"Reason is immortal, all else mortal." - Pythagoras
Fall 2023
Organized by Kamal Diki and Polona Durcik, sponsored by CECHA.
The talks will be held in a hybrid format, on Zoom (meeting ID: 99396752824) and/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), room KC 156 or KC 370.
Schedule
Friday, September 15, 3:00pm - 4:00pm, Keck 171
Alain Hénaut (Université de Bordeaux, France)
Variations on Implicit Planar Webs
Abstract: Web geometry deals with foliations in general position. In the complex setting, a planar $d$-web is given by the integral curves of an analytic or algebraic differential equation $F(x,y,y')=0$ with $y'$-degree $d$. New invariants of these configurations especially related to abelian relations, Lie symmetries and Godbillon-Vey sequences will be introduced. An analogue of the canonical map and an associated Gauss-Manin connection will be also investigated. Basic results and some open problems will be mentioned. As the initials of the talk title suggest ``VIP webs'', as important models, will illustrate the presentation.
Friday, November 17, 3:00pm - 4:00pm, Keck 156
Alain Yger (Université de Bordeaux, France)
About the approximation by Bernstein polynomials
Abstract: S. Bernstein (1935) and L. Kantorovitch (1931) proved that, given an holomorphic function $f$ in the interior of an ellipse $E$ with foci $\{0,1\}$, it admits in $H(E)$ the approximation by the sequence of polynomials $$B_N[f](z) = \sum\limits_{\nu=0}^N \binom{N}{\nu} z^\nu (1-z)^{N-\nu} f(\nu/N).$$ The fact (inherited from a probabilistic approach, established by S. Bernstein) that any continuous function on $[0,1]$ is uniformlly approximated on this segment by the sequence of polynomial functions $B_N[f]$ is a basic tool for numerical approximation, despite the fact the approximation error converges rather slowly ($\sim 1/N$) towards $0$. The phenomenom of superoscillation and its persistence in time through the evolution $(t,x)\mapsto \psi(t,x)$ under the chronological Schr\"odinger equation $(i\partial/\partial t +\partial^2/\partial x^2) (u(t,x))= V(x)$ with $u(0,x) = e^{i\omega x}$ (for some particular continuous or real-analytic potentials $V$) motivated the introduction of a particular class of continuous functions on an open interval $I$ of $\mathbb R$ with diameter $>1$, namely that of continuous functions on $I$ such that, if $$ \mathbb I = \{(x,\tau)\in \mathbb R\times I\,:\, \tau + [0,1] \subset I,\ x + \tau \in I\}, $$ the sequence of functions $$(x,\tau) \in \mathbb I \longmapsto \sum_{\nu=0}^N \binom{N}{\nu}x^\nu (1-x)^{N-\nu} f\Big(\tau + \frac{\nu}{N}\Big), \ N=1,2,...$$ converges on $\mathbb I$ towards $(x,\tau) \mapsto f(x+\tau)$ uniformly on any compact subset. I will discuss in this talk how far such concept could be from that of analyticity and formulate a certain number of natural questions, most of them conjectural, which seem not to appear (at least in such form) in the extremely rich literature about approximation theory. The talk will be mostly at a totally elementary level. I will also explain why the answers to such questions would be of extreme importance to clarify the phenomenom of superoscillation and its persistence under Schrodinger type evolution problems in optics or quantum physics. This is joint work from several years with Fabrizio Colombo, Irene Sabadini and Daniele Struppa.
Friday, December 8, 3:00pm - 4:00pm, BK 106
Uwe Kaehler (University of Aveiro, Portugal)
Harmonic analysis and Pseudo-differential operators over Spin groups
Abstract: In this talk we present a construction of a global symbol calculus of pseudo- differential operators in the sense of Ruzhansky-Turunen-Wirth on spin groups with emphasise on the case of Spin(4). Using representations of the Spin group we construct a group Fourier transform and establish the calculus of left-invariant differential operators and of difference operators over the group. Afterwards we apply this calculus to give criteria for the subellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of first and second order globally hypoelliptic differential operators are given, including some where the criteria of hypoellipticity will be reduced to the problem of distance between irrational and rational numbers. The talk is based on joint work with P. Cerejeiras, M. Ferreira, and J. Wirth.