IMSA Seminar
Dr. Philip Griffiths
University of Miami
Period Mapping at Infinity
Wednesday, May 6, 2020, 10:00am
Online
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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
Online
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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
Online
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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
Online
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
Online
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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
Online
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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 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.
