All talks will be in Room E208 of the Mathematics and Science Center (400 Dowman Drive, Atlanta GA 30322). Registration and refreshments will be in the second floor atrium.
Roman Fedorov (University of Pittsburgh)
On the conjecture of Grothendiek and Serre in mixed characteristic.
The conjecture of Grothendieck and Serre predicts that a torsor under
a reductive group scheme over a regular local ring is trivial, if it
is trivial generically. This has been known if the ring contains a
field since approximately 2014. This is still open in the mixed
characteristic case. I will survey some recent progress due to
Kestutis Cesnavicius and myself.
Evangelia Gazaki (University of Virginia)
Weak Approximation for zero-cycles.
A smooth projective variety over an algebraic number field is said to satisfy Weak Approximation if the set of rational points is dense inside the set of all adelic points. By the foundational work of Y. Manin, the Brauer group of the variety often obstructs Weak Approximation and for certain classes of varieties this obstruction is known or conjectured to fully explain the failure. In other words, every family of local points that is orthogonal to the Brauer group can be approximated by a global point. In this talk I will discuss analogs of these questions for the Chow group of zero-cycles. In the 1980's Colliot-Thelene, Sansuc, Kato and Saito conjectured that the Brauer group should give the only obstruction to Weak Approximation for zero-cycles when X is a general smooth projective variety. This conjecture has only been established for certain classes of rationally connected varieties under a strong assumption about the set of rational points, and there is some recent partial evidence for products of K3 surfaces. The purpose of this talk will be to give some unconditional evidence for a product of elliptic curves with complex multiplication. This talk will be based on a recent preprint with an Appendix by Angelos Koutsianas.
Angela Gibney (University of Pennsylvania)
Towards vector bundles on the moduli space of curves from strongly finite VOAs
Abstract: Given any vertex operator algebra V, Zhu defined an associative algebra A(V), and showed that to any A(V)-module, one can associate an admissible V-module. This gives rise to a functor taking n-tuples of A(V)-modules to a sheaf of coinvariants (and its dual sheaf of conformal blocks) on the moduli space of stable n-pointed curves of genus g. If V is strongly rational (in which case A(V) is finite and semi-simple), much is known about these sheaves, including that they are coherent and satisfy a factorization property. Factorization ultimately allows one to show the sheaves are vector bundles with Chern classes in the tautological ring. In this talk I will describe a program in which we are aiming for analogous results after removing the assumption of rationality. As a first step, we replace the standard factorization formula with an inductive one that holds for sheaves defined by modules over any VOA of CFT-type. As an application, we show that if V is strongly finite, then sheaves of coinvariants and conformal blocks are coherent. This is a preliminary description of new and ongoing joint work with Krashen and Damiolini, extending work with Damiolini and Tarasca.
Philippe Gille (CNRS, Université Lyon 1)
R-equivalence for group schemes
Abstract :This is a report on a joint work with Anastasia Stavrova (St Petersburg). For a group scheme G over a ring A, we define the R-equivalence on G(A) in a compatible way with the case of algebraic groups. We compute the invariant G(A)/R in the case of a local ring when G is a torus or G is isotropic semisimple simply connected.
Diego Izquierdo (École Polytechnique)
Milnor K-theory and zero-cycles over p-adic function fields
In 1986, Kato and Kuzumaki introduced a set of conjectures in order to characterize the cohomological dimension of fields in diophantine terms. The conjectures are known to be wrong in full generality, but they provide interesting arithmetical problems over various usual fields in arithmetic geometry. The goal of this talk is to discuss the case of function fields of p-adic curves. This is an ongoing work with G. Lucchini Arteche.
Martin Olsson (University of California, Berkeley)
Reconstruction of varieties.
Abstract: In the standard treatments of algebraic geometry, such as typically taught in a graduate course on the subject, a scheme is defined as a pair $X=(|X|, O_X)$ consisting of a topological space $|X|$ and a sheaf of rings $O_X$, satisfying various conditions. In this talk I will discuss joint work with János Kollár, Max Lieblich, and Will Sawin showing that in many cases the topological space $|X|$ alone determines $X$.
Karl Schwede (University of Utah)
Singularities in mixed characteristic
Abstract: It has been known for several decades that the classes
of singularities coming out of the minimal program over the complex numbers
(log canonical, Kawamata log terminal, etc.) are closely related to
singularities coming out of the theories of Frobenius splitting and tight
closure for schemes in characteristic p > 0 (F-pure, F-regular, etc.)
Building upon breakthroughs of Scholze, Andr\'e, Bhatt, Gabber and others, an
opportunity has been created to build a theory of singularities in mixed
characterstic (for general Noetherian schemes). I will talk about
recent joint work with various co-authors including Bhatt, Hacon, Lamarche, Ma,
Patakfalvi, Tucker, Waldron, and Witaszek where we begin to develop this theory
and applications.
Brooke Ullery (Emory University)
Cayley-Bacharach theorems and measures of irrationality
Given a finite set of points in projective space, a classical question, dating back to at least the 4th century, is how many of the conditions imposed by the points on degree d polynomials are independent? The so-called Cayley-Bacharach condition describes instead whether the conditions imposed by the points are affected when one point is removed. In this talk, I’ll discuss some recent and ongoing work with Juliette Bruce, Jake Levinson, and Ritvik Ramkumar about points satisfying the Cayley-Bacharach condition and show how they connect to several seemingly unrelated questions about the gonality of curves and measures of irrationality of algebraic varieties.