Abstract: The talk will overview some geometric constructions aiming to define the notion of the absolute point and  the arithmetic over it (joint with A. Connes).

Abstract: We review how supersymmetric quantum mechanics naturally leads to several standard constructions in homological algebra. We apply these ideas to 2d Landau-Ginzburg models with (2,2) supersymmetry to discuss wall crossing. Some aspects of the web formalism are reviewed and applied to the categorification of the Cecotti-Vafa wall-crossing formula for BPS invariants. We then sketch the generalization to include twisted masses. In the final part of the talk we sketch how some of these ideas give a natural framework for understanding a recent conjecture of Garoufalidis, Gu, and Marino and lead to potentially new knot invariants. The talk is based on work done with Ahsan Khan and the final part is the result of discussions with Ahsan Khan, Davide Gaiotto,

and Fei Yan. 

Abstract: This talk is devoted to   the subring of  the complex  cobordism ring which is generated by almost complex manifolds with trivial odd Chern classes. The motivation for this study is the theory of hyper-Kähler manifolds. Our main result says that punctual Hilbert schemes of K3 surfaces and generalized Kummer manifolds generate multiplicatively this ring with rational coefficient. This is joint work  with Georg Oberdieck and Jieao Song. 

Abstract: I will discuss a new proof of the projectivity of the moduli space of semistable quiver representations for an acyclic quiver, avoiding methods from geometric invariant theory. Similar to GIT-free proofs of the projectivity of moduli spaces of semistable vector bundles on curves (due to Faltings and Esteves--Popa) one can produce a determinantal line bundle on the good moduli space, and prove its ampleness by using the modular interpretation (instead of appealing to a quotient description). I will give a gentle introduction to moduli of quiver representations and highlight the parallels (and differences) with moduli of vector bundles on curves. This is joint work with Chiara Damiolini, Hans Franzen, Vicky Hoskins, Svetlana Makarova and Tuomas Tajakka. 

Abstract: I will review several problems of mathematical physics, with the moduli spaces of solutions (to some partial differential equations modulo symmetries) used in constructing the invariants of smooth, symplectic, hyperkahler manifolds. I will then pass to the discussion of the spaces (really, virtual spaces), which facilitate/geometrize the construction of those invariants, yet the partial differential equations whose solutions would have the moduli parametrized by these spaces, are not known. Examples include the ``moduli spaces of crossed and folded instantons''. If time permits, I will explain the algebraic structures behind the topology of these moduli spaces in search of a moduli problem 

Abstract : Landau-Ginzburg models are a family of physical theories described by some polynomial (or  ‘potential’) characterized by having an isolated singularity at the origin. Often appearing in mirror-symmetric phenomena, they can be collected in higher categories with nice properties that allow direct computations. In this context, it is possible to introduce an equivalence relation between two different potentials called `orbifold equivalence’. We will present some recent examples of this equivalence, and discuss the computational challenges posed by the search of new ones. Joint work with Timo Kluck.

 Abstract: The talk is based on https://arxiv.org/pdf/2108.10644.pdf. See abstract there.

Abstract: The classical theory of integration concerns integrals of differential forms over domains of integration. In geometric terms, this corresponds to a canonical pairing between de Rham cohomology and singular homology. For varieties defined over the reals, one can make use of complex conjugation to define a real-valued pairing between de Rham cohomology and its dual, de Rham homology. The corresponding theory of integration, that we call single-valued integration, pairs a differential form with a `dual differential form’. We will explain how single-valued periods are computed and give an application to superstring amplitudes in genus zero. This is joint work with Francis Brown. 

Abstract: We will give an overview of results on isomonodromic deformations of a class of irregular systems which are non-generic, in the sense that the deformation parameters, which are the eigenvalues of the leading matrix at an irregular singularity, vary in a domain containing a "coalescence" set (a set where some eigenvalues merge). This has a counterpart in the coalescence of Fuchsian singularities when a Laplace transform is applied to the irregular system. Some applications will be sketched, which include Dubrovin-Frobenius manifolds and Painlevé equations. 

