The Third Florida Analysis Seminar

Saturday, April 28, 2012

Florida Southern College, Lakeland FL

(follow the link above for map and directions)

For information about previous editions of the __Florida Analysis Seminar__, please visit the webpage.

Seminar speakers:

Peter Duren, University of Michigan

Title: “Sums for divergent series: A Tauberian adventure”

Abstract: A numerical infinite series that is not “too badly divergent” can be assigned a generalized sum in a variety of natural ways. An Abelian theorem is any statement to the effect that a method of summability assigns to each convergent series its ordinary sum. A Tauberian theorem goes in the opposite direction and asserts that every summable series which is not too badly divergent is actually convergent. Propelled by a problem that appeared in a recent Puzzles Column, we will discuss some Tauberian theorems of Hardy and Littlewood, together with Karamata’s elegant method of proof, based only on the Weierstrass approximation theorem.

The talk will be accessible to graduate students.

Alexander Tovbis, University of Central Florida

Title: “Painleve’ transcendents and universality of transitions at the point of gradient catastrophe for some integrable systems and some orthogonal polynomials with varying exponential weight”

Abstract: A point of gradient catastrophe is the point where the leading order asymptotic behavior loses smoothness (for example, derivatives are not square integrable). In the case of the small dispersion (semiclassical) focusing NLS, this is a point where a slowly modulated high frequency plane wave suddenly burst into rapid amplitudial oscillations (spikes). Adjusting the nonlinear steepest descent (Deift-Zhou) method for Riemann-Hilbert problems, we give the leading order description (with error estimates) of the point of gradient catastrophe in terms the tritronque’e solution to the Painleve’ I and rational breathers for the NLS. We prove that each spike corresponds to a pole of the tritronque’e and has the universal shape of a scaled rational breather. Similar phenomenon was studied for the asymptotic of orthogonal polynomials with complex varying weight exp[-N (z^2/2 + t z^4/4) near the critical value of the parameter t. We also study the global asymptotic regime for complex t.

Registration and talks will be held in Polk Science 152.

To register for this event, please e-mail the University of South Florida organizer, professor Dima Khavinson (at dkhavins@usf.edu), indicating your name, affiliation, and contact information. __Parking information__: please use the link above for the map and directions. Campus parking is free and lot VA is the most convenient for access to the seminar location.

Meeting program:

10:00 - 10:30 Registration and coffee reception. A flat charge of 15.00$ will cover registration costs, a boxed lunch and coffee breaks.

10:30 -11:30 Peter Duren

11:30 - 11:50 coffee break

11:50 - 12:50 Alexander Tovbis

1:00 - 2:30 lunch in the Honeyman Pavilion.

2:30 - Discussions.