Abstracts

The Deligne conjecture, many times a theorem, states that for a dg category C, the dg endomorphisms End(Id_C) of the identity functor - that is, the Hochschild cochains - carries a natural structure of 2-algebra. When C is endowed with a Calabi-Yau structure, then Hochschild cochains and Hochschild chains are identified up to a shift, and we may transport the circle action from Hochschild chains onto Hochschild cochains. The cyclic Deligne conjecture states that the 2-algebra structure and the circle action together give a framed 2-algebra structure on Hochschild cochains. We establish the cyclic Deligne conjecture,  as well as a variation that works for relative Calabi-Yau structures on dg functors D --> C, more generally for functors between stable infinity categories. We discuss examples coming from oriented manifolds with boundary, Fano varieties with anticanonical divisor, and doubled quivers with preprojective relation. This is joint work with Nick Rozenblyum.

This talk is based on the joint work with Eric T. Mortenson. All modular invariant representations of Kac-Moody algebras can be conjecturally classified by admissible highest weight representations (Kac-Wakimoto conjecture); for A_1^{(1)} the conjecture is known to be true. We study the string functions of admissible highest weight representations of A_1^{(1)}, which have certain rational levels and also have interpretation as the characters of the generalized parafermionic theories. We demonstrate that 1/2-level string functions are mixed mock modular forms, and their mock modular part can be evaluated in terms of Ramanujan's mock theta functions. In addition, we show that the negative level string functions can be evaluated in terms of false theta functions.

It is well-known that many interesting examples of discrete Painlevé equations can be obtained from QRT mappings via a deautonomization process. There is an algebraic approach of B. Grammaticos, A. Ramani, and their collaborators, that uses the notion of singularity confinement. Recently, together with A.S. Carstea and T. Takenawa, we introduced the notion of geometric deautonomization of QRT maps based on a choice of a fiber in the QRT fibration. In this talk we present some examples of geometric deautonomization using a particular QRT map that appears in discretization of Nahm equations.

In this talk, we will discuss a classification of finite subgroups of the group of autoequivalences of the derived category of coherent sheaves on a general K3 surface. We will provide the necessary background on Bridgeland stability conditions and mirror symmetry. This is a joint work with Kuan-Wen Lai.

A compact complex manifold X is called homogeneous if the group of its biholomorphic automorphisms acts on X transitively. Historically, the first examples of non-Kaehler compact complex manifolds were constructed in the class of homogeneous manifolds (for instance, Hopf manifolds, Calabi–Eckmann manifolds, Iwasawa manifold etc). In my talk I will survey known results (including some very recent ones) regarding the structure of automorphism groups of compact complex homogeneous manifolds.

We consider Liouville conformal blocks with degenerate field insertions and one operator in an irregular representation of the Virasoro algebra. Such blocks satisfy modified BPZ equations and admit integral representations with integration over non-compact Lefschetz cycles. We show how they satisfy an irregular version of KZ equations from which the modified BPZ equations follow. Finally, we argue that our irregular KZ equations follow from irregular Kac-Moody representations.

The theory of geography of log models plays an important role in the study of birational geography, especially the Sarkisov program. I will talk about the structure on the geography, and a similar phenomenon on the moduli of Landau-Ginzburg models.

I will describe the technology of homotopical enhancements for categories based on Grothendieck's idea of a "derivator". This technology is very close to the usual category theory, and obviates the need for model categories, simplicial sets and similar technical gadgets.
 

In this talk we will introduce new birational invariants. Many examples of obstruction to rationality and G rationality will be considered.

In this talk I will discuss a construction of a parameterizing space for ideal sheaves corresponding to involutive and formally integrable non-linear PDEs. We will indicate how this object might be used to extend calculations of deformation invariants to the setting of non-linear PDE geometry. 

A set of natural numbers A is called primitive if no element of A divides any other. Let Omega(n) be the number of prime divisors of n counted with multiplicity. Let f_z(A) be the sum of z^(Omega(n))/(n (log n)^z) over elements of A. Erdős proved in 1935 that f_1(A) is uniformly bounded over all choices of primitive sets A. It turns out that the same fact is true for f_z(A), when 0 < z < 2. We will discuss the supremum of  f_z(A) over all primitive sets, and the limit f_z(A) as z approaches 0, which generalize the notion of Dirichlet density. Also we will discuss the asymptotic of f_z(A), when A is the set of numbers with k prime divisors, as k approaches infinity.

