Titles and Abstracts

SCHEDULE OF EVENTS

Titles and Abstracts of Talks

James Lewis - University of Alberta Title: The Complexity of Higher Chow Groups Abstract: Let X be a projective algebraic manifold. We introduce two integral invariants, one of which is the level of the Hodge cohomology algebra of X, and the other involving a certain coniveau level of the higher Chow groups of X. We provide a convincing argument as to why these invariants ought to be the same. Link to the talk
Ajneet Dhillon- University of Western Ontario Title: Cartier and Mukai duality and commutative group stacks Abstract: This is a work in progress with Brett Nasserden. We will discuss some duality theorems for commutative groups stacks, aka Picard stacks that are variations and extensions on well known results. They are based upon the foundational work of S. Brochard Link to the talk
Ravindra Girivaru - University of Missouri Title: Lefschetz theorems for higher rank bundles. Abstract: A conjecture of Hailong Dao, subsequently proved by Kestutis Cesnavicius, states (among other things) that a vector bundle on a smooth complete intersection of dimension at least three splits into a sum of line bundles if its endomorphism bundle satisfies certain vanishing conditions. This generalizes the Grothendieck-Lefschetz theorem to arbitrary rank bundles. In this talk, which is based on joint work with Amit Tripathi (IIT Hyderabad), we will describe a geometric proof of this theorem by relating it to a vanishing theorem of Kempf (which was sharpened subsequently by Mohan Kumar and I. Biswas). We will also talk about a version of this theorem for complete intersection surfaces which generalizes the Noether-Lefschetz theorem. Link to the talk
Xi Chen - University of Alberta Title: Moduli of Curves on K3 Surfaces Abstract: We prove that on every projective complex K3 surface and every g > 0, there are infinitely many families of curves of geometric genus g with maximal moduli. In particular every K3 surface contains a curve of geometric genus 1 which moves in a non-isotrivial family. This implies a conjecture of Huybrechts on constant cycle curves and gives an algebro-geometric proof of a theorem of Kobayashi that a K3 surface has no global symmetric differential forms. This is a joint work with Frank Gounelas. Link to the talk

Fabien Morel - University of Munich Title: On the Poincar'e duality of the cellular A^1-chain complex of a smooth projective variety Abstract: This talk is based on joint work with Anand Sawant. In a previous work we introduced (accidentally) a simplified version of the A^1-chain complex (or motive) of a smooth k-scheme, that we called the cellular A^1-chain complex. For smooth k-schemes admitting a cellular structure, this chain complex is quite concrete and very computable (as well as its associated cellular A^1-homology). This construction can be conceptually understood and extended for general smooth k-schemes. This leads to a complex of pro-objects in the category of strictly A^1-invariant sheaves over k (related to the Gersten-Rost-Schmitt complexes), which can be seen as a "simplication" of the Motive, though it contains for instance the Chow groups. In this talk I will address the problem of proving Poincar'e duality in the most elementary possible way, and discuss some closely related questions: for instance are each cellular A^1-homology sheaves of any smooth k-scheme in fact constant? And what does imply the Poincar'e duality on the top dimensional cellular A^1-homology sheaf of a smooth k-scheme. Link to the talk
Deepam Patel - Purdue University Title : Push forward of Boundary Monodromy Abstract: Given a smooth proper morphism f: X —> D^, where X is a complex algebraic variety and D^ is a small punctured disk, it is a well-known theorem that the local monodromy of the local systems R^if_*C is quasi-unipotent. In this talk, I’ll discuss generalizations of this sort of result for arbitrary morphisms (over higher dimensional bases) and with coefficients in arbitrary constructible sheaves. If there is time, I’ll discuss applications to variation of local monodromy in abelian towers, and its relation to Gamma motives. This is based on joint work with Madhav Nori. Link to the talk
Charles Doran - Bard College/University of Alberta Title: Modularity of Landau-Ginzburg Models Abstract: Unlike the case of Calabi-Yau manifolds which are mirror to other Calabi-Yau manifolds, the mirror of a Fano manifold is a Landau-Ginzburg model, essentially a mapping from a non-compact manifold to the affine line. The fibers of this mapping can be compactified to become Calabi-Yau, and conjecturally these should be mirror to the anticanonical hypersurfaces in the Fano manifold. This expectation was known to be true for rank 1 Fano threefolds, where the anticanonical K3 surfaces possess various even polarizations and the mirror fibers are derived from modular families of high Picard rank lattice polarized K3 surfaces. We have now fully extended this story to the complete Mori-Mukai list of smooth Fano threefolds. As a corollary we obtain strong uniruledness results for the moduli spaces of lattice polarized K3 surfaces arising as fibers of the mirror Landau-Ginzburg models. Link to the talk
Connor Cassady - Ohio State University Title: Universal quadratic forms over semi-global fields Abstract: Given a quadratic form (homogeneous degree two polynomial) q over a field k, some basic questions one can ask are Does q have a non-trivial zero (is q isotropic)? Which non-zero elements of k are represented by q? Does q represent all non-zero elements of k (is q universal)? Over a global field F, the Hasse-Minkowski Theorem, which is one of the first examples of a local-global principle, allows us to use answers to these questions over the completions of F to form answers to these questions over F itself. In this talk, we will focus primarily on quadratic forms over semi-global fields (function fields of curves over complete discretely valued fields), and see how a local-global principle of Harbater, Hartmann, and Krashen can be used to study universal quadratic forms over semi-global fields. Link to the talk

