MenuEvents for 11/14/2024 from all calendars
Working Seminar on Banach and Metric Spaces
Time: 10:00AM - 11:00AM
Location: Zoom
Speaker: Thomas Speckhofer, Johannes Kepler University Linz
Title: Dimension dependence of factorization problems
Abstract: For a natural number n, let Yn denote the linear span of the first n+1 levels of the Haar system in a rearrangement-invariant function space or, more generally, in a Haar system Hardy space (this class also contains related spaces such as dyadic H1), and let IYn denote the identity operator on Yn. Fix δ, ε > 0 and a natural number n, and consider the following question: How large does N have to be chosen such that for every linear operator T: YN → YN with ||T||≤ 1 whose diagonal entries with respect to the Haar basis are at least δ, there exist operators A,B with ||A||||B||≤ (1+ε)/δ such that IYn = ATB? We show that in general, an inequality of the form N ≥ Cn^2 is sufficient, whereas if the Haar system is unconditional in the underlying space, then N≥ Cn suffices. This amounts to a quasi-polynomial or, respectively, polynomial dependence between dim YN and dim Yn.
Noncommutative Geometry Seminar
Time: 4:00PM - 5:00PM
Location: BLOC 302
Speaker: Boris L Tsygan, Northwestern University
Title: Basics of noncommutative calculus and the Gauss-Manin connection
Abstract: In 1957, Yu. I. Manin, then 19 years old, asked a question: for a parameter-depending algebraic curve (a.k.a. Riemann surface), its periods satisfy a linear differential equation, known as the Picard-Fuchs equation. Why? His answer : because the integrand is defined up to a differential of a function, and therefore lives in a finite-dimensional space. Differentiate enough times, and you will get a relation. But how to differentiate a parameter-dependent element of a finite-dimensional space when the space itself depends on a parameter? He gave a simple recipe that was later understood by Grothendieck as a consequence of basic algebraic properties of differential calculus. As Manin himself pointed out (quoting André Weil), the key is to consider our curves in relation to each other and to be able to compare them by mapping them to one another.
All these considerations are conducted in terms of the ring of algebraic functions on a space. Now consider any ring, commutative or not. In this talk, I will explain how to develop the basics of differential calculus in this generality starting from basic algebraic properties of rings and their maps. The talk requires no prior knowledge.
