"Reason is immortal, all else mortal." - Pythagoras
Spring 2021
Organized by Polona Durcik and Mihaela B. Vajiac, sponsored by CECHA.
The talks will be held online on Zoom (meeting ID: 99396752824) or on campus at Keck Center of Science and Technology (no. 30 on the Campus map, at the intersection of Walnut Ave and Center St).
Schedule
Friday, February 19, 3pm - 4pm
William Riley Casper (CSUF)
Commuting differential and integral operators and the adelic Grassmannian
Beginning with the work of Landau, Pollak and Slepian in the 1960s on time-band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory and integrable systems. In this talk, we will describe a close connection between commuting integral and differential operators and points in the adelic Grassmannian, which provides a commuting pair for each self-adjoint point in the Grassmannian. Central to this relationship is the Fourier algebra, a certain algebra of differential operators isomorphic to the algebra of differential operators on a line bundle over a rational curve.
Friday, March 5, 3pm - 4pm
Dan Volok (Kansas State U.)
Zeros of discrete analytic polynomials
We shall discuss some basic interpolation results for discrete analytic (in the sense of J. Ferrand and R.J. Duffin) functions on the integer lattice in the complex plane.
Friday, March 12, 3pm - 4pm
Mario Stipčić (U. of Zagreb)
L^p estimates for dyadic singular integral forms associated with hypergraphs and for ergodic-martingale paraproducts
We will identify T(1)-type conditions and other characterizations of L^p boundedness of entangled multilinear singular integrals associated with hypergraphs. After that, we will examine the convergence of ergodic-martingale paraproducts in the Lebesgue spaces with respect to the range of exponents.
Friday, April 16, 3pm - 4pm
Annina Iseli (UCLA)
Thurston maps with four postcritical points
A Thurston map is a branched covering map of the 2-sphere which is not a homeomorphism and for which every critical point has a finite orbit under iteration of the map. Frequently, a Thurston map admits a description in purely combinatorial-topological terms. In this context it is an interesting question whether a given map can (in a suitable sense) be realized by a rational map with the same combinatorics. This question was answered by Thurston in the 1980's in his celebrated characterization of rational maps. Thurston's Theorem roughly says that a Thurston map is realized if and only if it does not admit a Thurston obstruction, which is an invariant multicurve that satisfies a certain growth condition. However, in practice it can be very hard to verify whether a given map has no Thurston obstruction, because, in principle, one would need to check the growth condition for infinitely many curves.
In this talk, we will consider a specific family of Thurston maps with four postcritical points that arises from Schwarz reflections on flapped pillows (a simple surgery of a polygonal sphere). Using a counting argument, we establish a necessary and sufficient condition for a map in this family to be realized by a rational map. In the last part of the talk, we will discuss a generalization of this result which states that, given an obstructed Thurston map with four postcritical points, one can eliminate obstructions by applying a so-called blowing up operation. These results are joint with M. Bonk and M. Hlushchanka.
Friday, May 7, 3pm - 4pm
Oumar Wone (U. of Dakar, Senegal)
Frobenius determinants and toric varieties
Abstract
Friday, May 14, 3pm - 4pm
Trang Thi Thien Nguyen (U. of South Australia)
Non-homogeneous T(1) theorem for singular integrals on product quasimetric spaces
In the Calderón-Zygmund Theory of singular integrals, the T(1) theorem of David and Journé is one of the most celebrated theorems. It gives easily-checked criteria for a singular integral operator T to be bounded from L^2(R^n) to L^2(R^n). Since then, this classical result has been generalized to various settings, including replacing the underlying space R^n on which the operators act.
In this talk, I will present our work on generalizing the T(1) theorem, that brings together three attributes: 'product space', 'quasimetric' and 'non-doubling measure'. Specifically, we prove a T(1) theorem that can be applied to operators acting on product spaces equipped with a quasimetric and an upper doubling measure, which only satisfies an upper control on the size of balls.