Introductory speech by Jean-Pierre Bourguignon Room: Centre de conférences Marilyn et James Simons

Some Applications of Non-Abelian Hodge Theory

• Takuro Mochizuki (RIMS, Kyoto University)

Room: Centre de conférences Marilyn et James Simons We shall discuss ongoing investigations about some applications of non-abelian Hodge theory to the study of meromorphic flat bundles and more general holonomic D-modules, inspired by Kontsevich. The plan is as follows. First, we discuss a generalization of Barannikov-Kontsevich theorem about twisted de Rham cohomology. Then, after reviewing irregular Hodge filtration which was motivated by the study of Kontsevich complexes, we explain its application to an evidence for a conjecture of Kontsevich about cohomologically rigid holonomic D-modules. We would also like to discuss a boundedness of families of some families of meromorphic flat bundles.

Geometric Quantization of General Kähler Manifolds

• Jørgen E. Andersen (Odense University)

Room: Centre de conférences Marilyn et James Simons We will consider Geometric Quantization on general Kähler phase spaces and propose a program for compatible constructions of the quantization of functions, the Hilbert space structure, and the dependence on the choice of the Kähler structure (generalized Hitchin connection), fixing only the underlying symplectic manifold and a prequantum line bundle. We will in particular see explicitly how the curvature of the phase space modifies the quantization.

Exponential Volumes in Geometry and Representation Theory

• Alexander Goncharov (Yale University)

Room: Centre de conférences Marilyn et James Simons Let S be a topological surface with holes. The moduli space parametrising hyperbolic structures on S with geodesic boundary, and a given set L of lengths of boundary circles carries the Weil-Peterson volume form. Its volume is finite. Maryam Mirzakhani proved remarkable recursion formulas for these volumes, related to several areas of Mathematics. In particular the volumes are polynomials in L. Their leading coefficients are the volumes studied by Maxim Kontsevich in his proof of Witten's conjecture. However for a surface P with polygonal boundary, e.g. just a polygon, similar volumes are infinite. We consider a variant of these moduli spaces, and show that they carry a canonical exponential volume form. We prove that exponential volumes are finite, and satisfies unfolding formulas generalizing Mirzakhani's recursions. There is a generalization of these moduli spaces for any split simple real Lie group G, with canonical exponential volume forms. When the modular group of the surface P is finite, the exponential volumes are finite for any G. We show that when P are polygons, they can be used to define a commutative algebra of positive Whittaker functions for the group G. We define the tropical limits of the exponential volumes. The tropical limits for surfaces S with holes and SL(2) lead to the volumes studied by Kontsevich in his proof of Witten's conjecture. The tropical limits of the algebra of positive Whittaker functions for any group G give the algebra of spherical functions for the group G(C). A part of the talk is based on the joint work with Zhe Sun.

Homological Mirror Symmetry and Quantum Link Invariants

• Mina Aganagić (UC Berkeley)

Room: Centre de conférences Marilyn et James Simons There is a new family of homological mirror pairs, for which homological mirror symmetry can be understood as explicitly as in the simplest known examples. They categorify braid group representation coming from quantum groups. One application is to categorification of quantum link invariants.

Perverse Sheaves and Resurgence

• Mikhail Kapranov (IPMU)

Room: Centre de conférences Marilyn et James Simons The Borel summation procedure, common in resurgence theory, translates irregular behavior governed by asymptotic series and Stokes phenomena, in terms of multivalued functions in the Borel plane, which can be studied via perverse sheaves and their generalizations. The talk proposes a framework for resurgence based on perverse sheaves which are algebras with respect to additive convolution. The singularity structure of a perverse sheaf on C is given by its spaces of vanishing cycles and transport maps between them along various paths. The concept of alien derivatives adapts naturally to this context. An algebra with respect to convolution gives, after Fourier transform, an algebra with respect to tensor product, so its Stokes matrices are algebra automorphisms, i.e., in case of a commutative algebra, coordinate changes. Joint work in progress with Y. Soibelman.

Motivic Invariants of Moduli of Irregular Parabolic Higgs Bundles and Bundles with Connection

• Yan Soibelman (Kansas State University & IHES)

Room: Centre de conférences Marilyn et James Simons Motivic integration was introduced by Maxim Kontsevich in 1995. About 2007 in the joint work with Maxim, and based on the ideas of motivic integration, we introduced the notion of motivic Donaldson-Thomas invariants of a 3-dimensional Calabi-Yau category endowed with stability condition. In my talk I will overview how those ideas were used in the joint project with Roman Fedorov and Alexander Soibelman. We computed explicitly motivic Donaldson-Thomas invariants of the moduli stacks of semistable irregular parabolic Higgs bundles and bundles with connection on a smooth projective curve. Here the words "irregular parabolic" mean that the Higgs field and the connection can have poles of arbitrary order at the fixed points, and their "irregular parts" preserve flags of given types attached to the points.

