A tropical approach to the Gamma conjectures

The Gamma conjectures are a collection of conjectures relating the asymptotic behaviour of Iritani's integral structure on quantum cohomology to classical topology via a twisted square root of the Todd class (the Gamma class). Via mirror symmetry, the conjectures can be related to periods of the mirror varieties. I will present joint work with Ganatra–Iritani–Sheridan which reproves the most basic of the Gamma for Calabi–Yau hypersurfaces via a tropical approach. The basic idea is that the leading term of the asymptotic expansion of the volume of the real locus given by the tropical volume of the base, but that the subleading term can be expressed in terms of the tropical singularities, and recovers the Gamma conjecture predictions.

Applied quantization of Hitchin integrable system

The talks will focus on aspects of geometric Langlands duality related to other chapters of geometry and representation theory, such as quantization in positive characteristic, Bridgeland stability conditions and abelian categories of representation theoretic interest.

Non-Abelian Hodge Theory and Geometric Langlands

We will review the Geometric Langlands program, emphasizing its overlaps with homological mirror symmetry, non Abelian Hodge theory, and Hitchin's system, and will describe some recent results on the construction of automorphic sheaves in several geometrically accessible cases.

Slides

What is complex algebraic geometry?

Algebraic geometry is the study of the geometry of algebraic varieties, defined as the solutions of a system of polynomial equations over a field \(\mathrm{k}\). When \(\mathrm{k} = \mathbb{C}\) the earliest deep results in the subject were discovered using analysis, and analytic methods (complex function theory, PDEs and differential geometry) continue to play a central and pioneering role in algebraic geometry. The objective of these talks is to present an informal, historical, and illustrative account of some answers to the question in the title. Every attempt will be made to have the talks accessible to an audience of graduate students and post docs.

Slides: Part 1, Part 2, Part 3, Part 4

Mirror symmetry for Higgs bundles

The Strominger–Yau–Zaslow approach to mirror symmetry is well adapted to the Special Lagrangian fibration given by the Higgs bundle integrable system. The two talks will discuss various aspects of this including the semi-flat metric, duality of the fibres, and the questions raised by the conjectured symmetry between BAA branes, which are holomorphic Lagrangian submanifolds and BBB branes, which are supported on hyperkahler submanifolds.

Slides: Part 1, Part 2

Duality with corners

I will describe an algebraic structure which appears in several setups:

1. Poincare duality on manifolds with corners (here is the origin of the name),
2. Serre duality on an algebraic variety endowed with anticanonical divisor with normal crossings,
3. Interaction of mixed Hodge structures and Poincare duality on cohomology of smooth varieties with normal crossing divisors (by Goncharov),
4. A new clean construction of Fukaya and Fukaya-Seidel categories.

The structure under the title is a variant of pre-Calabi-Yau structure by Vlassopoulos and mine, and of relative Calabi-Yau structure by Brav and Dyckerhoff.

Hodge Theory in Homotopy Theory

In the first lecture we will review classical Hodge theory for the cohomology of compact Kahler manifolds and show how it is a consequence of d-dbar lemma. We will then review Sullivan's theory of differential algebras and rational homotopy theory and deduce the formality of the rational homotopy type of a compact Kahler manifold. In the second lecture, we review Deligne's notion of Mixed Hodge Structures and reformulate the result of the first lecture in terms of a Mixed Hodge Structure on the homotopy type. We then show that the homotopy type of an open smooth complex algebraic varieties has a Mixed Hodge Structure extending that constructed by Deligne on the cohomology and deduce homotopy theoretic consequences.

Slides: Part 1, Part 2

Homological Mirror Symmetry and the mirror map for del Pezzo surfaces

I will discuss the general mirror symmetry question for symplectic del Pezzo surfaces in a setup that goes beyond the Hori-Vafa construction. I will explain how homological mirror symmetry considerations lead to an explicit description of the mirror map and will discuss some consequences that can be checked directly. This is a joint work with Auroux, Katzarkov and Orlov.

Mirror symmetry for some homogeneous spaces

I will give an overview of mirror symmetry for homogeneous spaces \(G/P\) from a Lie-theoretic perspective. The picture is particularly nice for co-minuscule \(G/P\).

Asymptotics of the monodromy of local systems — WKB problems and harmonic mappings to buildings

We'll discuss families of connections or Higgs bundles approaching the divisor at infinity in the respective moduli spaces, and what kind of behavior is expected for the corresponding monodromy representation. This goes under the name "WKB problem". In turn, the exponents that appear are governed by a harmonic map to a building (Parreau). We'll then discuss our results with Katzarkov, Pandit and Noll: the derivative is given by the spectral curve associated to the WKB problem (due to Mochizuki for the Higgs case), and we have a program going towards the construction of a versal harmonic map to a pre-building that depends only on the spectral curve.

Towards the construction of stability conditions for rank 3 spectral curves

This is about current work in progress with Haiden, Katzarkov and Pandit. We would like to consider Fukaya–Seidel type categories of sections of a constant sheaf of categories such as \(A_2\) over Lagrangian graphs in the complex plane with boundary points. Following Kontsevich's idea of interpreting this situation as a limiting Fukaya category for a fibration, one expects that there should be a stability condition, combining base and fiber, whose stable objects have "special Lagrangian" representatives that are basically the Gaiotto–Moore–Neitzke spectral networks. We'll report on the current status of our progress in extracting the first destabilizing subobject of the Harder–Narasimhan filtration in this setting.