Arithmetic, K-Theory and Algebraic Cycles - Titles and Abstracts.

DIRECTIONS TO CAMPUS

CONFERENCE ON ALGEBRAIC CYCLES, 2008

COLUMBUS WEATHER

SCHEDULE OF EVENTS

CONFERENCE POSTER

Titles and Abstracts of Talks

Thomas Brazelton - Harvard University. Contributed talk. Title: Rank two bundles on smooth affine fourfolds. Abstract: Over a finite CW complex there only finitely many isomorphism classes of complex vector bundles with some prescribed Chern classes. Given a smooth affine variety of finite Krull dimension over a field, we can ask an analogous question for algebraic bundles, and the answer is mostly unknown. However Morel's "affine representability" theorem indicates that these sorts of algebraic questions are amenable to techniques from motivic obstruction theory. Following work of Mohan Kumar, Murthy and others in the 20th century, as well as the contemporary research program of Asok and Fasel, we understand that the complexity of these sorts of questions are governed by two factors: (1) the 2-cohomological dimension of the base field, and (2) the corank (dimension of base minus rank of bundle). In joint work with Opie and Syed we explore corank two in the first interesting setting. We explore to what extent Chow-valued Chern classes and Chow-Witt-valued Euler classes uniquely classify algebraic vector bundles over smooth affine fourfolds over an algebraically closed field.
Gunnar Carlsson - Stanford University. Title: Representation rings of profinite groups and descent in algebraic K-theory. Abstract: The Quillen-Lichtenbaum and Bloch-Kato conjectures give an attractive description of a spectral sequence (due to Bloch-Lichtenbaum) converging to the algebraic K-theory of fields. What would be very attractive is an explicit geometric method of constructing the actual spectrum (rather than a spectral sequence) from only the absolute Galois group and the K-theory of algebraically closed fields. We'll describe such a construction which achieves this for geometric fields (away from the characteristic), and discuss methods for extending to all fields. It depends on a specific property which holds for absolute Galois group, as well as work on general descent for algebraic group actions. This is joint work with Roy Joshua.
Evangelia Gazaki- University of Virginia. Title: Zero-cycles on smooth surfaces over p-adic fields. Abstract: For a smooth projective variety X over a field k the Chow group CH₀(X) of zero-cycles modulo rational equivalence is a generalization to higher dimensions of the Picard group of a smooth projective curve. When X is a curve having a k-rational point, its Picard group can be fully understood by the Abel-Jacobi map, which gives an isomorphism between the degree zero elements of the Picard group and the k-points of the Jacobian variety of X. In higher dimensions however the situation is much more chaotic. Although there is still an Abel-Jacobi map, this map generally has a kernel, which over large fields like C, R or Q_p can be enormous. In this talk we will focus on the case when the variety X is defined over a finite extension of the p-adic field Q_p. In this case, a famous conjecture of Colliot-Thélène predicts that the kernel of the Abel-Jacobi map decomposes as a direct sum of a divisible group and a finite group. In this talk I will report on some recent joint work with Jonathan Love, where we establish the conjecture for some new classes of surfaces, including the very first examples of K3 surfaces. Moreover, I will propose a similar conjecture for smooth quasi-projective surfaces, by replacing the Chow group CH₀(X) with a larger class group of zero-cycles, namely Suslin's singular homology group. This part is based on work in progress joint with Jitendra Rathore.
David Harbater - University of Pennsylvania Title: Toward higher dimensional patching. Abstract: Patching provides an approach to the study of function fields over complete discretely valued fields. In the context of one-dimensional function fields, it has yielded results about Galois theory, differential Galois theory, quadratic forms, and local-global principles. Recent joint work with Julia Hartmann and Daniel Krashen concerns the generalization of this approach to higher dimensional function fields.
