Mixed Period Maps: Definability and Algebraicity
Dr. Jacob Tsimerman
Toronto
9:00am - 10:00am
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Abstract: In this pair of lectures, we will explain how to develop an o-minimal geometry allowing for nilpotents, that we call "definable analytic spaces". We explain how to use this theory to prove a definable GAGA statement, and how one can use this to prove a conjecture of Griffiths that the images of period maps are algebraic. We will also discuss the analogous o-minimal in the setting of variations of mixed Hodge structures, and a generalization of Griffiths conjecture to this setting.
Differential Equations and Mixed Hodge Structures
Dr. Matt Kerr
University of Washington
10:15am - 11:15am
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Abstract: We report on a new development in asymptotic Hodge theory, arising from work of Golyshev-Zagier and Bloch-Vlasenko, and connected to the Gamma Conjectures in Fano/LG-model mirror symmetry. The talk will focus exclusively on the Hodge/period-theoretic aspects through two main examples.
Given a variation of Hodge structure M on a Zariski open in P^1, the periods of the limiting mixed Hodge structures at the punctures are interesting invariants of M. More generally, one can try to compute these asymptotic invariants for iterated extensions of M by "Tate objects", which may arise for example from normal functions associated to algebraic cycles.
The main point of the talk will be that (with suitable assumptions on M) these invariants are encoded in an entire function called the motivic Gamma function, which is determined by the Picard-Fuchs operator L underlying M. In particular, when L is hypergeometric, this is easy to compute and we get a closed-form answer (and a limiting motive). In the non-hypergeometric setting, it yields predictions for special values of normal functions; this part of the story is joint with V. Golyshev and T. Sasaki.
Fixed Points in Toroidal Compactifications and Essential Dimension of Covers
Dr. Patrick Brosnan
University of Maryland
2:00pm - 3:00pm
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Abstract: Essential dimension is a numerical measure of the complexity of algebraic objects invented by J. Buhler and Z. Reichstein in the 90s. Roughly speaking, the essential dimension of an algebraic object is the number of parameters it takes to define the object over a field. For example, by Kummer theory it takes one parameter to define a mu_n torsor, so the essential dimension of the functor of mu_n torsors (or the essential dimension of the group mu_n for short) is 1.
In a preprint from 2019, Farb, Kisin and Wolfson (FKW) prove theorems about the essential dimension of congruence covers of Shimura varieties using arithmetic methods. In many cases, they are able to prove that the congruence covers are incompressible, that is, they are not obtainable by base change from varieties of strictly smaller dimension. In my talk, I will discuss recent work with Najmuddin Fakhruddin, where we recover many (but definitely not all) of the results of FKW, by geometric arguments using a new fixed point theorem. This also allows us to extend the incompressibility results of FKW to Shimura varieties of exceptional type to which the arithmetic methods of FKW do not apply. I will also discuss a general conjecture we make on the essential dimension of congruence covers arising from Hodge theory. (With some caveats, we conjecture roughly that it is equal to the dimension of the image of the period map.)
Archimedean Height Pairings for Higher Cycles
Dr. Greg Pearlstein
University of Texas A&M
3:30pm - 4:30pm
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Abstract: By the work of Richard Hain, the archimedean height pairing on ordinary algebraic cycles can be interpreted as an invariant of an associated mixed Hodge structure. In this talk, we will present a similar construction for higher cycles in the Bloch complex. Families of higher cycles produce admissible variations of mixed Hodge structure. We will describe the asymptotic behavior of the height pairing in the case where the associated variation of mixed Hodge structure is Hodge-Tate. This is joint work with J. Burgos Gil and S. Goswami.
