Rivière-Fabes Symposium on Analysis and PDE

April 5-7, 2002

Speakers

D. Jerison

N. Katz

M. Lacey

W. Schlag

T. Toro

For further information and (free) registration: reitich@math.umn.edu

Average high/low (degrees F): 50/31.
Hotels

Partial financial trip support is available for a limited number of out-of-town graduate students and postdocs. More about support.

Schedule

Friday afternoon, April 5:

Registration, coffee/refreshments
3:00-3:30pm, VinH 120

Wilhelm Schlag (Department of Mathematics, Princeton University)
Energy growth for Schroedinger equations with Markovian forcing
3:30-4:30pm, VinH 16

Coffee Break
4:30-4:50pm, VinH 120

Michael Lacey (School of Mathematics, Georgia Institute of Technology)
Product BMO and second order commutators
4:50-5:50pm, VinH 16

Reception
6:00-7:30pm, VinH 120

Saturday, April 6:

Registration, coffee/refreshments
9:00-9:30am, VinH 120

Michael Lacey (School of Mathematics, Georgia Institute of Technology)
Carleson's theorem with quadratic phase
9:30-10:30am, VinH 16

Coffee Break
10:30-11:00am, VinH 120

David Jerison (Department of Mathematics, Massachusetts Institute of Technology)
Global energy minimizers for free boundary problems and full regularity in three dimensions
11:00am-12:00pm, VinH 16

Lunch
12:00-2:00pm

Tatiana Toro (Department of Mathematics, University of Washington)
Free boundary regularity below the continuous threshold
2:00-3:00pm, VinH 16

Coffee Break
3:00-3:30pm, VinH 120

Nets Katz (Department of Mathematics, Washington University)
Stickiness in the 3-dimensional Kakeya problem
3:30-4:30pm, VinH 16

Symposium Dinner
6:30pm

Sunday morning, April 7:

Registration, coffee/refreshments
9:00-9:30am, VinH 120

David Jerison (Department of Mathematics, Massachusetts Institute of Technology)
Carleman inequalities and the absence of embedded eigenvalues
9:30-10:30am, VinH 16

Coffee Break
10:30-11:00am, VinH 120

Wilhelm Schlag (Department of Mathematics, Princeton University)
Dispersive estimates for solutions of Schroedinger equations with slowly decaying and time-dependent potentials
11:00am-12:00pm, VinH 16

Abstracts

David Jerison, Global energy minimizers for free boundary problems and full regularity in three dimensions

David Jerison, Carleman inequalities and the absence of embedded eigenvalues

Nets Katz, Stickiness in the 3-dimensional Kakeya problem

Michael Lacey, Product BMO and Second Order Commutators
Abstract: A classical result considers the commutator between the Hilbert transform and multiplication by a function $b$. The commutator maps $L^2$ to itself iff $b$ is in $BMO$. We consider a version of this same theorem in which the product $BMO$ of S.Y.~Chang and R.~Fefferman is charaterized. It is as follows. For a function $b$ on the plane let $M_b$ be the operator of multiplication by $b$. Let $H_j$ be the one dimensional Hilbert transform acting on the $j$th coordinate, for $j=1,2$. Then, the commutator $$ [[M_b,H_1],H_2] $$ maps $L^2( R^2)$ to itself iff $b$ is in product $BMO$. As for the classical result, there are further consequences, which we shall review. This is joint work with Sarah Ferguson.

Michael Lacey, Carleson's theorem with quadratic phase
Abstract: We will discuss a proof of the $L^2$ boundedness of the maximal function which includes as special cases, the Theorem of Carleson on Fourier series, and an observation of E.M.~Stein. The maximal operator is $$ \sup_{a,b}\int f(x-y) e^{i(ay+by^2) } {dy \over y} $$ The supremum of $a$ alone is the Theorem of Carleson, and that of $b$ alone an observation due to Stein.

Wilhelm Schlag, Energy growth for Schroedinger equations with Markovian forcing
Abstract: In joint work with Burak Erdogan and Rowan Killip we study a Schroedinger equation on the circle with a Markovian random potential. We prove a conjecture of Tom Spencer and Zakharov concerning the growth of the higher Sobolev norms in this case.
Copies of the slides for this presentation can be found at http://www.its.caltech.edu/~schlag

Wilhelm Schlag, Dispersive estimates for solutions of Schroedinger equations with slowly decaying and time-dependent potentials
Copies of the slides for this presentation can be found at http://www.its.caltech.edu/~schlag

Tatiana Toro, Free boundary regularity below the continuous threshold
Abstract: In this talk we will show that the "weak" regularity of the Poisson kernel of a domain determines the regularity of its boundary. We will explain how techniques from Geometric Measure Theory and PDE come together to produce these results. This is joint work with C. Kenig.

This symposium is held in memory of Nestor M. Rivière and Eugene B. Fabes.
The organizing committee consists of Naresh Jain, Carlos Kenig (Chicago), Nicolai Krylov, Walter Littman, Fernando Reitich and Mikhail Safonov (Chair).

THE Rivière-Fabes SYMPOSIUM: Background

The Symposium was established in memory of our colleagues Nestor M. Rivière and Eugene B. Fabes. Both of them were analysts and did their graduate work together at the University of Chicago. After finishing his Ph.D. under Alberto Calderón in 1966, Nestor joined the School of Mathematics the same year. Gene finished his Ph.D. under Antoni Zygmund in 1965 and spent two years at Rice University before coming to Minnesota in 1967. The two started a new era in classical analysis at Minnesota.

Unfortunately for us, cancer claimed Nestor's life at the young age of 38 in 1978, ending a brilliant career. The department established the Nestor M. Rivière Lecture in his memory. Every year a highly distinguished mathematician delivered the lecture in the broad area of analysis. Gene usually took care of the organizational work and the Rivière Lecture was supported by a fund established by donations from friends of Nestor. In 1997 another tragedy struck. Gene passed away just after he turned sixty and was still at the peak of his productive career. A list of his mathematical achievements can be found in the obituary in the Amer. Math. Soc. Notices, v. 45 (1998), pp. 706-708, and in the Journal of Fourier Analysis and Appl., v. 4, no. 4/5 (1998). Former colleagues, students and friends of Nestor and Gene from all over the world expressed the sentiment that we should establish an annual symposium in their memory. Families of Nestor and Gene fully endorsed the idea of turning the Nestor M. Rivière Lecture into the Rivière-Fabes Symposium. With financial support from interested people the symposium was formally established in 1998.

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