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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.
Bio: Bio: 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.