Birational and Singularity Invariants from nc Hodge Theory

• Tony Pantev (University of Pennsylvania)

Room: Centre de conférences Marilyn et James Simons I will explain how a natural amalgam of classical Hodge theory with the nc Hodge structures arising from Gromov-Witten theory gives rise to new additive invariants of smooth projective varieties called Hodge atoms. Combined with Iritani's blow-up formula, Hodge atoms provide obstructions to birational equivalence and novel invariants of singularities. I will discuss applications to the Lüroth rationality problem and singularity theory. This is a joint work with L.Katzarkov, M.Kontsevich, and T.Y.Yu.

Combinatorics and Geometry of the Amplituhedron

• Lauren Williams (Harvard University)

Room: Centre de conférences Marilyn et James Simons The amplituhedron is a geometric object introduced by Arkani-Hamed and Trnka to compute scattering amplitudes in N=4 super Yang Mills theory. It generalizes interesting objects such as cyclic polytopes and the positive Grassmannian. It has connections to tropical geometry, cluster algebras, and combinatorics (plane partitions, Catalan numbers). I’ll give a gentle introduction to the amplituhedron, then survey some recent progress on some of the main conjectures about the amplituhedron: the Magic Number Conjecture, the BCFW tiling conjecture, and the Cluster Adjacency conjecture.

$A$ Infinity Functor in Symplectic Geometry and Gauge Theory (remote)

• Kenji Fukaya (SCGP)

Room: Centre de conférences Marilyn et James Simons I want to explain several works in progress to use the notion of representability of $A$ infinity functor in the study of Floer theory in symplectic geometry and gauge theory.

Universally Counting Curves in Calabi-Yau Threefolds

• John Pardon (SCGP)

Room: Centre de conférences Marilyn et James Simons Enumerating curves in algebraic varieties traditionally involves choosing a compactification of the space of smooth embedded curves in the variety. There are many such compactifications, hence many different enumerative invariants. I will propose a "universal" (very tautological) enumerative invariant which takes values in a certain "Grothendieck group of 1-cycles". It is often the case with such "universal" constructions that the resulting Grothendieck group is essentially uncomputable. But in this case, the cluster formalism of Ionel and Parker shows that, in the case of threefolds with nef anticanonical bundle, this Grothendieck group is freely generated by local curves. This reduces the MNOP conjecture (in the case of nef anticanonical bundle and primary insertions) to the case of local curves, where it is already known due to work of Bryan--Pandharipande and Okounkov--Pandharipande.

An Update on SYZ Mirror Symmetry and Family Floer Theory

• Denis Auroux (Harvard University)

Room: Centre de conférences Marilyn et James Simons The Strominger-Yau-Zaslow approach to homological mirror symmetry starts from a Lagrangian torus fibration on the complement of an anticanonical divisor, and constructs the mirror as a moduli space of weakly unobstructed objects of the Fukaya category supported on the fibers. However, in the presence of holomorphic discs of negative Maslov index, the geometry of the mirror may be deformed beyond the familiar world of Landau-Ginzburg models. We propose a Morse-theoretic construction of the Fukaya-Floer algebra of a family of Lagrangian tori, which recovers a (suitably deformed) Cech model for the algebra of polyvector fields on the mirror, as well as a functor from Lagrangian sections of the SYZ fibration to modules over this algebra.

Holomorphic-Topological Twists and their Applications

• Davide Gaiotto (Perimeter Institute)

Room: Centre de conférences Marilyn et James Simons We review the holomorphic-topological twist of supersymmetric quantum field theories and some of their mathematical properties and applications. These include some general formality-like results or lack thereof, a discussion of categorical wall-crossing in four dimensional gauge theories and a categorical approach to the 't Hooft expansion.

Modular Forms and Differential Equations

• Don Zagier (MPI Bonn & ICTP)

Room: Centre de conférences Marilyn et James Simons The theory of automorphic forms originated in the late 19th and early 20th century (works of Klein, Fricke, Poincaré and many others) from the study of differential equations, but this aspect has become somewhat forgotten in the course of the years. In the lecture, I will talk about the many connections that exist between modular forms and differential equations of various types, especially linear (like the ones used in Apéry's famous proof of the irrationality of $\zeta(2)$ and $\zeta(3)$, or the "modular linear differential equations" that have become important in conformal field theory and the theory of vertex operator algebras) , but also non-linear. The latter include the so-called Chazy differential equation occurring in the theory of Painlevé equations and also various operators arising from the theory of Frobenius manifolds. I will talk about some of these connections and their applications.

Prospects for Spectral Mirror Symmetry

• Mohammed Abouzaid (Stanford University)

Room: Centre de conférences Marilyn et James Simons Fukaya categories are constructed from moduli spaces of discs of virtual dimension 0 and 1. The higher dimensional moduli spaces can in principle be used to define variants the Fukaya category which have coefficients in spectral rings, lifting the current construction over discrete rings. However, outside the setting of exact symplectic manifolds, the problem of curvature arises, and one needs to resolve the anomaly problem without appealing to the tools of ordinary algebra. I will discuss joint work with Blumberg on formulating this generalisation, with McLean and Smith on showing that Lagrangian Floer theory fits within this new framework, and with Bottman on Floer-theoretic constructions that yield mirror varieties in this setting.

