Titles and Abstracts

Titles and Abstracts of Talks

James Lewis - University of Alberta Title: Abel-Jacobi Maps Associated to Algebraic Cycles, II Abstract: For a smooth complex quasiprojective variety, we give an explicit description of the Bloch cycle class map from the higher Chow groups into Beilinson's absolute Hodge cohomology. We then arrive at an explicit formula for a weight filtered Abel-Jacobi map. This is based on joint work with Matt Kerr.
Donu Arapura - Purdue University Title: Hodge structure on the fundamental group revisited. Abstract: Given a CW complex X with a map from its fundamental group to a group G, one gets a map X to K(G,1), which induces a map from the cohomology of G to X. When X is a variety, I would like to discuss a Hodge theoretic/motivic analogue of this, where G would be the group associated to the Tannakian category of variations of mixed Hodge structure/motivic sheaves on X. While I'm at it, I'd like to compare this "Hodge structure" on the fundamental group with those of Hain, Morgan, and Simpson.
Gregory Pearlstein - Michigan State University Title: On the zero locus of a normal function.
Burt Totaro - Cambridge University Title: Moving codimension-one subvarieties over finite fields Abstract: Let C be a curve with self-intersection zero on a smooth projective surface. We study when some multiple of C moves, meaning that C belongs to a family of disjoint curves that cover the surface. We give positive results, related to the abundance conjecture, as well as negative answers to questions by Mumford and Keel.
Chad Schoen - Duke University Title: Calabi-Yau threefolds with vanishing third Betti number Abstract: Calabi-Yau threefolds with vanishing third Betti number do not exist over base fields of characteristic zero. In recent years examples have been constructed in finite characteristics. We will sketch some of the constructions by Horikado, Schroeer, Ekedahl, and the speaker and discuss some open questions in the field.

Matthias Flach - Caltech, Title: Weil-etale cohomology and the Tamagawa number conjecture Abstract: We discuss the equivariant Tamagawa number conjecture on motivic L-functions from the point of view of Weil-etale cohomology, a recent idea of Lichtenbaum. We focus on the fact that the conjecture always describes the leading Taylor coefficient of a motivic L-function and give some illustrating examples involving Dirichlet L-functions.
Karen Yeats - Boston University Title: Where are the transcendentals in Dyson-Schwinger equations? Abstract: Recent progress on rearranging Dyson-Schwinger equations, the recursive equations describing quantum field theory, has mostly progressed by sweeping the transcendental content under the rug. I will try to illuminate their hiding places in this context. Time and circumstances permitting I will finish by briefly highlighting the mixed Hodge structure approach to renormalization following new work of Bloch and Kreimer.
Jean-Louis Colliot-Thélène - CNRS et Université de Paris-Sud Title: Chow groups of zero-cycles. Summary : This is a survey talk. Special emphasis will be laid on varieties over a local field.
Stephen Lichtenbaum - Brown University Title: Special values of L-functions of 1-motives Abstract: We propose a conjecture that for any motive over a number field, there exist a finite number of cohomology complexes such that the order of the zero of the L-function of the motive at s = 0 is given by the sum of the ranks of the complexes, and the leading term at s = 0 is given by the procuct of the Euler characteristics of the same complexes. These complexes, or rather their homology, will be made very explicit in the case of 1-motives.
Henri Gillet - University of Illinois, Chicago Title: K-Correspondences and complexes of Motives Abstract: I shall discuss how to use algebraic K-theory to construct an enrichment of the category of varieties over the category of chain complexes of rational vector spaces, and how to use this to define maps between the homological weight complexes of singular varieties (such as pull back with respect to morphisms of finite tor-dimension). (Joint work with C. Soule)

