Location: 25 Park Place, Room 1441

Colloquium: Recent progress in Ramsey Theory

Speaker: Jacques Verstraete, University of California, San Diego

Title: Recent progress in Ramsey Theory

Abstract: The Ramsey number r(s,t) denotes the minimum N such that in any red-blue coloring of the edges of the complete graph K_N, there exists a red K_s or a blue K_t. While the study of these quantities goes back almost one hundred years, to early papers of Ramsey and Erdos and Szekeres, the long-standing conjecture of Erdos that r(s,t) has order of magnitude close to t^{s - 1} as t goes to infinity remains open in general.

It took roughly sixty years before the order of magnitude of r(3,t) was determined by Jeong Han Kim, who showed r(3,t) has order of magnitude t^2/(\log t). In this talk, we discuss a variety of new techniques which lead to the lower bound in the following statement: for some constants a,b > 0 and t \geq 2,

\[ a\frac{t^3}{(\log t)^4} \leq r(4,t) \leq b\frac{t^3}{(\log t)^2}.\]

This solves a conjecture of Erd\H{o}s. We also come close to determining related quantities known as Erdos-Rogers functions,

as well as determine the current best bounds for other graph Ramsey numbers.

Joint work in part with Sam Mattheus and Dhruv Mubayi.

Host: Yi Zhao (yzhao6@gsu.edu)

]]>The keynote speaker is Dr. Michael Kosorok, W.R. Kenan, Jr. Distinguished Professor, Department of Biostatistics, and Professor, Department of Statistics and Operations Research, University of North Carolina at Chapel Hill and the current President, Institute of Mathematical Statistics.

The workshop is sponsored by the National Science Foundation, Georgia Chapter of the American Statistical Association, Lifetime Data Science Section of the American Statistical Association, Institute of Mathematical Statistics, National Institute of Statistical Sciences, International Chinese Statistical Association, SRCOS, and the Department of Mathematics and Statistics in the GSU.

For more information, please visit the website https://math.gsu.edu/yichuan/2024Workshop/.

Graduate students are welcome to submit the title/abstract to the poster session.

The keynote speaker is Dr. Michael Kosorok, W.R. Kenan, Jr. Distinguished Professor, Department of Biostatistics, and Professor, Department of Statistics and Operations Research, University of North Carolina at Chapel Hill and the current President, Institute of Mathematical Statistics.

The workshop is sponsored by the National Science Foundation, Georgia Chapter of the American Statistical Association, Lifetime Data Science Section of the American Statistical Association, Institute of Mathematical Statistics, National Institute of Statistical Sciences, International Chinese Statistical Association, SRCOS, and the Department of Mathematics and Statistics in the GSU.

For more information, please visit the website https://math.gsu.edu/yichuan/2024Workshop/.

Graduate students are welcome to submit the title/abstract to the poster session.

The keynote speaker is Dr. Michael Kosorok, W.R. Kenan, Jr. Distinguished Professor, Department of Biostatistics, and Professor, Department of Statistics and Operations Research, University of North Carolina at Chapel Hill and the current President, Institute of Mathematical Statistics.

The workshop is sponsored by the National Science Foundation, Georgia Chapter of the American Statistical Association, Lifetime Data Science Section of the American Statistical Association, Institute of Mathematical Statistics, National Institute of Statistical Sciences, International Chinese Statistical Association, SRCOS, and the Department of Mathematics and Statistics in the GSU.

For more information, please visit the website https://math.gsu.edu/yichuan/2024Workshop/.

Graduate students are welcome to submit the title/abstract to the poster session.

Location: 25 Park Place

Algebra Seminar: Monomial Ideals Fixed by Differential Operators

Speaker: William D Taylor, Tennessee State University

Title: Monomial Ideals Fixed by Differential Operators

Abstract: In this talk we will look at monomial ideals in polynomial rings, or more generally semigroup rings, and differential operators on those rings. We say an ideal is “compatible” with an operator if the operator sends the ideal to itself, and an ideal is “fixed” by an operator if in addition the ideal is the image of itself under the operator. We will ask the following questions: Which differential operators fix some monomial ideal, and which monomial ideals are fixed by some differential operator? These questions are motivated by analogies between Cartier maps in positive characteristic and differential operators in any characteristic, and by similar results on fixed ideals of Cartier maps. We will give a complete, and satisfyingly constructive, answer to both questions in the case of a homogeneous differential operator on a normal complex semigroup ring. Our main tools will be a characterization of such operators due to Saito and Traves and identifying the monomials of the ring with a lattice of points in Euclidean space.

Host: Florian Enescu (fenescu@gsu.edu)

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