**Upcoming talk**

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**Title: **Algebraic formulae for differential equations of polylogarithms: genus 0, 1, and beyond?

**Speaker: **Prof. Tiago J. Fonseca (IMECC - Unicamp)

**Time: **Fri., 8:00-9:00 am, June 14, 2024

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

**Abstract: **

The classical (multiple) polylogarithm functions play, through their special values, a prominent role in a number of arithmetic questions related to periods and motives. Crucially, these functions satisfy an algebraic differential equation, the KZ equation, from which many of their properties are derived. In particular, they can be written as iterated integrals of certain algebraic differential forms with logarithmic singularities on the complex projective line. One may ask what can be said for higher genus curves.

In this talk, I will survey some of the classical genus 0 theory and discuss joint work with Nils Matthes one the genus 1 case. We describe explicit algebraic formulae for the analogue of the KZ equation in the genus 1 case, the so-called elliptic KZB equation, relying on algebro-geometric properties of universal vector extensions of elliptic curves. If time permits, I will also speculate on a possible direction to approach the higher genus case.

**Past talk**

**Title: **Differential equations, hypergeometric families, and Beilinson's conjectures

**Speaker:** Prof. Matt Kerr（Washington University in St. Louis）

**Time:** Tues., 9:00-10:00 am, May 28, 2024

**Venue：**Zoom Meeting ID: 687 513 9542 Passcode: YMSC

**Abstract: **

We discuss two approaches to solving inhomogeneous equations of the form L(.)=t^{1/d}, where L is a hypergeometric differential operator attached to a family of CY varieties. The first is by elementary complex analysis, using so-called Frobenius deformations, and gives an explicit series solution. The second is via normal functions attached to algebraic cycles (both "classical" and "higher") on a base-change of the family.

I will briefly review regulator maps, their relation to inhomogeneous Picard-Fuchs equations, and the relevant cases of the Beilinson conjecture. Turning then to the CY3 examples classified by Doran-Morgan, I will explain how to identify which types of cycles arise (viz., K_0, K_2, or K_4 classes), and how to use the Frobenius deformations to make the conjecture more explicit. This is joint work with Vasily Golyshev.

**Title:** Rational K\pi 1 property for Reimann surfaces

**Speaker: **Prof. Tomohide Terasoma（Hosei University）

**Time: **Fri., 9:00-10:00 am, May 10, 2024

**Venue：**Online Zoom Meeting ID: 4552601552 Passcode: YMSC

**Abstract:**

In the study of mixed elliptic motives, we noticed the importance for rational K\pi 1 property on some bar complex arising from mixed elliptic motives. Since its bar complex is some what complicated, it will be useful to consider some easier case, the case for Riemann surfaces.

Rational fundamental group of Riemann surface is desribed by Sullivan's old paper, but it seems that rational K\pi 1 property seems to be not well discussed. As for arrangements of hyperplanes, several technic and results are know by several person, including the case for supersolvable case,etc.

In our talk, we consider some technic to show the rational K\pi 1 property for Riemann surfaces, which might be applicable to mixed elliptic motives.

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Rational_Kpi_1_property_for_Reimann_surfaces.html

**Title:** Techniques to move algebraic cycles, old and new

**Speaker：**Wataru Kai（Tohoku University）

**Time：**Thur., 15:00-16:00, April 25, 2024

**Venue：**Online Zoom Meeting ID: 4552601552 Passcode: YMSC

**Abstract:**

Moving lemmas are statements about how to move given closed subsets of varieties into generic, well-behaved positions. They are often technical keys to important results on algebraic cycles and cohomology theories. I will review some historically significant moving lemmas and mention a few that I have proven in recent years.

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

**Title: **Cohomology of local systems on the moduli of abelian varieties and Siegel modular forms

**Speaker：**Gerard van der Geer (University of Amsterdam)

**Time：**Thur., 16:30-17:30, Jan. 18, 2024

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

**Abstract:**

The cohomology of local systems on the moduli of principally polarized abelian varieties naturally harbours modular forms and their motives. The talk explains joint work with Faber and Bergström on the cohomology of local systems on the moduli of abelian varieties of dimension 2 and 3 and shows how it provides a lot of knowledge about Siegel modular forms.

Personal Website：http://van-der-geer.nl/~gerard/

**Title:** Is the GT Lie bialgebra motivic?

**Speaker:** Richard Hain (Duke University)

**Time:** Tues., 9:00-10:00 am, January 9, 2024

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

**Abstract: **

The Goldman--Turaev Lie bialgebra of an oriented 2-manifold $X$ is a Lie bialgebra structure on the free abelian group spanned by the conjugacy classes of its fundamental group. Its structure encodes how isotopy classes of immersed loops on $X$ intersect each other and themselves. When $X$ is a smooth complex curve, a suitable completion of the GT-Lie bialgebra carries a natural mixed Hodge structure. The bracket and cobracket are both morphisms (after a suitable twist). This raises the question of whether (when $X$ is defined over a number field) of whether the bracket and cobracket are motivic and, if so, how they are related to algebraic cycles.

In this talk I will define the Goldman bracket and Turaev cobracket, as well as their extensions by Kawazumi and Kuno. I'll survey what is known about the MHS and Galois actions on the completed GT-Lie bialgebra and indicate some connections to the Ceresa cycle when $g(X) > 2$.

Personal Website： https://scholars.duke.edu/person/hain

**Title: **Feynman periods as Apéry limits

**Speaker: **Erik Panzer (University of Oxford)

**Time: **Thur., 4:00 pm-5:00 pm, Dec. 21, 2023

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

**Abstract: **

To prove that zeta(3) is irrational, Apéry realized this number as the limit of the ratio of two solutions of a linear recurrence with polynomial coefficients. Similar recurrences and their "Apéry limits" have been studied in mirror symmetry. In this talk, I will sketch a method to realize Feynman integrals as Apéry limits, using a combinatorial graph invariant from arXiv:2304.05299. Examples include fourth order recurrences for zeta(3,5) and a third order recurrence for zeta(5) and zeta(9). The mechanism behind this method is a general theory (applicable beyond Feynman integrals) based on a Mellin transform, and furthermore connecting "diagonal" coefficients of powers of a polynomial (combinatorics) with point-counts over finite fields (arithmetic). This is work in progress together with Francis Brown.

Personal website: https://people.maths.ox.ac.uk/panzer/

**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.

Slides:

Talk_Online_seminar_on_periods_and_motives.pdf

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

**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/