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Texas A&M University
Mathematics

Maxson Lecture Series

Spring 2023

 

Date:April 10, 2023
Time:4:00pm
Location:BLOC 117
Speaker:Kannan Soundararajan, Stanford University
Title:Divisibility properties of the character values of the symmetric group
Abstract:The values taken by characters of the symmetric group have long been known to be integers. But only recently A. Miller made the surprising empirical observation that most of the character values are even integers, and indeed most seem to be divisible by any given prime number. Miller's conjecture that most character values are even was established by Sarah Peluse in 2020, and subsequently she and I have shown that almost all entries in the character table of the symmetric group are divisible by any fixed integer. I will discuss some of the ideas behind this and related questions. The techniques will combine combinatorial ideas together with an understanding of the structure of random partitions of integers.

Date:April 11, 2023
Time:4:00pm
Location:BLOC 117
Speaker:Kannan Soundararajan, Stanford University
Title:Covering integers using quadratic forms
Abstract:How large must \Delta be so that we can cover a substantial proportion of the integers below X using the binary quadratic forms x^2 +dy^2 with d below \Delta? Problems involving representations by binary quadratic forms have a long history, going back to Fermat. The particular problem mentioned here was recently considered by Hansen and Vaughan, and Diao. In ongoing work with Ben Green, we resolve this problem, and identify a sharp phase transition: If \Delta is below (log X)^{log 2-\epsilon} then zero percent of the integers below X are represented, whereas if \Delta is above (log X)^{log 2 +\epsilon} then 100 percent of the integers below X are represented. I will give a gentle introduction to some of the ideas involved.