Moduli Spaces of Points on the Projective Line and Other Varieties with Many Symmetries

• Yuri Tschinkel (SCGP & New York University)

Room: Centre de conférences Marilyn et James Simons I will discuss several recent results and constructions in equivariant birational geometry.

Periodic pencils of flat connections and their $p$-curvature

• Pavel Etingof (MIT)

Room: Centre de conférences Marilyn et James Simons A periodic pencil of flat connections on a smooth algebraic variety $X$ is a linear family of flat connections $\nabla(s_1,...,s_n)=d-\sum_{i=1}^r\sum_{j=1}^ns_jB_{ij}dx_i$, where $\lbrace x_i\rbrace$ are local coordinates on $X$ and $B_{ij}: X\to {\rm Mat}_N$ are matrix-valued regular functions. A pencil is periodic if it is generically invariant under the shifts $s_j\mapsto s_j+1$ up to isomorphism. I will explain that periodic pencils have many remarkable properties, and there are many interesting examples of them, e.g. Knizhnik-Zamolodchikov, Dunkl, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. I will also explain that in characteristic $p$, the $p$-curvature operators $\lbrace C_i,1\le i\le r\rbrace$ of a periodic pencil $\nabla$ are isospectral to the commuting endomorphisms $C_i^*:=\sum_{j=1}^n (s_j-s_j^p)B_{ij}^{(1)}$, where $B_{ij}^{(1)}$ is the Frobenius twist of $B_{ij}$. This allows us to compute the eigenvalues of the $p$-curvature for the above examples, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.

Shimurian Analogs of Barsotti-Tate Groups (remote)

• Vladimir Drinfeld (University of Chicago)

Room: Centre de conférences Marilyn et James Simons I will first recall Grothendieck's notion of n-truncated Barsotti-Tate group. Such groups form an algebraic stack over the integers. The problem is to give an illuminating description of its reductions modulo powers of p. A related problem is to construct analogs of these reductions related to general Shimura varieties with good reduction at p. Some time ago I formulated conjectures which address these problems. The conjectures have been proved by Z.Gardner, K.Madapusi, and A.Mathew. In particular, they developed a modern version of Dieudonné theory.

Geometry from Donaldson-Thomas invariants

• Tom Bridgeland (University of Sheffield)

Room: Centre de conférences Marilyn et James Simons Our aim is to use the DT invariants to of a CY3 triangulated category to define a geometric structure on the space of stability conditions. So far we only know how to do this in a few simple examples. In the talk I will describe the expected geometry, which involves a hyperkahler structure, and discuss the main class of examples, which are some kind of complexified Hitchin systems.

Counting in Calabi-Yau Categories

• Fabian Haiden (Syddansk Odense University)

Room: Centre de conférences Marilyn et James Simons I will discuss a replacement for homotopy cardinality in situations where it is a priori ill-defined, including Z/2-graded dg-categories. A key ingredient are Calabi-Yau structures and their relative generalizations. As an application we obtain a Hall algebra for many pre-triangulated dg-categories for which it was previously undefined. Another application is the proof of a conjecture of Ng-Rutherford-Shende-Sivek expressing the ruling polynomial of a Z/2m-graded Legendrian knot (which is part of the HOMFLY polynomial if m=1) in terms of the homotopy cardinality of its augmentation category. All this is joint work with Mikhail Gorsky.

Billiards, Arithmetic and Hodge Theory

• Curtis McMullen (Harvard University)

Room: Centre de conférences Marilyn et James Simons What are the slopes of periodic billiard paths in a regular polygon? We will connect this question and others to: - cusps of thin groups, - curves on Hilbert modular varieties, - heights from Jacobians with real multiplication, and - a spectral gap for the Galois orbits of triangles. Along the way we will encounter issues of chaos and decidability, first appearing in polygons with 7 and 12 sides.

Noncommutative Elliptic Poisson Structures on Projective Spaces

• Alexander Odesskii (Brock University)

Room: Centre de conférences Marilyn et James Simons We review noncommutative Poisson structures on affine and projective spaces over ${\mathbb C}$. This part of the talk is based on ideas of Maxim Kontsevich from his paper "Formal non-commutative symplectic geometry". We also construct a class of examples of noncommutative Poisson structures on ${\mathbb C} P^{n-1}$ for $n>2$. These noncommutative Poisson structures depend on a modular parameter $\tau\in{\mathbb C}$ and an additional descrete parameter $k\in{\mathbb Z}$, where $1 \leq k < n$ and $k,n$ are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras $q_{n,k}(\tau)$. This talk is based on a joint paper with Vladimir Sokolov.