Prof. Griffiths
Abstracts of Prof. Griffiths's lectures can be found here.
Jim Carlson - Transcendence degree of the field of periods
Abstract: The field generated by periods of integrals
for the cohomology of a projective variety is at most
the square of the dimension of the cohomology.
Hodge cycles in the tensor algebra of the Hodge structure
reduce this dimension, a fact which is best expressed
via the Mumford-Tate group. Thus, in the case of elliptic
curves, the transcendence degree is two if the curve
has complex multiplication, and it is conjecturally equal
to four in the contrary case. We will discuss some
higher dimensional examples, in particular, the case
of doubly cyclic cubic threefolds. This is joint
work with Domingo Toledo.
Eduardo Cattani - Asymptotics of the Period Map
Abstract: In this introductory lecture we will review the asymptotic behavior
of a variation of Hodge structure.
Wushi Goldring - Algebraicity of Automorphic Representations
Abstract: The Langlands Program and related conjectures predict that
a special class of automorphic representations of reductive groups
admits a number of different algebraicity properties. We shall first
define this special class and explain the algebraicity conditions it
is conjectured to satisfy. Then we shall explain the two methods that
have been used to prove that an automorphic representation has
algebraic properties: (1) Realizing the representation in algebraic
geometry and (2) Applying Langlands functoriality to the
representation. We will review the cases where (1) and (2) have been
successful. Finally, we shall explain why (i) limits of discrete
series, (ii) arithmetic quotients of non-classical Mumford-Tate
domains and (iii) the relationship
between them, play a pivotal role in understanding the remaining cases.
Mark Green - Review of Real and Complex Compact and Semisimple Lie Groups and Finite Dimensional Representation Theory
Abstract: This will be a 90 minute lecture to provide some introductory material for Prof. Griffiths' main lecture series.
Aroldo Kaplan - Topics from Griffith's Lecture 3
Abstract: A review of the Hodge Theorems, Lefschetz decomposition, Hodge-Riemann bilinear relations and Griffiths' transversality.
Matt Kerr - Representations of SL_2, part I
Abstract: We will review representations of the Lie algebra sl_2 and the
classical theory of modular forms, and begin to explore the link between
them.
Matt Kerr - Representations of SL_2, part II
Abstract: The second talk will consider how modular forms lead to
discrete series representations of SL_2(R) in the setting of automorphic
forms, and discuss other representations that show up in this context
(such as those generated by Maass forms). We will conclude by relating
this to Lie algebra cohomology in anticipation of material in Griffiths's
lectures.
James Lewis - Hodge Type Conjectures and the Bloch-Kato Theorem
Abstract: We will discuss a version of the Hodge
conjecture for higher K-groups, and explore some
consequences of an affirmative answer to this conjecture
for Abel-Jacobi maps. We further explain the
impact of the Bloch-Kato theorem on the cycle class map
at the generic point, in the Milnor K-theory case. This talk
is based on joint work with Rob de Jeu.
Gregory Pearlstein - Boundary components of Mumford-Tate domains
Abstract: Mumford-Tate groups arise as the natural symmetry groups
of Hodge structures and their variations. I describe recent work
with Matt Kerr on computing the Mumford-Tate group of the Kato-Usui
boundary components of a degeneration of Hodge structure.
Colleen Robles - Schubert integrals and invariant characteristic cohomology of the
infinitesimal period relation
Abstract: I will describe the Schubert varieties that are integrals of
the infinitesimal period relation (IPR) governing variations of Hodge
structure. Their Poincare duals form a basis for the invariant
characteristic cohomology (ICC) of the IPR. As a consequence, the
projection of the Hodge bundle Chern classes to the ICC is given by
Bernstein-Gelfand-Gelfand operators.
Domingo Toledo - Period Domains and Kahler Manifolds
Abstract: It is known that period domains that are not Hermitian symmetric spaces carry invariant indefinte Kahler metrics but no invariariant positive definite Kahler metrics. This talk will review an old result with Jim Carlson to the effect that compact quotients of such period domains are not homotopy equivalent to Kahler manifolds. It will also present a natural unsolved conjecture on fundamental groups of compact Kahler manifolds suggested by this theorem, and the present status of the evidence for this conjecture.
Sampei Usui - Log Mixed Hodge Theory
Abstract: Log Mixed Hodge Theory is a joint work of K. Kato, C. Nakayama, and S. Usui. Feeling relation with mirror symmetry, I will explain a core of this theory: fundamental diagram which relates various kind of partial compactifications of an arithmetic quotient of a classifying space of mixed Hodge structures.