Tag - Mirror symmetry

Simion Filip: Degenerations of Kähler forms on K3 surfaces, and some dynamics

4th June, 2021

K3 surfaces have a rich geometry and admit interesting holomorphic automorphisms. As examples of Calabi-Yau manifolds, they admit Ricci-flat Kähler metrics, and a lot of attention has been devoted to how these metrics degenerate as the Kähler class approaches natural boundaries. I will discuss how to use the full automorphism group to analyse the degenerations and obtain certain canonical objects (closed positive currents) on the boundary. While most of the previous work was devoted to degenerating the metric along an elliptic fibration (motivated by the SYZ picture of mirror symmetry) I will discuss how to analyse all the other points. Time permitting, I will also describe the construction of canonical heights on K3 surfaces (in the sense of number theory), generalizing constructions due to Silverman and Tate.

Daniel Pomerleano: Intrinsic mirror symmetry and categorical crepant resolutions

19th February, 2021

Gross and Siebert have recently proposed an "intrinsic" programme for studying mirror symmetry. In this talk, we will discuss a symplectic interpretation of some of their ideas in the setting of affine log Calabi-Yau varieties. Namely, we describe work in progress which shows that, under suitable assumptions, the wrapped Fukaya category of such a variety X gives an intrinsic "categorical crepant resolution" of Spec(SH0(X)). No background in mirror symmetry will be assumed for the talk.

Cheol-Hyun Cho: Fukaya category for Landau-Ginzburg orbifolds and Berglund-Hübsch homological mirror symmetry for curve singularities

21st September, 2020

For a weighted homogeneous polynomial and a choice of a diagonal symmetry group, we define a new Fukaya category based on wrapped Fukaya category of its Milnor fiber together with monodromy information. It is analogous to the variation operator in singularity theory. As an application, we formulate a complete version of Berglund-Hübsch homological mirror symmetry and prove it for two variable cases. Namely, given one of the polynomials f=xp+yq, xp+xyq, xpy+xyq and a symmetry group G, we use a Floer-theoretic construction to obtain the transpose polynomial ft with the transpose symmetry group Gt as well as an explicit A∞-equivalence between the new Fukaya category of (f,G) to the matrix factorization category of (ft,Gt). In this case, monodromy is mirror to the restriction of LG model to a hypersurface. For ADE singularities, Auslander-Reiten quiver for indecomposable matrix factorizations were known from the 1980s, and we find the corresponding Lagrangians as well as surgery exact sequences.

Liana Heuberger: Constructing Fano varieties via mirror symmetry

7th September, 2020

An LMS online lecture course in mirror symmetry.

Mirror symmetry conjecturally associates to a Fano orbifold a (very special type of) Laurent polynomial. Laurent inversion is a method for reversing this process, obtaining a Fano variety from a candidate Laurent polynomial. We apply this to construct new Fano 3-folds with terminal Gorenstein quotient singularities.

In this series of talks, I will go through some of the basics of toric geometry to showcase how one can use combinatorial data to systematically build geometric objects. We will restrict our attention to the well-studied Fano case, for which there is concrete evidence that the mirror theorem holds in many cases.

Denis Auroux: Mirrors of curves and their Fukaya categories

22nd May, 2020

Homological mirror symmetry predicts that the derived category of coherent sheaves on a curve has a symplectic counterpart as the Fukaya category of a mirror space. However, with the exception of elliptic curves, this mirror is usually a symplectic Landau-Ginzburg model, i.e. a non-compact manifold equipped with the extra data of a 'stop' in its boundary at infinity. Most of the talk will focus on a family of Landau-Ginzburg models which provide mirrors to curves in (C*)2 or in toric surfaces (or more generally to hypersurfaces in toric varieties), and their fiberwise wrapped Fukaya categories (joint work with Mohammed Abouzaid). I will then discuss more a speculative way of constructing mirrors of curves without Landau-Ginzburg models, involving a new flavour of Lagrangian Floer theory in trivalent configurations of Riemann surfaces (joint work with Alexander Efimov and Ludmil Katzarkov).

Yuan Gao: Wrapped Fukaya categories and functors

23rd October, 2017

Inspired by homological mirror symmetry for non-compact manifolds, one wonders what functorial properties wrapped Fukaya categories have as mirror to those for the derived categories of the mirror varieties, and also whether homological mirror symmetry is functorial. Comparing to the theory of Lagrangian correspondences for compact manifolds, some subtleties are seen in view of the fact that modules over non-proper categories are complicated. In this talk, the story concerning the fundamental construction of Fourier-Mukai type functors of wrapped Fukaya categories is discussed, under slightly modified framework of wrapped Floer theory. Applications of the relevant techniques to be presented include the Kunneth formula and restriction maps.