Roy Joshua- Ohio State University Title: Equivariant Algebraic K-theory and Derived completion II: the case of equivariant homotopy K-theory and applications Abstract : This is a report on work in progress with Gunnar Carlsson and Pablo Pelaez. Derived completion is a technique that originated slightly over 20 years ago. In the first part of this work, we proved a derived completion theorem for equivariant G-theory, extending the work of Atiyah and Segal in the topological case and that of Thomason for equivariant G-theory with finite coefficients and with the Bott-element inverted. In the present work we extend these results to equivariant homotopy K-theory and discuss applications to equivariant Riemann-Roch problems. Link to the talk
Gregory Pearlstein - University of Pisa Title: Infinitesimal Torelli and rigidity results for a remarkable class of elliptic surfaces Abstract: I will discuss joint work with Chris Peters which extends rigidity results of Arakalov, Faltings and Peters to period maps arising from families of complex algebraic varieties which are non-necessarily proper or smooth. Inspired by recent work with P. Gallardo, L. Schaffler, Z. Zhang, I will discuss two classes of elliptic surfaces which can be presented as hypersurfaces in weighted projective spaces which have a unique canonical curve. In each case, we will show that infinitesimal Torelli fails for H^2 of the compact surface, but is restored when one considers the period map for the complement of the canonical curve. Link to the talk
Ursula Whitcher - American Math Society Title: Highly symmetric Calabi-Yau Grassmannian hypersurfaces Abstract:We use computational techniques based on work of Fatighenti and Mongardi to study the Hodge structure of families of Calabi-Yau varieties realized as quotients of Grassmannian hypersurfaces by abelian groups. We show that, in contrast to a proposal of Coates, Doran, and Kalashnikov, these families do not yield a direct generalization of the classical Greene-Plesser mirror construction, but they do provide a new perspective on the Calabi-Yau landscape. Link to the talk
Pablo Pelaez - UNAM, Mexico Title: Incidence equivalence and the Bloch-Beilinson filtration Abstract: We will present an approach to the Bloch-Beilinson filtration via Voevodsky’s triangulated category of motives, and show that the second step of the filtration is given by cycles incident equivalent to zero. Link to the talk

