**Upcoming talk**

**Title:** Motivic Galois theory for algebraic Mellin transforms

**Speaker：**Clément Dupont, Université de Montpellier, France

**Time: **Fri., 4:00 pm-5:00 pm, Dec. 8, 2023

**Venue：**Zoom Meeting ID: 276 366 7254 Passcode: YMSC

**Abstract: **

This talk will discuss series expansions of algebraic Mellin transforms, and the periods that appear as their coefficients. The basic example is Euler's beta function, whose series expansion features values of the Riemann zeta function at integers. I will explain how the motivic Galois group acts on series expansions of algebraic Mellin transforms, and give examples. As an application, we obtain a ''cosmic Galois theory'' (prophesized by Cartier) for Feynman integrals in dimensional regularization. This is joint work with Francis Brown, Javier Fresán, and Matija Tapušković.

Clément Dupont，Personal Website：https://imag.umontpellier.fr/~dupont/

**Past talk**

**Title:** Kummer extensions of Hodge structures

**Speaker: **Spencer Bloch (University of Chicago)

**Time: **Thur., 8:30 am, Nov. 23, 2023

**Venue：**Online Zoom: 271 534 5558 Passcode: YMSC

**Abstract:**

Kummer extensions of Hodge structures are extensions of the form 0 -> Z(1) -> K -> Z(0) ->0. Associated to a Kummer extension is the extension class, which has a tendency to be algebraic, and a unique period associated to the Hodge structure on K. The period is the log of the extension class. Kummer extensions are too simple to be of interest in and of themselves, but they often serve as the "platter" on which the meat and potatoes are served. I will discuss two examples.

1. Higher cross-ratios and functions on Hilbert schemes. Here the Kummer extension is actually a degenerate biextension. Biextensions are mixed Hodge structures with weight graded structure Q(1), H, Q(0), where H is a pure Hodge structure of weight -1. It can happen that H=(0), in which case the biextension becomes a Kummer extension. A class of such degenerate biextensions arises from the study of algebraic cycles on varieties with vanishing odd dimensional Betti cohomology. In this case, the extension class becomes an algebraic function on the Hilbert scheme.

2. The Gross-Zagier conjecture. This is a conjecture about values of suitable Green's functions where the t_i are CM points on a modular curve. The conjecture gives G(t_1,t_2) = D log(a) where D is a product of discriminants for the t_i, and a is an algebraic number. Since periods of Kummer extensions are always of the form D log(a), it is natural to look for a Kummer extension somewhere.

Spencer Bloch （University of Chicago）, Personal Website：https://www.math.uchicago.edu/~bloch/

Video： http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Kummer_extensions_of_Hodge_structures.html

**Title: **Combing the n-sphere over a field

**Speaker：**Marc Levine (Duisburg-Essen University)

**Time: **Wed., 4:00 pm, Nov. 8, 2023

**Venue：**Zoom Meeting ID: 271 534 5558 Passcode: YMSC

**Abstract: **

This is a report on a joint work with Alexey Ananyevskiy. It is a classical result in differential topology that every vector field on the 2n-sphere must vanish somewhere, while the 2n+1-sphere admits nowhere zero vector fields. Viewing the n-sphere as the affine hypersurface \sum_{i=1}^{n+1}x_i^2=1 in R^{n+1}, one can ask the same question over an arbitrary field k, or more generally, for the hypersurface \sum_{i=1}^{n+1}a_ix_i^2=1, with the a_i units in k. In the case of odd n, one can write down an explicit nowhere zero section of the tangent bundle. For even n and assuming the equation \sum_{i=1}^{n+1}a_ix_i^2=1 has a solution with the x_i in k, and taking k to be perfect, we have a number of equivalent criteria for the existence of a nowhere zero section of the tanget bundle, which are often easy to check. For example, in this case, the tangent bundle has a nowhere zero section if and only if -1 is in the subgroup of the units of k generated by the non-zero values of the quadratic form \sum_{i=1}^{n+1}a_ix_i^2, with the x_i in k. The proof uses a mixture of some recent results on quadratic Euler characteristics, properties of Euler classes in the Chow-Witt ring, and some useful facts and constructions involving quadratic forms.

Personal Website：https://www.esaga.uni-due.de/marc.levine/