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
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]
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 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.
Notes available here and a recording here. 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: This paper.
A recording is available here.
In this talk Nick introduced the central object in Homological Mirror Symmetry - the Fukaya Category. This talk gives context for other talks.
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.
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.