I plan to discuss several related arithmetic and analytic dualities governed by the Möbius function and the sawtooth function. While these dualities—whether pointwise (twisted or untwisted), or functional in the L^2 or continuous setting—are formally valid, identifying conditions under which they become genuine identities is a subtle and challenging problem. It draws on deep results in analytic number theory, including Vinogradov-type bounds for exponential sums over primes (Davenport; de la Bretčche–Tenenbaum), as well as Tauberian theory (Wintner; Ingham).
Arithmetic and analytic dualities governed by the Mobius and sawtooth functions
Descriptive combinatorics is a field concerned with graph theoretic problems on nice (definable) infinite graphs. In this first half of the talk, I will introduce the field of descriptive combinatorics and discuss some of the basic results of the area. Then, I will turn to the descriptive version of complexity problems, in particular, the CSP Dichotomy: it turns out that the finitary complexity landscape is only partially reflected in the descriptive context.
CSPs in the descriptive context
Feb. 24, 2026 2:30pm (MATH 3…
Lie Theory
Richard Green (CU)
X
The E8 root system is a highly symmetric configuration of vectors, corresponding to the 240 spheres that touch a given one in the densest possible packing of spheres in 8-dimensional Euclidean space. Despite this eightness, some combinatorial features of the root system are much easier to understand in terms of 9-dimensional coordinates. We will use this point of view to gain insight into a certain tightly structured partially ordered set, by using the combinatorics of perfect matchings and Fano planes.
This is based on work in progress with Tianyuan Xu (University of Richmond).
Arithmetic and analytic dualities governed by the Mobius and sawtooth functions