Analysis and Geometry Seminar
Center of Excellence in Complex and Hypercomplex Analysis
Schmid College of Science and Technology
Chapman University

"Reason is immortal, all else mortal." - Pythagoras

Fall 2021
Organized by Polona Durcik, Mario Stipčić, and Mihaela B. Vajiac, sponsored by CECHA.

The talks will be held online 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).

Schedule

• Friday, September 17, 3pm - 4pm
  Kamal Diki (Chapman University)
  An approach to the Gaussian RBF kernels via Fock spaces
  
  In this talk we use methods from the Fock spaces theory in order to prove several results on the Gaussian RBF kernels in the complex case. The latter is one of the most used kernels in modern machine learning kernel methods, and support vector machines (SVMs) classification algorithms. It turns out that complex analysis techniques allow us to consider several notions linked to the RBF kernels like the feature space and the feature map, using the so-called Segal-Bargmann transform. We show also that the RBF kernels can be related to some important operators in quantum mechanics and time frequency analysis, specifically, we prove different connections of such kernels with creation, annihilation, Fourier, translation, modulation and Weyl operators. A semi-group property will be proved in the case of Weyl operators. This is a joint work with Daniel Alpay, Fabrizio Colombo and Irene Sabadini.
  

• Friday, October 8, 3pm - 4pm
  Izchak Lewkowicz (Ben-Gurion University of the Negev, Beer-Sheva, Israel)
  Passive Linear Time-invariant Systems - Characterization through Structure

Passivity is a basic physical property. We here show that the family linear time-invariant passive systems may be characterized by the structure of the whole set. A refined description of strict dissipativity will be presented as well.

• Friday, November 12, 3pm - 4pm
  Paata Ivanisvili (UC Irvine)
  Learning low degree functions in logarithmic number of random queries.

Perhaps a very basic question one asks in learning theory is as follows: we have an unknown function f on the hypercube {-1,1}^n, and we are allowed to query samples (X, f(X)) where X is uniformly distributed on {-1,1}^n. After getting these samples (X_1, f(X_1)), ..., (X_N, f(X_N)) we would like to construct a function h which approximates f up to an error epsilon (say in L^2). Of course h is a random function as it involves i.i.d. random variables X_1, ... , X_N in its construction. Therefore, we want to construct such h which approximates f with probability at least (1-delta). So given parameters epsilon, delta in (0,1) the goal is to minimize the number of random queries N. I will show that around log(n) random queries are sufficient to learn bounded "low-complexity" functions. Based on joint work with Alexandros Eskenazis.

• Wednesday, November 17, noon - 1pm
  Alain Yger (Université Bordeaux)
  Revisiting syzygies, hence division or interpolation problems, in terms of residue and principal value currents

A joint paper I wrote together with M. Passare and August Tsikh in 2000 (ideas there coming from my unfortunately last joint paper with Carlos Berenstein in 1998) inspired since then the construction of what reveals to be a very powerful method to attack interpolation or division problems in Cn or Pn(C) (also on Stein manifolds) by solving them through explicit closed formulae. The beautiful idea which was introduced by Mats Andersson since 2004 consists in the following: attach to any generically exact complex of hermitian bundles over a complex analytic space both a Principal Value current and a residue current, the last one precisely encoding the lack of exactness of the complex of holomorphic bundles one started with. Time has now come, despite the technicity inherent to such construction, to popularize such tool facing general questions such as Hilbert’s nullstellensatz, the surprising (and curiously not so-well known) Brian ̧con-Skoda theorem (even in the polynomial setting), Euler-Ehrenpreis- Palamodov Fundamental Principle, or spectral synthesis problem in (ad hoc) weighted algebras of entire functions. I will try to explain this in general terms, avoiding as far as I can technicity by cheating a little, and will illustrate with few concrete examples the novelty and efficiency of such approach.