Abstract: Topological quantum field theories (TQFTs) often serve as a bridge between physics and mathematics. In this talk, I will illustrate how TQFTs that arise in physics can help to shed light on 1) the quantization of moduli spaces 2) quantum invariants of 3-manifolds, and 3) smooth structures on 4-manifolds.

Abstract: After some introduction, the first part of the talk will concern intersection theory of the divisor of special cubics inside the moduli space of cubic fourfolds. We will give some necessary conditions so that (up to) 20 divisor intersect each other, and we will describe the K3 surfaces associated to these cubics, in Hodge theoretical sense. We will then apply this construction to build new families of cubics with finite dimensional Chow motive, and of abelian type. Finally, we will consider some HyperKähler varieties associated to cubic fourfolds (the Fano varieties of lines inside the fourfold, the LLSvS 8 fold, etc.) and we will show that in some cases our preceding results imply that also these HK varieties have a finite dimensional, abelian Chow motive. 

Abstract:  We give the notion of a CLWX 2-algebroid and show that a  QP-structure (symplectic NQ structure) of degree 3  gives rise to a CLWX 2-algebroid. This is the higher analogue of the result that a  QP-structure of degree 2 gives rise to a Courant  algebroid. A CLWX 2-algebroid can also be viewed as a categorified Courant algebroid. We give a detailed study on the structure of a transitive Lie 2-algebroid and describe a transitive Lie 2-algebroid using a morphism from the tangent Lie algebroid TM to a strict  Lie 3-algebroid constructed from derivations. Then we introduce the notion of a quadratic Lie 2-algebroid and define its first Pontryagin class, which is a cohomology class in H^5(M). Associated to a CLWX 2-algebroid, there is a quadratic Lie 2-algebroid naturally. Conversely, we show that the first Pontryagin class of a quadratic Lie 2-algebroid is the obstruction class of the existence of a CLWX-extension. 

Abstract: This is a joint work with Roman Travkin and Alexander Braverman. D.Gaiotto conjectured that the category of finite dimensional representations of U_q(gl(N-1|N)) is equivalent to the category of q-monodromic GL(N-1,C[[t]])-equivariant perverse sheaves on the determinant line bundle on the affine

Grassmannian of GL(N). I will explain an approach to this via the category of factorizable sheaves. 

Abstract: I will review some recent results on existence and obstructions for semiorthogonal decomposition of varieties with ordinary double points. The constructions originate from geometry of blow ups and Sarkisov links, while the obstructions come from the negative K-theory groups. The theory is primarily developed for nodal Fano threefolds, with the index two case completely understood.  The talk is based on several papers joint with Kalck, Karmazyn, Kuznetsov, Pavic. 

Abstract: The talk will focus on finite-dimensional DG algebras and categories of perfect complexes over such DG algebras. These categories can be considered  as proper derived noncommutative schemes. We are going to discuss basic properties of these noncommutative schemes and their geometric realizations. 

Abstract:  In the talk we will introduce the main players of the geometric Langlands equivalence (with its different flavors: de Rham, Betti and l-adic) and explain its connections with the classical Langlands conjectures.  

Abstract: Using the Gelfand-Kapranov-Zelevinski system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana amplitudes with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel Γˆ-class evaluation in the ambient spaces of the mirror, while the imaginary part of the amplitude in this regime is determined by the Γˆ-class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the amplitudes, when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogenous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the amplitude as well as other master integrals with raised powers of the propagators in very short time to very high numerical precision for all values of the physical parameters. Using a recent p-adic analysis of the periods we determine the value of the maximal cut equal mass four-loop amplitude at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

Abstract: Quantum cohomology is a deformation of the cohomology of a projective variety governed by counts of stable maps from a curve into this variety. Quantum K-theory is in a similar way a deformation of K-theory but also of quantum cohomology, It has recently attracted attention in physics since a realization in a physical theory has been found. Currently, both the structure and examples in quantum K-theory are far less understood than in quantum cohomology. We will give an introduction into the properties of quantum K-theory and we will discuss the examples of projective space and the quintic

hypersurface in P^4. 

