Short Courses and Seminars

Celebration of Alexander Efimov's EMS Prize in Mathematics
Noncommutative Geometry Conference

Thursday, December 17th, 2020, 5:30am (1:30pm Moscow)
Via Zoom
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Abstract: In 2020 Alexander Efimov, a Research Fellow at the International Laboratory for Mirror Symmetry and Automorphic Forms (HSE University) and a Senior Researcher at the Algebraic Geometry Section of Steklov Mathematical Institute of RAS, has been awarded by the European Mathematical Society’s Prize. This event is organized by ILMS NRU HSE and Steklov Mathematical Institute of RAS.


Women in Mathematics in South-Eastern Europe

December 10th-11th, 2020, 3:00am (10:00am Bulgaria)
Via Zoom
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Abstract: This event is organized by the International Center for Mathematical Sciences – Sofia (ICMS-Sofia). To view more information regarding this webinar, click here


IMSA Seminar

Dr. Ludmil Katzarkov
University of Miami & IMSA

Old and New Birational Invariants

Monday, September 28th, 2020, 5:00pm
Via Zoom
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Abstract: In this talk we will propose new birational invariants based on combining Hodge theory and Symplectic Geometry.


IMSA Seminar

Dr. Paul Horja
IMSA

A Categorical Interpretation of the GKZ D-Module

Tuesday, September 1st, 2020, 5:00pm
Via Zoom
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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
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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
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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
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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
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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
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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. 


IMSA Seminar

Dr. Philip Griffiths
University of Miami

Period Mapping at Infinity

Wednesday, May 6, 2020 10:00am
Via Zoom
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Abstract: Hodge theory provides a basic invariant of complex algebraic varieties. For algebraic families of smooth varieties the global study of the Hodge structure on the cohomology of the varieties (period mapping) is a much studied and rich subject. When one completes a family to include singular varieties the local study of how the Hodge structures degenerate to limiting mixed Hodge structures is also much studied and very rich. However, the global study of the period mapping at infinity has not been similarly developed. This has now been at least partially done and will be the topic of this talk. Sample applications include:

  • new global invariants of limiting mixed Hodge structures
  • a generic local Torelli assumption implies that moduli spaces are
  • log canonical (not just log general type); and
  • a proposed construction of the toroidal compactification of the image of a period mapping

The key point is that the extension data associated to a limiting mixed Hodge structure has a rich geometric structure and this provides a new tool for the study of families of singular varieties in the boundary of families of smooth varieties.

Presentation

*Joint work with Mark Green and Colleen Robles


IMSA Seminar

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

Quantum Toric Geometry II Non-commutative Geometric Invariant Theory

Wednesday, April 29, 2020, 10:00am
Via Zoom
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Abstract: In this talk I will explain how to develop the quantum version of geometric invariant theory that gives a generalization of the classical GIT for the non-commutative case.

This is joint work with L. Katzarkov, L. Meersseman, and A. Verjovsky.


IMSA Seminar

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

Quantum Toric Geometry I

Wednesday, April 22, 2020, 10:00am
Via Zoom
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Abstract: In this talk I will give a bird's eye view of the field of Quantum Toric Geometry (QTG). QTG is a generalization of Classical Toric Geometry where the various classical tori appearing in the usual theory are replaced by quantum tori (also known as non-commutative tori). As a result the new theory can be though of as a deformation of the usual theory and hence, it permits the construction of a remarkable moduli space of toric varieties. I will try to convey the basic ideas required to understand this story in this first talk. 

This is joint work with L. Katzarkov, L. Meersseman, and A. Verjovsky.


IMSA Seminar

Dr. Kyoung-Seog Lee
University of Miami

Seiberg-Witten Gauge Theory and Complex Surfaces

Thursday, April 16, 2020, 5:00pm
Via Zoom

Abstract: Most part of this talk will be a survey about Seiberg-Witten gauge theory and how it can be understood for complex smooth projective surfaces. I will discuss several interesting examples and raise some questions.


IMSA Seminar

Dr. Benjamin Gammage
University of Miami

Mirror Symmetry and Cluster Varieties

Thursday, April 2, 2020, 5:00pm
Via Zoom
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Abstract: We discuss the symplectic geometry of cluster varieties with applications to mirror symmetry, including an explanation of homological mirror symmetry for Gross-Hacking-Keel cluster varieties. This is based on work in progress with Ian Le.


IMSA Seminar

Dr. Tokio Sasaki
University of Miami

A Construction of Apéry Constants from Landau-Ginzberg Models

Thursday, March 26, 2020, 5:00pm
Via Zoom
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Abstract: The irrationality of the Riemann zeta function at 3 was historically proven by R. Apéry by finding a rapidly converging sequence which is consisted of two sequences in integers and rationals satisfying certain recursive relations. Nowadays it is known that this sequence is obtained from the power series expansion of the holomorphic period function of a family of K3 surfaces, and the recurrences arise from the Picard-Fuchs differential equation.

For some Fano threefolds with Picard rank 1, V. Golyshev obtained similar special values of L-functions as Apéry limit of the quantum differential equations. If one believes the mirror symmetry also preserves these arithmetic special values, there should be a "mirror" construction in the B-model side. In this talk, as an evidence I introduce constructions of geometric higher normal functions on the mirror Landau-Ginzberg models of the above Fano threefolds. Limiting values of these normal functions toward singular fibers reconstruct the Apéry constants computed in the A-model side. With Mukais classification of the Fano threefolds, the results for V_10, V_12, V_16, V_18 are shown by M. Kerr and G. Silva Jr. A partial result for the V_14 case is given by the speaker.


IMSA Special Evening

Dr. Mina Teicher
Dr. Gabriela Olmedo
University of Miami

Mathematics & Biomedicine

Thursday, January 30, 2020, 5:30pm
Newman Alumni Center, First Floor

Registration is required.
Click here to view videos

Dr. Mina Teicher's Abstract: Mathematics is everywhere! In manmade systems and in God made systems. In this talk Dr. Teicher will demonstrate appearance of mathematical forms and logic in nature (humans, plants, stones,...). Moreover, she will describe practical application of math to medicine and to the understanding of the human brain.

Dr. Gabriela Olmedo's Abstract: Why would researchers from biology, physics and mathematics join to study invisible microorganisms? Bacteria organize typically in complex communities that are not only essential for all the biogeochemical cycles on Earth and our health. They also hold clues to the story of life. Two crucial questions in biology are: How can thousands of species coexist in communities and what the rules are for their organization.


IMSA Seminar

Dr. R. Paul Horja
University of Miami

D-modules and Toric Schobers

Tuesday, January 21, 2020, 5:00pm
Ungar Room 528B

Abstract: I will present a translation of the classical mirror symmetry point of view into the more recent language of schobers. A conjecture on a categorical interpretation of the quantum toric D-module naturally appearing in mirror symmetry will be discussed.


IMSA Seminar

Dr. Ludmil Katzarkov
University of Miami

Categorical Linear Systems

Tuesday, January 14, 2020, 5:00pm
Ungar Room 528B