Short Courses and Seminars 2020

IMSA Seminar

Dr. Ludmil Katzarkov
University of Miami & IMSA

Old and New Birational Invariants

Monday, September 28th, 2020, 5:00pm
Via Zoom
Click here to view video

Abstract: In this talk we will propose new birational invariants based on combining Hodge theory and Symplectic Geometry.


IMSA Seminar

Dr. R. Paul Horja
IMSA

A Categorical Interpretation of the GKZ D-Module

Tuesday, September 1st, 2020, 5:00pm
Via Zoom
Click here to view video

Abstract: I will explain a proposal for the B-side category in toric homological mirror symmetry along the strata of the characteristic cycle of the associated GKZ D-module. Various consistency checks will be presented. The construction builds on the string theoretical work by Aspinwall-Plesser-Wang.


Joint IMSA & ICMS HSE Event

Dr. Ernesto Lupercio
Center for Research and Advanced Studies of the National Polytechnic Institute (Cinvestav-IPN)

An Introductory Mini Course into Quantum Toric Geometry: Lecture II

Friday, August 28th, 2020, 9:00am
Via Zoom
Click here to view video

Abstract: We will introduce the foundations of Quantum Toric Geometry as developed by Katzarkov, Lupercio, Meersseman and Verjovsky, Quantum toric geometry is a generalization of toric geometry where irrational fans correspond to non-commutative spaces called quantum toric varieties. As non-commutative spaces, Quantum toric varieties are to usual toric varieties what the Quantum torus is to the usual torus.

Lecture II

  1. LVM theory
  2. Quantum GIT
  3. Moduli spaces of toric varieties

References: It is useful to be familiar with toric varieties (for example the book of Fulton).

Click here to view this presentation. 


Joint IMSA & ICMS HSE Event

Dr. Artan Sheshmani
Harvard University CMSA/University of Miami/IMSA

Atiyah Class and Sheaf Counting on Local Calabi-Yau 4 Folds

Friday, August 28th, 2020, 8:00am
Via Zoom
Click here to view video

Abstract: We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface. We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.

Click here to view this presentation. 


Joint IMSA & ICMS HSE Event

Dr. Tony Yue YU
Laboratoire de Mathématiques d'Orsay

Frobenius Structure Conjecture and Moduli of Calabi-Yau Pairs

Friday, August 28th, 2020, 7:00am
Via Zoom
Click here to view video

Abstract: I will explain the Frobenius structure conjecture of Gross-Hacking-Keel in mirror symmetry, and an application towards the moduli space of Calabi-Yau pairs. I will show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a simple way, a mirror family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form.

The structure constants of the algebra are constructed via counting non-archimedean analytic disks in the analytification of U. Furthermore, I will introduce a generalization of the Gelfand-Kapranov-Zelevinsky secondary fan, and show that the mirror family admits a natural compactification and extension over the toric variety associated to the secondary fan, which generalizes the families previously studied by Kapranov-Sturmfels-Zelevinsky and Alexeev in the toric case.

We conjecture that this gives rise to a (nearly uni) versal family of polarized Calabi-Yau pairs (embedded in the moduli space of KSBA stable pairs), and has a surprising consequence that such moduli space is unirational. We prove the stability in dimension two. This is based on arXiv:1908.09861 joint with S. Keel, and arXiv:2008.02299 joint with Hacking and Keel.

Click here to view this presentation.


Joint IMSA & ICMS HSE Event

Dr. Ernesto Lupercio
Center for Research and Advanced Studies of the National Polytechnic Institute (Cinvestav-IPN)

An Introductory Mini Course into Quantum Toric Geometry: Lecture I

Thursday, August 27th, 2020, 9:00am
Via Zoom
Click here to view video

Abstract: We will introduce the foundations of Quantum Toric Geometry as developed by Katzarkov, Lupercio, Meersseman and Verjovsky, Quantum toric geometry is a generalization of toric geometry where irrational fans correspond to non-commutative spaces called quantum toric varieties. As non-commutative spaces, Quantum toric varieties are to usual toric varieties what the Quantum torus is to the usual torus.

Lecture I

  1. Introduction
  2. Stacks and non-commutative spaces
  3. The quantum torus
  4. Quantum toric varieties

References: It is useful to be familiar with toric varieties (for example the book of Fulton).

Click here to view this presentation. 


Joint IMSA & ICMS HSE Event

Dr. Artan Sheshmani
Harvard University CMSA/University of Miami/IMSA

Stable Higher Rank Flag Sheaves on Surfaces

Thursday, August 27th, 2020, 8:00am
Via Zoom
Click here to view video

Abstract: We study moduli space of holomorphic triples f: E_{1}—>E_{2}, composed of (possibly rank > 1) torsion-free sheaves (E_{1}, E_{2}) and a holomorphic map between them, over a smooth complex projective surface S. The triples are equipped with Schmitt stability condition. We prove that when Schmitt stability parameter becomes sufficiently large, the moduli space of triples benefits from having a perfect relative and absolute obstruction theory in some cases (depending on rank of holomorphic torsion-free sheaf E_{1}).

We further generalize our construction to higher-length flags of higher rank sheaves by gluing triple moduli spaces, and extend the earlier work, with Gholampur and Yau, where the obstruction theory of nested Hilbert schemes over the surface was studied. Here we extend the earlier results to the moduli space of flags E_{1}——>E_{2}—>...——>E_{n}, where the maps are injective (by stability). There is a connection, by wallcrossing, between the theory of such higher rank flags, and the theory of Higgs pairs on the surface, which provides the means to relate the flag invariants to the local DT invariants of threefold given by a line bundle over the surface, X :=Tot(L —> S).

Click here to view this presentation.


Joint IMSA & ICMS HSE Event

Dr. Tony Yue YU
Laboratoire de Mathématiques d'Orsay

Frobenius Structure Conjecture and Moduli of Calabi-Yau Pairs

Thursday, August 27th, 2020, 7:00am
Via Zoom
Click here to view video

Abstract: I will explain the Frobenius structure conjecture of Gross-Hacking-Keel in mirror symmetry, and an application towards the moduli space of Calabi-Yau pairs. I will show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a simple way, a mirror family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form.

The structure constants of the algebra are constructed via counting non-archimedean analytic disks in the analytification of U. Furthermore, I will introduce a generalization of the Gelfand-Kapranov-Zelevinsky secondary fan, and show that the mirror family admits a natural compactification and extension over the toric variety associated to the secondary fan, which generalizes the families previously studied by Kapranov-Sturmfels-Zelevinsky and Alexeev in the toric case.

We conjecture that this gives rise to a (nearly uni) versal family of polarized Calabi-Yau pairs (embedded in the moduli space of KSBA stable pairs), and has a surprising consequence that such moduli space is unirational. We prove the stability in dimension two. This is based on arXiv:1908.09861 joint with S. Keel, and arXiv:2008.02299 joint with Hacking and Keel.

Click here to view this presentation.


IMSA Seminar

Dr. Maxime Kontsevich
University of Miami & IMSA

Integral PL Actions from Birational Geometry

Monday, August 17th, 2020, 9:00am
Via Zoom
Click here to view video

Abstract: The group of birational automorphisms of a N-dimensional algebraic torus, preserving the standard logarithmic volume element, acts (by tropicalization) on N-dimensional real vector space by homogeneous piece-wise linear homeomorphisms. A similar construction exists for any compact Calabi-Yau variety over a non-archimedean field, through the notion of the "essential EEEE". I'll talk about examples coming from generalized cluster varities, and from Calabi-Yau varieties parameterizing linkages of regular graphs.