Charles Weibel - Rutgers University Title: The Bloch-Kato Conjecture
Eric Friedlander - Northwestern University Title: Constructing vector bundles on certain singular projective varieties
Roy Joshua - The Ohio State University Title: Motivic $E_{\infty}$-algebras and the motivic dga Abstract: In this talk we will construct explicit $E_{\infty}$-structures on the motives of smooth schemes, on the motivic complex and on complexes defining etale cohomology. This is contrasted with the construction of $E_{\infty}$-structures on complexes defining singular homology and cohomology of complex algebraic varieties -several similarities and some surprising differences will emerge. Applications of such $E_{\infty}$-structures include the construction of a category of mixed Tate motives for a large class of schemes over number fields and the construction of "classical" cohomology operations in motivic cohomology with finite coefficients.
Joseph Ayoub - University of Paris XIII Title: Operations on the singular cohomology of algebraic varieties. Abstract: We study the endomorphisms of the Betti realization functor from Voevodsky motives to $Q$-vector spaces. This will lead to a natural definition of the motivic Galois group of a field of characteristic zero.
Christian Haesemeyer - University of Illinois, Chicago Title: Invariants of singularities and a question of Bass. Abstract: In 1972, H. Bass asked if $K(R) = K(R[t])$ implied that $K(R) = K(R[s,t])$ for a ring $R$. We explain joint work with Cortinas, Walker and Weibel answering this question for commutative algebras over the rationals. It turns out that the answer is "no" over number fields and "yes" over sufficiently large fields. Finding a counterexample relies on computing parts of Nil K-theory in terms of cedrtain classical invariants of singularities.

Marc Levine - Northeastern University Title: Comparing algebraic cobordism theories. Abstract: With Fabien Morel, we have defined a theory of algebraic cobordism, $\Omega^$, on smooth varieties over a field $k$ of characteristic zero. There is also a bi-graded theory of algebraic cobordism, due to Voevodsky, defined using the algebraic Thom complex $MGL$; the universal property of $\Omega$ gives a natural map $\vartheta_X:\Omega^(X) --> MGL^{2,}(X)$ for all smooth $X$ over $k$. Relying on results of Hopkins-Morel, we show that $\vartheta_X$ is an isomorphisms for all smooth $X$; in fact, $\vartheta_X$ extends to a natural isomorphism of oriented Borel-Moore homology theories on finite type $k$-schemes.
Claudio Pedrini - University of Genova Title: On the transcendental part of the motive of a surface.
Andrea Miller - Harvard University Title: Chow motives of some mixed Shimura varieties Abstract: We construct Chow-Kuenneth projectors for universal families over some Shimura varieties thus proving a conjecture of Murre for this case.
Reza Akhtar - Miami University Title: The Beilinson-Soule vanishing conjecture with finite coefficients. Abstract: The Beilinson-Soule vanishing conjecture asserts that for any variety $X$ over a field $k$, the motivic cohomology groups $H^i(X, \mathbb{Z}(n))$ vanish for $i<0$ and any $n$. We discuss a "finite coefficients" variation on this conjecture and give an elementary proof under the hypothesis that the base field contains an algebraically closed field.
Mona Mocanasu - Northwestern University Title: Push-Forward Maps in Algebraic Oriented Theories Abstract: The existence of a push-forward structure for an algebraic oriented theory on smooth pairs determines its ability to study singular schemes. We describe the needed properties for push-forward maps in a general set-up and discuss the known push-forward structures of the classical theories. Since Chern classes can be constructed from the push-forward maps, this leads to the existence of a Verdier-type theorem for the associated Borel-Moore homology.

Li Guo - Rutgers University Title: Algebraic continuation of multiple zeta values.
Mark Walker - The University of Nebraska Title: Equivariant K-theory of toric varieties Abstract: This is joint work with Suanne Au and Mu-wan Huang. I will present results concerning the equivariant K-theory $K_*^T(X)$ of a (not necessarily smooth) toric variety $X = X(\Delta)$, where $\Delta$ is a fan, that allow one to compute it somewhat explicitly from the purely combinatorial data provided by $\Delta$. These results shed light, in particular, on the group $K_0^T(X$), and on the Chern character map from this group to the equivariant operational Chow group of $X$.
Herbert Gangl - The University of Durham Title: Polygons and mixed Tate motives Abstract: We relate two "polygonal" approaches to mixed Tate motives over a field: one approach arising from work of Brown in the study of periods of moduli spaces M_{0,n}, the other one from joint work with Levin and Goncharov in connection with algebraic cycles and iterated integrals. (Report on joint work with Francis Brown and Andrey Levin)