# Number theory seminar this fall

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### Bianca Viray

Oct 4, 2022, 1:06:55 AM10/4/22
to ntuw
Hi everyone,

This fall we will have two number theory seminars.

The first is in person, tomorrow at 11am in C401, and the speaker is Seungki Kim from University of Cincinnati.  Jayadev sent the title and abstract late last week, but I've included it for reference below.

The second talk is next week (October 11) at 11am on Zoom.  The speaker is Debanjana Kundu from UBC and she'll be speaking on "Studying Hilbert's 10th problem via explicit elliptic curves".  https://washington.zoom.us/j/92671346962

I'll be in touch later in the quarter about the schedule for winter and spring.

Hope you see you tomorrow in C401 and next week on Zoom!

Best,
Bianca

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October 4
Seungki Kim from University of Cincinnati
Lattice problems and the Siegel/Rogers integral formula Abstract: This talk is an introduction to the classical lattice problems of sphere packing and covering in arbitrarily high dimensions, and the Siegel and Rogers integral formula developed in the 1940-50's to study them. They still remain as one of the strongest approaches to the family of such problems, and the records established back then remain unbeaten to this day. I present some past and recent works in the literature, as well as a few works of mine, that are in some sense reflections on the difficulty of these problems.

Debanjana Kundu from UBC
Studying Hilbert's 10th problem via explicit elliptic curves
Abstract: In 1900, Hilbert posed the following problem: "Given a Diophantine equation with integer coefficients: to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in (rational) integers."

Building on the work of several mathematicians, in 1970, Matiyasevich proved that this problem has a negative answer, i.e., such a general `process' (algorithm) does not exist.

In the late 1970's, Denef--Lipshitz formulated an analogue of Hilbert's 10th problem for rings of integers of number fields.

In recent years, techniques from arithmetic geometry have been used extensively to attack this problem. One such instance is the work of García-Fritz and Pasten (from 2019) which showed that the analogue of Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. In joint work with Lei and Sprung, we improve their proportions and extend their results in several directions.

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Bianca Viray (she/her)
Craig McKibben and Sarah Merner Professor of Mathematics
University of Washington
Seattle, WA 98195