, to random matrices, Gromov-Witten invariants, Hurwitz numbers, Mirzakhani’s hyperbolic volumes of moduli spaces, knot polynomials. A recursion needs an initial data: a spectral curve, and the recursion defines the sequence of invariants of that spectral curve. There is a duality in topological recursion which allows one to obtain closed formulas for the invariants of the recursion and which has implications in free probability theory and integrable hierarchies. In the talk I will survey recent progress in the topic with the examples from Hurwitz numbers theory, Hodge integrals and combinatorics of maps.

The talk is based on the joint works with A. Alexandrov, P. Dunin-Barkowski, M. Kazarian and S. Shadrin.

Lara—Srinivas—Stix, building on joint work with Esnalut, have recently shown that the etale fundamental group of a connected proper scheme over an algebraically closed field is topologically finitely presented, thus answering a question raised in SGA. The proof relies on a finite presentation criterion of Lubotzky for profinite groups, resolutions of singularities/alterations, a theorem of Deligne—Ilusie on the Euler characteristic, as well as other modern and classical results in (arithmetic) algebraic geometry.

Part of the talk is expository: I will explain how supersingular isogeny graphs can be used to construct cryptographic hash functions and survey some of the rich mathematics involved. Then, with this motivation in mind, I will discuss two recent theorems by Jonathan Love and myself. The first concerns the generation of maximal orders by elements of particular norms. The second states that maximal orders of elliptic curves are determined by their theta functions.

The lecture will start with a few useful (and probably new!) theorems on adic completion of commutative rings and modules. Then I will discuss derived adic completion, in its two flavors: the idealistic and the sequential. The weak proregularity (WPR) condition on an ideal \a in a ring A, which is a subtle generalization of the noetherian condition on the ring A, is a necessary and sufficient condition for the two flavors of derived completion to agree. WPR occurs often in the context of perfectoid theory, and I will finish the talk with theorems relating WPR to prisms.

Typed notes available soon at the bottom of this web page:

https://sites.google.com/view/amyekut-math/home/lectures

Equidistribution of “special” points is a theme of both analytic and geometric interest in number theory: in this talk I plan to deal with the case of CM points on Shimura curves.

The first part will be devoted to a geometric description of the aforementioned curves and of their moduli interpretation.Subsequently, I plan to sketch an equidistribution result of reduction of CM points in the special fiber of a Shimura curve associated to a ramified quaternion algebra. Time permitting, I will mention how Ratner’s theorem and subconvexity bounds on Fourier coefficients of certain theta series can be used to obtain two different equidistribution results.

In this talk, I will discuss the irreducibility problem of Severi varieties on toric surfaces. The classical Severi varieties were introduced by Severi almost 100 years ago in the context of Severi’s attempt to provide an algebraic treatment of the irreducibility problem of the moduli spaces of curves. Although, the irreducibility of the moduli spaces was achieved algebraically by Deligne and Mumford in 1969 using completely different techniques, Severi varieties remained in the focus of study of many algebraic geometers including Harris, Fulton, Zariski and others.

In this talk I will present the main result of my M.Sc. Thesis providing a complete description of the irreducible components of the genus-one Severi varieties on toric surfaces.

This work was done under the supervision of Professor Ilya Tyomkin.

Many estimates for the “size” of a subset of the natural numbers (a “size” usually expressed by some notion of density) come from “local” conditions, like reduction modulo prime powers. The idea can be formalized in terms of the Haar measure on the profinite completion of Z or, in a more refined way, via distributions on this profinite ring. This approach can be easily generalized by replacing Z with the ring of S-integers of any global field. In this talk (based on a number of joint works with L. Demangos and F.M. Saettone), I will discuss how to use these ideas to extend classical results and reformulate long standing conjectures in profinite terms.

The stable module category is obtained from the category of modules over a ring by factoring out the projective modules. In this setting, the syzygy of a module becomes a well defined functor, so for instance, the classical Hilbert’s syzygy theorem, can be stated as saying this functor is nilpotent. In this talk we present some new properties of the syzygy functor over a commutative noetherian ring. We then explain how to associate to the stable category a stabilization, obtaining the singularity category of a ring (or a scheme). Finally, we explain how relations between the stable category and the singularity category are related to some homological conjectures in noncommutative algebra.

With each simple connected graph $G$ with $n$ vertices one can associate a generalized configuration space $Conf_{G}(n,X)$ consisting of $n$ points $(p_1,\ldots,p_n)$ on $X$, with $p_i\neq p_j$ whenever vertices $i$ and $j$ are connected by an edge. For $X=\mathbb{C}$ the generalized configuration space admits a compactification that coincides for a complete graph with Deligne-Mumford compactification of moduli spaces of rational curves with $n$ marked points. The latter is known under the name modular compactification. I will explain what kind of natural algebraic structure exists in the union of these spaces and how one can extract information about the Hilbert series of cohomology rings for different collections of graphs. Surprisingly, the same method can be used to obtain the generating series for different combinatorial data assigned with a graph: such as the number of Hamiltonian paths, Hamiltonian cycles, Acyclic orientations and Chromatic polynomials. The talk is based on the joint work with my student D.Lyskov: https://arxiv.org/abs/2406.05909