James Hotchkiss - Columbia University. Title: The topological period-index conjecture. Abstract: The period-index problem is an elementary question about central simple algebras over a field, whose origins lie in the calculation of the Brauer groups of number fields from the early part of the twentieth century. The problem is widely open. In recent years, there has been some interest in an analogue of the problem where the field is replaced by a topological space. I will explain a solution to the topological problem for spaces underlying smooth complex projective varieties.
Matt Kerr - Washington University, St.Louis. Title: Two vignettes on hypergeometric regulators. Abstract: Among the more explicitly computable examples of normal functions are those taking values in Jacobians of hypergeometric variations of Hodge structure. (These can be thought of as regulators on families of algebraic K-theory classes.) The goals of this talk are to (i) explain how such normal functions lead to number-theoretic identities of a type recently considered by Guillera, and (ii) to give a short proof of a conjecture of Golyshev on algebraic hypergeometrics, aided by an argument in the spirit of Lefschetz's (1,1) theorem.
Eoin Mackall- University of California, Santa Cruz. Title: Deformations of Azumaya algebras with quadratic pair. Abstract: Quadratic pairs on Azumaya algebras are algebraic gadgets that, in a precise way, correlate to forms of quadric bundles (inside forms of projective bundles). We'll talk about the deformation theory of quadratic pairs relative to the deformation theory of Azumaya algebras. More precisely, we ask: does there exist a variety X and an Azumaya algebra with quadratic pair on X which has obstructed deformations to a thickening X' of X and is such that the underlying Azumaya algebra deforms to X'? We'll explain why the answer is "no" if the characteristic of the base field is not 2 and we'll give an example to show that the answer is "yes" in characteristic 2. This talk is based on joint work with Cameron Ruether.
Akhil Mathew - University of Chicago. Title: Hyperdescent and \'etale K-theory. Abstract: I'll describe joint work with Dustin Clausen proving general hyperdescent results for \'etale and Selmer K-theory (generalizing results going back to Thomason). The main new inputs here are results (joint with Clausen and Morrow) on the relationship between K-theory and topological cyclic homology.
Anubhav Nanavaty - University of California, Irvine. Contributed talk. Title: Feynman Integrals and Symmetric Matrices. Abstract: Since the foundational work of Broadhurst and Kreimer, there has been a significant push to understand the amplitudes of Feynman Integrals as periods, or integrals of algebraic functions over algebraic domains. Work of Brown has related these periods to the homology of Kontesevich's graph complex, and therefore the cohomology of the moduli space of curves by work of Chan, Galatius and Payne. Central to the story are the Borel classes - GLn(Z) equivariant cohomology classes on the space of projective symmetric matrices of full rank. I first show that, in a very general setting, the Voevodsky motive of this space splits into a direct sum of Tate Motives. I conclude by work in-progress with collaborators, where we aim to compute the weights of the Borel classes viewed as differential forms on the space of projective complex symmetric matrices of full rank and use this computation for the study of multiple zeta values.
Martin Olsson - University of California, Berkeley. Title: Representability of flat cohomology and applications. Abstract: I will discuss the structure of cohomology of finite flat group abelian group schemes in positive characteristic and explain analogs for such cohomology groups of classical results in \'etale cohomology, including finiteness results. This is joint work with Daniel Bragg.

