Talk

The chromatic perspective organizes elements in p-local homotopy groups into periodic families whose period depends on a height n. A simplified form of the redshift conjecture of Ausoni and Rognes asks if algebraic K-theory increases height by exactly one. In joint work with J. Hahn and D. Wilson, we prove this in the case of the prime skew fields in homotopy theory known as Morava K-theory. In my talk, I will mostly spend time motivating this result and perhaps mention a new tool from arithmetic geometry that is key to the proof.

The local Langlands correspondence conjecturally partitions the irreducible representations of a p-adic group into the so-called L-packets. Such a partition is conjecturally to be characterized by the stability condition, which is proven in many cases (when a construction of the local Langlands correspondence for certain representations is available) using the theory of endoscopy. In this talk, we will show that for elliptic L-parameters, the construction of Fargues-Scholze satisfies the stability condition. Using a formula of Hansen--Kaletha--Weinstein, we will reduce the problem of stability to showing equi-distribution properties of the weight multiplicities of highest weight representations of an algebraic group. Our proof of equi-distribution properties might be of independent interest.

Knots which bound embedded discs in the 4-ball are called slice, and such knots have been studied for several decades. More generally one can ask which knots bound embedded discs, i.e. are slice, in an arbitrary 4-manifold with 3-sphere boundary. E.g. if one finds a knot which is slice in a homotopy 4-ball but not in the 4-ball, this would disprove the 4-dimensional Poincare conjecture. In this talk I will discuss recent work in this area, including but not limited to my joint work with Kasprowski, Powell, and Teichner; with Miller, Kjuchukova, and Sakalli; and with Marengon, Miller, and Stipsicz. I will also state some of my favourite open problems in this topic.

In this talk, I will introduce the concept of a separable commutative algebra in the setting of tt-geometry and discuss its connection to Mathew’s work on Galois theory. Building on this, I will present several classification results for separable algebras in derived commutative algebra and chromatic homotopy theory. In the final part of the talk I will focus attention to the problem of classifying separable algebras in the category of genuine G-spectra for a finite group G.

Let $s(n)$ denote the sum of proper divisors of an integer $n$. The function $s(n)$ has been studied for thousands of years, due to its connection with the perfect numbers. In 1992, Erdös, Granville, Pomerance, and Spiro (EGPS) conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then $s^{-1}(\mathcal{A})$ also has asymptotic density zero. This has been confirmed for certain specific sets $\mathcal{A}$, but remains open in general. In this talk, we will give a survey of recent progress towards the EGPS conjecture. This talk is based on joint work with Paul Pollack and Carl Pomerance, and also on joint work with Kübra Benli, Giulia Cesana, Cécile Dartyge, and Charlotte Dombrowsky.