Homological Mirror Symmetry

Mirror symmetry for moduli of flat bundles and non-abelian Hodge theory

Tony Pantev
University of Pennsylvania
April 14, 2017
I will discuss a construction of the homological mirror correspondence on algebraic integrable systems arising as moduli of flat bundles on curves. The focus will be on non-abelian Hodge theory as a tool for implementing hyper Kaehler rotations of objects in the Fukaya category. I will discuss in detail a specific example of the construction building automorphic sheaves on the moduli space of rank two bundles on the projective line with parabolic structure at five points. This is a joint work with Ron Donagi.

Mirror symmetry for moduli of flat bundles and non-abelian Hodge theory

Tony Pantev
University of Pennsylvania
April 12, 2017
I will discuss a construction of the homological mirror correspondence on algebraic integrable systems arising as moduli of flat bundles on curves. The focus will be on non-abelian Hodge theory as a tool for implementing hyper Kaehler rotations of objects in the Fukaya category. I will discuss in detail a specific example of the construction building automorphic sheaves on the moduli space of rank two bundles on the projective line with parabolic structure at five points. This is a joint work with Ron Donagi.

Two rigid algebras and a heat kernel

Amitai Zernik
Member, School of Mathematics
April 7, 2017
Consider the Fukaya A8 algebra $E$ of $RP^{2m}$ in $CP^{2m}$ (with bulk and equivariant deformations, over the Novikov ring). On the one hand, elementary algebraic considerations show that E admis a rigid cyclic minimal model, whose structure constants encode the associated open Gromov-Witten invariants. On the other hand, in a recent paper another rigid minimal model was computed explicitly, using fixed-point localization for A8 algebras. In this talk I'll discuss these two models and explain how to use the heat kernel on $RP^{2m}$ to relate them.

Speculations about homological mirror symmetry for affine hypersurfaces

Denis Auroux
University of California, Berkeley; Member, School of Mathematics
March 31, 2017
The wrapped Fukaya category of an algebraic hypersurface $H$ in $(C*)^n$ is conjecturally related via homological mirror symmetry to the derived category of singularities of a toric Calabi-Yau manifold $X$, whose moment polytope is determined by the tropicalization of $H$. (The case of the pair of pants discussed in the first talk is a special case of this construction).

Homological mirror symmetry for the pair of pants

Denis Auroux
University of California, Berkeley; Member, School of Mathematics
March 29, 2017
Homological mirror symmetry postulates a derived equivalence between the wrapped Fukaya category of an exact symplectic manifold and a category of coherent sheaves or matrix factorizations on a mirror space. This talk will provide an introduction to the relevant concepts and illustrate the statement on one simple example: the pair of pants. We will describe explicitly the wrapped Fukaya category of the pair of pants, and relate it to algebraic geometry on the mirror. (This is based on joint work with Abouzaid, Efimov, Katzarkov and Orlov).

Real Lagrangians in toric degenerations

Bernd Siebert
University of Hamburg
March 17, 2017
Abstract: Real loci of the canonical toric degenerations constructed from integral affine
manifolds with singularities in the joint work with Mark Gross, provide an ample source of
examples of Lagrangians that conjecturally are amenable to algebraic-geometric versions of
Floer theory. In the talk I will discuss joint work with Hülya Argüz on how the topology of the real
locus can be understood by means of the affine geometry and by Kato-Nakayama spaces
associated to log spaces.

Equivariant geometry and Calabi-Yau manifolds

Daniel Halpern-Leistner
Columbia University
March 16, 2017
Abstract: Mirror symmetry has led to deep conjectures regarding the geometry of Calabi-Yau
manifolds. One of the most intriguing of these conjectures states that various geometric
invariants, some classical and some more homological in nature, agree for any two Calabi-Yau
manifolds which are birationally equivalent to one another. I will discuss how new methods in
equivariant geometry have shed light on this conjecture over the past few years, ultimately

Descent and equivalences in non-commutative geometry

Tony Pantev
University of Pennsylvania
March 16, 2017
Abstract: I will describe descent formalism in categorical non-commutative geometry which is
geared towards constructions of Fourier–Mukai functors. The formalism allows one to carry out
descent constructions in general algebraic and analytic frameworks without resorting to
generators. I will discuss various applications, such as the connection to the classical Zariski and
flat descents, constructions of Fukaya categories, and homological mirror symmetry. This is a
joint work with Katzarkov and Kontsevich.

Central charges of B-branes at non geometric phases

Mauricio Romo
IAS
March 16, 2017
Abstract: I'll give an overview of basic concepts about B-branes and their central charges and
how they arise in physics and mathematics. In particular I'll present the gauge linear model
approach which allows for defining quantities in the full stringy Kahler moduli M of certain
Calabi-Yaus (CY). Then, I'll show some examples and motivate an intrinsic definition of the central
charges on phases (some sub-regions of M), based on field theory data. I'll put particular