A quadric bundle is a flat morphism with fibers isomorphic to quadric hypersurfaces. I will talk about the operations of hyperbolic reduction and extension for quadric bundles and check that they do not change the main invariants of the quadric bundle such as the discriminant divisor and the Brauer data. I will also show that these operations can be used to transform any quadric bundle over a projective plane to a quadric bundle contained in the projectivization of a direct sum of line bundles and any quadric bundle over a higher-dimensional projective space to a quadric bundle contained in the projectivization of a vector bundle that enjoys some cohomology vanishing.

Let A be any commutative associative ring. C. Contou-Carrére defined the bimultiplicative antisymmetric pairing on the  multiplicative group of the ring of Laurent series A(t). This pairing  is functorial with respect to the ring A. This pairing is called now the Contou-Carrére symbol. If the ring A is a field, then the Contou-Carrére symbol coincides with the tame symbol. Moreover, using the Contou-Carrére symbol, one can construct many other pairings known in the number theory such as the residue of a differential form or the Witt pairing. D. Osipov and X. Zhu for n = 2, and later S. Gorchinskiy and D. Osipov for arbitrary n, defined the n-dimensional  Contou-Carrére symbol. They proved that the higher-dimensional Contou-Carrére symbol is invariant under continuous automorphisms of the ring A(t). This invariance propery  generalizes the known analogous properties of the tame symbol, the residue and the Witt pairing. We prove that every higher-dimensional n+1-multiplicative pairing which is functorial with respect to the ring A and satisfies the above invariance property is the power of the n-dimensional Contou-Carrére symbol up to a sign.

The integral homology of a rational homogeneous variety has a free additive basis of Schubert classes. A Schubert class is called rigid if it can only be represented by Schubert varieties. In this talk, I will discuss the rigidity problem in partial flag varieties and its generalizations.

I will recall how B-type topological string appeared and how it is related to Kodaira-Spencer equations and BCOV theory. Then I will describe generalized complex structures (in the sense of Barannikov-Kontsevich) and how KS euations and BCOV theory is generalized (in particular, I will recall LG generalization that appears in mirror symmetry for toric manifolds).

Then I will explain geometry of Nielsen-Berkovits view on supersymmetry in d=10, and show how derived manifold appears there. I propose to treat superstrings (in the limit when string mass goes to infinity) as a generalized BCOV theory for such manifold and give supporting arguments for this conjecture.

In this talk, I will introduce the machinery of toric exoflops and explain how they can be used to study derived categories of coherent sheaves. In particular, I will be discussing how to construct categorical resolutions to the derived categories associated to complete intersections in certain toric varieties.

This will be a nontechnical talk about our work with David Kazhdan on the spectrum of the Laplace and Hecke operators in the span of Eisenstein series. This very classical problem in the spectral theory of automorphic forms goes back to at least Langlands, who also introduced the analytic techniques that allow to find the spectrum in principle. To replace an in-principle answer by a concrete answer, we found it very useful to have a certain geometric interpretation of the multivariate contour integrals in Langlands' setup. My goal in this talk will be to explain this geometric interpretation.
 

We discuss the geometric characterisation of arithmetic varieties (i.e., a Belyi-type theorem) for smooth complete intersections of general type in weighted projective spaces and generalised Grassmannians. We also propose an explicit approach to its generalisation to complete  intersections in weighted generalised Grassmannians.

We are going to present in this talk two sub-gradient estimates for the quaternionic contact (qc) heat equation on a compact qc manifold of dimension 4n+3, provided some positivity conditions are satisfied. These are qc versions of the prominent Li-Yau gradient estimate in Riemannian geometry. Another goal of this talk is to exhibit two Perelman-type entropy formulas for the qc heat equation on a compact qc-Einstein manifold of dimension 4n+3 with non-negative qc scalar curvature (e.g. compact 3-Sasakian manifold), as well as an integral sub-gradient estimate for the positive solutions of the qc heat equation.

