Activities This Week

If $X$ is a curve of genus at least $2$ defined over the rational numbers, we know by Faltings’s Theorem that the set $X(Q)$ of rational points is finite but we don’t know how to systematically compute this set. In 2005, Minhyong Kim proposed a new framework for studying rational (or S-integral) points on curves, called the Chabauty–Kim method. It aims to produce $p$-adic analytic functions on $X(Q_p)$ containing the rational points $X(Q)$ in their zero locus. We apply this method to solve the S-unit equation for S={2,3} and computationally verify Kim’s Conjecture for many choices of the auxiliary prime $p$.

In the past, the discovery of ultra-rare compute bugs such as incorrect divisions by Pentium chips or cryptographic hash collisions have set headlines and rocked stock markets. All while the Table Maker’s Dilemma bug, which causes many (but not all) mathematical functions to unexpectedly return slightly incorrect results, remains largely unknown. In my talk I hope to shed some light on this widespread yet poorly understood bug. I will outline a new theoretical framework for modeling its behavior in “the real world”, and raise some open questions.

In this talk we describe two seemingly unrelated results on the symmetric group $S_n$. A family F of permutations is called t-intersecting if any two permutations in F agree on at least t values. In 1977, Deza and Frankl conjectured that for all $n>n_0(t)$, the maximal size of a t-intersecting subfamily of $S_n$ is $(n-t)!$. Ellis, Friedgut and Pilpel (JAMS, 2011) proved the conjecture for all $n>exp(exp(t))$ and conjectured that it holds for all $n>2t$. We prove that the conjecture holds for all $n>ct$ for some constant c. A well-known problem asks for characterizing subsets A of $S_n$ whose square $A^2$ contains (=”covers”) the alternating group $A_n$. We show that if A is a union of conjugacy classes of density at least $exp(-n^{2/5-\epsilon})$ then $A_n \subset A^2$. This improves a seminal result of Larsen and Shalev (Inventiones Math., 2008) who obtained the same with 1/4 in the double exponent. The common feature of the two results is the main tool we use in the proofs, which is (perhaps surprisingly) analytic - hypercontractive inequalities for global functions. We shall discuss the new tool (introduced recently by Keevash, Lifshitz, Long and Minzer, JAMS 2024) and other directions in which it may be applied.

Based on joint works with Noam Lifshitz, Dor Minzer, and Ohad Sheinfeld