Pablo Pelaez - St. Petersburg University. Title: Voevodsky's motives and finite filtrations on the Chow groups. Abstract : We will discuss how Voevodsky's triangulated category of motives provides a natural framework for constructing adequate equivalence relations on the Chow groups of smooth projective varieties, and in particular finite filtrations satisfying several of the properties of the conjectural Bloch-Beilinson filtration.
Bjorn Poonen - MIT. Title: TBA. Abstract:
N. Ramachandran - University of Maryland. Title: Brauer groups and algebraic curves. Abstract: This talk will review some recent results on the relation between the derived category of a genus one curve and the Brauer group of its Jacobian. If time permits, we will also discuss a new construction of the Deligne line bundle attached to a pair of line bundles on a relative curve. (Joint works with Rosenberg and Aldrovandi.)
Vivek Sadhu- IISER, Bhopal, India. Contributed talk. Title: Homotopy invariance of twisted K-theory and Weibel's conjecture. Abstract: This talk mainly discusses two fundamental (twisted) Algebraic K-theory questions: homotopy invariance and the vanishing of (twisted) negative K-groups. We first discuss Weibel's K-dimension conjecture for twisted K-theory. We will also discuss the twisted K-theory of Prufer rings.
Anand Sawant - Tata Institute, Mumbai. Title: Cellular A^1-homology Abstract: I will introduce Cellular A^1-homology for arbitrary smooth varieties over a perfect field, which gives the motivic analogue of cellular homology in classical topology. I will discuss results and conjectures about Poincare duality in this setting. I will also describe a few examples elaborating the analogy with classical Poincare duality for smooth manifolds in classical topology, with emphasis on smooth projective rational surfaces and generalized flag varieties. The talk is based on joint work with Fabien Morel.
Marco Schlichting - University of Warwick, UK. Title: Symmetric versus genuine symmetric forms in Hermitian K-theory. Abstract: In their work on Hermitian K-theory of Poincare infinity categories, Calmès-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle [CDH+] introduced a new version of Hermitian K-theory based on a homotopical notion of symmetric forms. The new theory, (somewhat confusingly) called Hermitian K-theory of symmetric forms, has many good properties, among which are Nisnevich descent and homotopy invariance on regular rings. In this talk, I will discuss the comparison map from the classical Hermitian K-theory of genuine symmetric forms to that of symmetric forms of [CDH+] and show that for finite dimensional regular Noetherian rings that contain a field or are smooth over a Dedekind domain, the comparison map is an isomorphism in degrees ≥−1 and a monomorphism in degree −2, generalising the comparison result of [CDH+] from fields to a large class of regular rings. In particular, the spaces of Hermitian K-theory of genuine symmetric forms and the symplectic K-theory space are homotopy invariant for such rings and are representable in Morel-Voevodsky's A1-homotopy theory. This is based on my preprint available at arXiv:2503.14288
V. Srinivas - State University of New York, Buffalo. Title: Enriched Hodge Structures and cycles on complex analytic thickenings. Abstract: This talk is a report on an ongoing project with Madhav Nori and Deepam Patel. We consider triples (X, A, B) where X is a complex analytic space, A, B are closed analytic subspaces such that A is a proper algebraic variety, and X \ B is a complex manifold, and A \ B is a submanifold. We view this as defining a representative of a germ of an analytic neighbourhood of A (the “thickening” of A). If ι : A → X and j : X \ B → X are the inclusions, we may consider cohomology groups H^m(A, ι⁻¹Rj_∗Z) (and Tate twists). Our goal is to define a variant of Deligne-Beilinson cohomology for such objects, using Enriched Hodge structures (Bloch-Srinivas), which are “enhanced” versions of Mixed Hodge structures. We expect that our “Enriched D-B Cohomologies” would be the targets of regulators defined on suitable K-groups associated to such germs, and these would detect interesting elements in the K-theory of the germs. An example is when X is a small ball around A = {0} in Cⁿ, and B = ∅, which corresponds to the K-groups of the ring of convergent power series in n complex variables; here the underlying MHS has no information, while the “enriched” version has content. In this talk, we will indicate how the EHS’s are constructed, what the corresponding Enriched DB-cohomology looks like, and discuss some simple examples.
Suresh Venapally - Emory University. Title: Rost injectivity for classical groups over function fields of of p-adic curves. Abstract: Let F be a field of characteristic 0 and G an absolutely simple simply connected linear algebraic group defined over F . Rost defines a degree three coho- mological invariant for G torsors which yields a map R_G : H¹(F, G) → H³(F, Q/Z((2))) known as the Rost invariant. If F is the function field of a curve over a p-adic field, G is of classical type and p is a 'good' prime for G, then we prove that the Rost invariant map R_G is injective.
Charles Weibel - Rutgers University. Title: K_2-regularity and normality. Abstract: We show that K_2-regular affine algebras over fields of characteristic zero are normal. Combining this result with K-theoretic calculations of the authors with Cortinas and Walker and algebraic-geometric results of Mustata and Popa, we improve on Vorst's conjecture, showing that much sharper bounds are possible, at least in the case of local complete intersections.
Yilong Zhang - Purdue University. Contributed talk. Title: Correspondence between elliptic-elliptic surfaces and K3 surfaces. Abstract: An elliptic-elliptic surface is an elliptic surface over a genus one curve and has p_g=1. Its polarized H^2 carries a K3-type Hodge structure, which is isometric to a Hodge substructure of a K3 surface with an E8 polarization. Hodge conjecture predicts that the correspondence on their transcendental Hodge structures is algebraic. In other words, two surfaces are geometrically related. In a joint work with Arapura and Greer, we show this is true for certain examples arising from rational double cover of Kummer surfaces. The construction generalizes the work of Shioda-Inose in the 70's.