Marco Schlichting - University of Warwick Title: On the relation of Milnor-Witt K-theory and Hermitian K-theory Abstract:Let R be a commutative local ring. There is a canonical multiplicative homomorphism K_n^{WM}(R) -> GW^n_n(R) from Milnor-Witt K-theory of R to its Hermitian K-theory. I will explain recent improvements on homology stability of symplectic groups over local rings with infinite residue fields and use this to construct a map GW^n_n(R) -> K_n^{MW}_n(R) when n=2,3 mod 4. Finally, we compute the composition K_n^{WM}(R) -> GW^n_n(R) -> K_n^{MW}_n(R). Using A1-homotopy theory, Asok, Fasel and Deglise have obtained similar results when R is in addition regular. Link to the talk
Aylet Lindenstrauss - Indiana University Title: On the K-theory of division algebras over local fields Abstract: I will discuss joint work with Lars Hesselholt and Michael Larsen in which we calculate the $p$-adic algebraic K-theory groups of a central division algebra of finite index $d$ over the fraction field of a complete discrete valuation ring with a finite residue field of odd characteristic $p$. The result is that the $p$-adic algebraic K-theory of the division algebra is isomorphic to that of the center, just like Suslin and Yufryakov showed for the $\ell$-adic K-theory for primes $\ell\neq p$. There is a norm map between the $p$-adic K-theory of the division algebra and that of the center, but it is not an isomorphism if $p$ divides $d$; we construct a homomorphism so that $d$ times it is the norm, using topological Hochschild homology and topological cyclic homology. Link to the talk
Jens Hornbostel - University of Wuppertal Title: Real topological Hochschild homology of schemes Abstract: In recent years, several people studied real topological Hochschild homology THR as a C_2-equivariant refinement of THH. This is in analogy with real or hermitian K-theory refining usual complex or algebraic K-theory. We recall the definitions and basic facts and computations about THH and THR. Then we discuss some recent results like base change and descent for the Z/2-isovariant \'etale topology. We also provide computations of THR for the projective line (with and without involution) and higher dimensional projective spaces. This is joint work with Doosung Park. Link to the talk
Morgan Opie - UCLA Title : Topological vector bundles on complex projective spaces Abstract: Given two complex topological bundles over $\mathbb CP^n$, one can ask whether the bundles are equivalent. The first test is to compare their Chern classes since equivalent bundles have the same Chern data. The converse fails in general, which leads to the following question: given $n$ and $k$ positive integers, what invariants beyond Chern classes are needed to distinguish complex rank $k$ topological bundles on $\mathbb CP^n$, up to topological equivalence? In this talk, I will discuss the subtleties of using methods from stable homotopy theory to answer this question. I'll start by explaining how Atiyah--Rees classified all complex rank 2 topological vector bundles on $\mathbb CP^3$ via an invariant valued in the generalized cohomology theory of real K theory. I will then discuss my work classifying complex rank 3 topological vector bundles on $\mathbb CP^5$ using a generalized cohomology theory called topological modular forms. As time allows, I will discuss work in progress (joint with Hood Chatham and Yang Hu) to address other ranks and dimensions. Link to the talk
Soumya Sinha Babu - University of Georgia Title: K_2 and Quantum Curves Abstract: The goal of this talk is to understand the implications of Codesido-Grassi-Marino conjecture in the ’t Hooft limit. In this case, Kashaev-Marino-Zarkany deduced that the limiting values of the local mirror map at the maximal conifold point are given by Bloch-Wigner dilogarithms of algebraic arguments. I will demonstrate these assertions by calculating regulator periods of a K_2 class on mirror curves attached to 3-term quantum operators. Consequently, numerous series identities are established. This is joint work with C. Doran and M. Kerr and part of ongoing work with P. Bousseau. Link to the talk

Lemarie-Rieusset-Clementine- University of Bourgogne Title: Motivic knot theory Abstract: In this talk I will present a new application of motivic homotopy theory: motivic knot theory. More specifically, I will present counterparts in algebraic geometry to the linking number of two oriented disjoint knots (the number of times one of the knots turns around the other knot). I will mostly focus on oriented couples of closed immersions of the affine plane minus the origin in the affine 4-space minus the origin (which play the role of couples of oriented knots). I will also consider, more generally, oriented couples of closed immersions of smooth models of motivic spheres (which are counterparts in algebraic geometry to classical topological spheres). Link to the talk
Bernhard Koeck - University of Southampton Title: Comparison of operations on higher K-theory Abstract: Operations such as exterior power operations provide additional structure on K-theory that has been very useful for many purposes (e.g. in Riemann-Roch theory). Many a priori different definitions of them In the context of higher algebraic K-theory of schemes have been given over the last decades. In this talk, I will explain how to prove that these constructions yield the same operations. This is joint work with F. Zanchetta. Link to the talk
Charles Weibel - Rutgers University Title: The K-theory of polynomial-like rings Abstract: By a polynomial-like ring R I mean the ring of an affine toric variety or a truncated polynomial ring in many variables. We show that its relative K-theory is a direct sum over the rays in R of continuous modules over the ring of big Witt vectors. This is joint work with Christian Haesemeyer. Link to the talk