Abstract: This talk will discuss a recent result, joint with F. Pop, describing the absolute Galois group of the rationals in terms of fundamental groups of complements of hyperplane arrangements.

After a brief introduction to the subject, most of the talk will focus on the key step in the argument, which gives a description of hyperplanes and their incidence structure in terms of certain explicit group-theoretical relations in fundamental groups.

Abstract:. Mutations of potentials of toric surfaces exhibit a striking phenomenon discovered about 10 years ago by Galkin and Usnich for the case of P^2, and by Cruz Morales and Galkin for the case of other Del Pezzo surfaces. Several important advances in 4-dimensional symplectic geometry from the last decade are directly or indirectly related to these mutations. In the talk I'll discuss a recent joint work with Joé Brendel and Felix Schlenk continuing the series of symplectic applications of the mutations with a construction of exotic embeddings between symplectic cubes. 

Abstract: Mirror symmetry was a phenomenon discovered by physicists around 1989: they observed that certain kinds of six-dimensional geometric objects known as Calabi-Yau manifolds seemed to come in pairs, with a strange relationship between different kinds of geometric objects on the pairs. Since then, the subject has blossomed into a vast field, with many different approaches and philosophies. I will give a brief introduction to the subject, and explain how one of these approaches, developed with Bernd Siebert, has led to a general construction of mirror pairs.

Abstract: One of the main points of Grothendieck's Esquisse d'un Programme was about giving a combinatorial/topological description of the absolute Galois group of the field rational numbers. Both GT (Grothendieck-Teichmuller Group) and I/OM (Ihara Question/Oda-Matsumoto Conjecture) are developments which aimed at Grothendieck's quest. In this talk I will give a quick introduction to the two questions and their relationship, and present some recent results and work in progress, the latter being collaboration with Adam Topaz.

Abstract: Motivic Donaldson-Thomas invariants were introduced in my joint work with Maxim Kontsevich more than 10 year ago.

Originally they were defined with the purpose to give a meaning of the ``number" of semistable objects  in a 3-dimensional Calabi-Yau category which have  fixed class in the Grothendieck group. Since their birth in 2008 motivic DT invariants have found many applications in very distant areas of mathematics and physics, including representation theory of quivers, integrable systems, Mirror Symmetry as well as in the gauge theory or theory of supersymmetric black holes.

In this talk I would like to explain my two recent joint works with Roman Fedorov and Alexander Soibelman on computation of motivic DT invariants, but this time  in the framework of 2-dimensional Calabi-Yau categories. More precisely, we considered the categories of Higgs bundles and bundles with connections on a smooth projective curve over a field of characteristic zero (including the parabolic case). Among other things, our work upgrades to the level of motivic invariants some earlier

results of Mellit, Mozgovoy and Schiffmann about the number of points of the moduli spaces (stacks) of Higgs bundles on the curve over a finite field.  It also has some interesting speculative connections to the affine Grassmannians and Kac-Moody algebras over local fields.

Abstract: An involution of a projective hyperkähler manifold is called antisymplectic if it acts as (-1) on the space of global holomorphic 2-forms. I will present joint work with Laure Flapan, Kieran O'Grady, and Giulia Saccà on antisymplectic involutions associated to polarizations of degree 2; in particular, how to determine the number of connected components of the fixed loci and to study their geometry.

Abstract: Fano and Calabi-Yau varieties play a fundamental role in algebraic geometry, differential geometry, arithmetic geometry, mathematical physics, etc. The notion of log Calabi-Yau fibration unifies Fano and Calabi-Yau varieties, their fibrations, as well as their local birational counterparts such as flips and singularities. Such fibrations can be examined from many different perspectives. The purpose of this talk is to introduce the theory of log Calabi-Yau fibrations, to remind some known results, and to state some open problems.  

Abstract: Categories with actions of an (affine) braid group arise in representation theory, algebra and algebraic geometry. I will review some of these constructions and discuss their intriguing properties and generalizations. Based on joint works with Anno, Losev, Mirkovic and Riche. 