I will discuss Fano threefolds with terminal singularities in relation to rationality question. In particular, I will present several examples of non-rational terminal Fano threefolds of large Fano index.

I will review the concept of quantum trace for hyperbolic knot complements in S^3 and its relation with SL(2,C) Chern-Simons theory and the length conjecture.

We discuss construction of a derived Lagrangian intersection theory of closely related moduli spaces of perfect complexes, with support on divisors on compact Calabi Yau threefolds. Our goal is to compute deformation invariants associated to a fixed linear system of divisors in CY3. We degenerate the CY3 into a normal-crossing singular variety composed of Fano threefolds meeting along their anti-canonical divisor. The derived Lagrangian intersection of the corresponding “Fano moduli spaces” provides one with categorification of DT invariants over the special fiber (of degenerating family). We then elaborate on the existence of a “Getzler-Gauss-Manin” connection on the Z/2-graded periodic cyclic homology associated to these categorified DT invariants. The latter provides the deformation invariance property, needed to relate the categorical DT invariants of the special fiber to the one over the generic fiber. If there is time, we also show how in terms of “non-derived” virtual interscetion theory these DT invariants relate to counts of D4-D2-D0 branes which are expected to have modularity property by the S-duality conjecture. This talk is based on joint work with Vladimir Baranovsky, Ludmil Katzarkov and Maxim Kontsevich.

We reformulate the time-independent Schrödinger equation as a Maurer-Cartan equation on the superspace of eigensystems of the former equation. We then twist the differential so that its cohomology becomes the space of solutions with a set energy. A perturbation of the Hamiltonian corresponds to a deformation of the twisted differential, leading to a simple recursive relation for the eigenvalue and eigenfunction corrections.

The moduli spaces of semistable Higgs bundles on a curve have been inspiring a lot of mathematicians and physicists for decades. Among others, the studies on non-abeilian Hodge theory, the P=W  conjecture, the SYZ style topological Mirror symmetry for the Hitchin fibrations, and the Langlands program have prompted the fascinating interactions between mathematicians and physicists over the years. In this talk, I would like to speak about interesting analogues of them on a surface, strongly motivated by the Kapustin-Witten and Vafa-Witten theories from topologically twisted versions of N=4 super Yang-Mills one.

In this talk, I will start with the review of holomorphic Ray-Singer torsion, BCOV torsion for compact Kahler manifolds, the corresponding anomaly equations for family case and physical meaning. Then I will switch to the Landau-Ginzburg model which is defined via a pair (X,f). I will focus on the simple case, where X=C^n, and f is a non-degenerate quasi-homogeneous polynomial. For this part, I will talk about the twisted laplacian, Hodge theorem, heat kernel analysis, and construction of LG type torsion invariant. Finally, I will end with some related work on this topic.

In this talk I will present joint work with Artan Sheshmani (BIMSA). The goal of the talk is to discuss the relationship between possible choices of tensor category structures one can put on the derived category of a smooth projective variety over a field k. We will see that under the hypothesis of the Bondal-Orlov reconstruction theorem one is able to deduce an uniqueness result for the choice of tensor category structures. We will also discuss the relative version of this theorem where the base is a qcqs scheme S by means of dg-categorical enhancements of both the derived categories and the bifunctors appearing on each tensor structure. 

The Jordan property of an infinite group G informally states that finite subgroups in G are "not very complex", and Jordan constants measure this "complexity". It turns out that many infinite groups of geometric nature satisfy this very property, for example, linear algebraic groups.  In 2009, J.-P. Serre proved that the Cremona group of rank 2 over the field of complex numbers satisfies the Jordan property, and in 2016, E. Yasinsky computed Jordan constants of this group over algebraically closed fields, over fields of real and rational numbers. I will explain how to generalize the latter result to the case of arbitrary fields of characteristic 0.

 In this talk we will discuss the most symmetric smooth cubic surfaces over a field of arbitrary characteristic. We present the classification of the largest automorphism groups of smooth cubic surfaces over arbitrary fields. We give a full description of smooth cubic surfaces with action of the largest automorphism groups.