This is the page for the Mirror Symmetry and Algebraic Geometry reading group. We meet weekly via zoom. Our current goal is understanding the basics of the Gross-Siebert program (all the talks below relate to this goal). Understanding something about intrinsic mirror symmetry and non-Archimedean mirror symmetry are also on the agenda (although there is not a talk by talk plan for achieving these goals yet!).

[CK] = Mirror Symmetry and Algebraic Geometry, David Cox and Sheldon Katz

Introduction Patrick Kennedy-Hunt 26/08/2021

Notes are available here and a recording here. I give this talk in an initial meeting. We discussed the big picture in mirror symmetry as presented in the first chapter of [CK]. This provides context for other talks.

Curve Counting on the Quintic Threefold Patrick Kennedy-Hunt 02/09/2021

Notes are available here and a recording here. In this talk I discuss the enumerative meaning of the instanton numbers appearing in the definition of the Yukawa coupling. We then discuss an explicit example of mirror symmetry in action. Appealing to the mirror theorem we will be able predict curve counts for the quintic threefold.

References: See these notes on lectures by Auroux and Chapter 2 of [CK]

Batyrev Mirror Symmetry Wanlong Zheng 09/09/2021

Notes are available here. In this talk Wanlong taught us how to construct the mirror to a hypersurface in a toric variety using combinatorics. He also discussed constructing the mirror to a complete intersection.

References: Batyrev's original paper, a survey article (another version), Mark explains how this fits in with GS here, Cox and Katz Chapter 4.

An Invitation to Toric Degenerations I Robert Crumplin 30/09/2021

Notes are available here. In this talk Rob recalled the polytope perspective on toric varieties. He went on to explain the Mumford construction, give the Fermat Curve example and sketched how these ideas become important in the Gross-Siebert program.

References: This article, and Rob's expertise

An Invitation to Toric Degenerations II Ajith Kumaran 07/10/2021

Notes are available here. Ajith explained how to build a toric degeneration from a polytope. He explained the first step in generalising this story: instead of a polytope (an "affine manifold") we can instead consider an "affine manifold with singularities". To such an affine manifold with singularities we associate a complex algebraic variety. This variety is "locally toric", just as our affine manifold is "locally a polytope".

References: the Gross-Siebert survey article here.

SYZ mirror symmety, theta functions and Jagged paths Patrick Kennedy-Hunt 18/10/2021

This talk will draw themes from the previous two together to give a picture of the Gross Siebert program. I will start by explaining the SYZ conjecture/approach. I will put the key themes of the last two talks in this context. I will move on to discuss theta functions which play a central roll in the Gross Siebert program.

References: A good starting point is wikipedia. Relevant lectures in this course. Cox/Katz chapter 8 also has a discussion.

Scattering Diagrams 21/10/2021

By this point we understand what an affine manifold is and their importance in mirror symmetry. Ajith explained how to associate a complex variety to this affine manifold in the simplest case. In this talk we explain a complication arising when handling in more complicated affine manifolds. These complications lead to scattering diagrams. These diagrams appear elsewhere in enumerative geometry so this talk may be of independent interest.

References: Section 4 onwards of this article is a good starting point. Breifly explaining an example from Section 3 would be a good warmup.

The Cubic Hypersurface 28/10/2021

This is an example of the program which has been explained in the previous four talks. It will round out our first glimpse of the Gross Siebert program.

The SYZ conjecture and its relation to the Gross Siebert program (2 talks?)

This starts to take us towards the intrinsic story that people are still thinking about. Source is Mark's survey.

Intrinsic Mirror Symmetry

An expert can introduce us to this then we can deicide where to go? This is where punctured invariants enter the picture. This was announced here and details appear here.