Abstract: I will talk on a project which aims  to develop a unified theory of modular and automorphic forms. It  encompasses most of the available theory of modular forms in the literature, such as those for congruence groups, Siegel and Hilbert modular forms, many types of automorphic forms on Hermitian symmetric domains, Calabi-Yau modular forms, with its examples such as Yukawa couplings and topological string partition functions, and even go beyond all these cases. Its main ingredient is the so-called ‘Gauss-Manin connection in disguise’. The talk is based on the author's book with the same title.

It is  available on his webpage.

Abstract: Matroids are combinatorial gadgets that reflect properties of linear algebra in situations where this latter theory is not available. This analogy prescribes that the moduli space of matroids should be a Grassmannian over a suitable base object, which cannot be a field or a ring; in consequence usual algebraic geometry does not provide a suitable framework. In joint work with Matt Baker, we use algebraic geometry over F1, the so-called field with one element, to construct such moduli spaces. As an application, we streamline various results of matroid theory and find simplified proofs of classical theorems, such as the fact that a matroid is regular if and only if it is binary and

orientable.

We will dedicate the first half of this talk to an introduction of matroids and their generalizations. Then we will outline how to use F1-geometry to construct the moduli space of matroids. In the last part, we will explain why this theory is useful to simplify classical results in matroid theory. 

Abstract: The talk is a survey on my Torelli type conjecture for  isolated hypersurface singularities. It says that such a singularity is determined up to coordinate changes by its Brieskorn lattice + topological data. It is proved for the  singularities with modality less or equal to 2. An infinitesimal and a variational Torelli result hold, too. A stronger version uses a classifying space for Brieskorn lattices and a moduli space for marked singularities.

Abstract: In 2008 Simpson conjectured the existence of a holomorphic Lagrangian foliation in the de Rham moduli space of holomorphic $G$-connections for a complex reductive group $G$.

The purpose of the talk is to establish the existence of a holomorphic Lagrangian foliation in the de Rham moduli space of holomorphic $SL_2(C)$-connections defined on a smooth connected projective curve $C$ of genus at least 2. The conjectural holomorphic Lagrangian foliation does not seem to constitute a holomorphic Lagrangian fibration. I will present an algebraic geometry description of the Lagrangian correspondence of conformal limits, based on the work of Simpson. This talk is based on joint work with Jennifer Brown and Motohico Mulase.

Abstract: A large part of the lecture concerns the introduction of: Isomonodromy, the formal and analytic theory of irregular singularities of linear differential equations, Stokes matrices and Painlevé equations. How this works in detail is demonstrated for the classical Painelevé equations $P_6$ and  $P_1$.

Case studies are concerned with the sport of finding new Painlevé type equations. Some known ones are rediscovered and for some ``new cases'' computations and formulas are shown. 

Abstract: We describe tropical hyperelliptic spin curves, i.e. tropical hyperelliptic curves equipped with a theta characteristic. We provide a formula for their combinatorial rank, and show that, modulo 2, the rank is a quadratic form as it happens for Riemann surfaces. The Z2-quadratic forms arising in this way only depend on the genus of the curve. We relate all this to moduli of spin curves. This is joint work with Marco Pacini. 

Abstract: We will describe the Langlands program, in its arithmetic and geometric incarnations, as a conjectural non abelian analogue of well known  "abelian" results: class field theory on the arithmetic side, and the  Abel-Jacobi theory of Riemann surfaces and their Jacobians on the geometric side. One powerful approach to the geometric Langlands conjectures involves abelianization via Hitchin's integrable system and  its spectral curves. Input from physics suggests that this geometric Langlands conjecture can be viewed as a statement in quantum field theory. This has a classical limit which has now been proved, at least generically. The relationship between the classical and quantum versions is deep and mysterious. The quantum version can of course be studied as a deformation of the classical one.  But there is tantalizing evidence - from quantum field theory as well as non abelian Hodge theory - that the full quantum version can also be understood as a twistor rotation of the classical version. I will try to present and relate the various strands of this story, and illustrate it in a couple of concrete low dimensional cases.