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[Lecture] Beijing-Saint Petersburg Mathematics Colloquium
Oct. 06, 2022
Lecture 1

Title: Aspects of the Bogomolov conjectures



Speaker: Xinyi Yuan, Beijing International Center for Mathematical Research

Time: 20:00-21:00 pm, October 6, 2022, GMT+8

Venue: Zoom Meeting ID: 831 9733 8053 Password: 654321

Abstract:

The original Bogomolov conjecture (proved by Ullmo) asserts that a projective curve of genus greater than 1 over a number field has only finitely many points of small Neron-Tate heights. We will introduce this conjecture and the recent developments on other versions of it, including the geometric Bogomolov conjecture and the uniform Bogomolov conjecture.

Biography:

Xinyi Yuan is currently a chair professor at Peking University. He got his Bachelor’s Degree from Peking University in 2003, and got his PHD from Columbia University in 2008. Before joining Peking University in 2020, he was an associate professor at UC Berkeley. He specializes in number theory and arithmetic geometry.

Lecture 2

Title: On a conjecture due to J.-L. Colliot-Thelene



Speaker: Ivan Panin, St. Petersburg Department of Steklov Mathematical Institute

Time: 21:00-22:00 pm, October 6, 2022, GMT+8

Venue: Zoom Meeting ID: 831 9733 8053 Password: 654321

Abstract:

Let $R$ be a regular local ring containing a field, $K$ be its fraction field, $a\in R^{\times}$ be a unit, $n\geq 1$ be an integer, $1/2$ is in $R$. Particularly, we prove the following result. Suppose a is a sum of n squares in K. Then a is a sum of n squares in R. This is a partial case of a conjecture due to J.-L. Colliot-Thelene (1979). The conjecture is solved in positive for regular local rings containing a field.

In more details. If R contains rational numbers, then the conjecture is solved by the speaker in his Inventiones paper (2009). If R contains a finite field and the residue field of R is infinite, then the conjecture is solved by the speaker jointly with K.Pimenov in 2010 in their Doc. Math. paper. If R contains a finite field, then the conjecture is solved by S. Scully in 2018 in his Proceedings of the AMS paper. If time permits, very recent progress in the topic will be discussed.

Biography:

Ivan Panin is a Chair of Algebra and Number Theory at Steklov Mathematical Institute at Sankt-Petersburg. He is a Corresponding Member of the Russian Academy of Sciences since 2003. He got his PhD in 1984 under the supervision of Andrei Suslin. Ivan Panin got his Habilitation in 1995. He was an invited speaker at ICM-2018 in Rio de Janeiro. He solved the Grothendieck--Serre conjecture on principal G-bundles (for regular local rings containing a field). He invented (jointly with A. Smirnov) a topic of oriented cohomology theories on algebraic varieties, stated and proved a Riemann--Roch type theorem for oriented cohomology theories. Jointly with G. Garkusha, he realised a project due to V.Voevodsky producing a machinery for computing motivic infinite loop spaces.

Source: SRMC