ICMS »

Oscillatory integrals in harmonic analysis

Jun 06, 2011 - Jun 10, 2011

15 South College Street, Edinburgh

Organisers

Name	Institution
Kenig, Carlos	University of Chicago
Nagel, Alexander	University of Wisconsin-Madison
Seeger, Andreas	University of Wisconsin-Madison
Wright, Jim	University of Edinburgh

Bringing together an international group of experts in harmonic analysis to discuss recent research and open problems in the area, and focussing on developments related to oscillatory integrals this workshop will focus on:

Oscillatory integrals of Calreson-Sjolin type and related topics
Oscillatory and Fourier integral operators with degenerate canonical relations
The role of affine invariant measures
Uniformity and stability issues for oscillatory integrals, and related problems

During the last decade there has been tremendous progress in this very active area of analysis. The aim of the workshop is to foster interaction and collaboration amongst participants.

Arrangements

Particpation
There are now a limited number of places available to apply for. The application form is linked at the top right of this page, in the green box. Applications must be received by 1 May. Notification of outcome will be sent out on 3 or 4 May.

Venue
The workshop will take place in the Newhaven lecture theatre at ICMS, 15 South College Street, Edinburgh, Scotland.

Lecture theatre facilities
The lecture theatre is fitted with data projection, visualiser (the new generation of over-head), smart-board and blackboard. Speakers may wish to note that since the lectures will be filmed, it is inadvisable to use blackboards as presentation medium simply because they don't come out clear on film. Please use alternative methods. Please also note that blackboard and data projection cannot be used simultaneously. More information about this is available form Helene Frossling.

Accommodation
Accommodation details have been sent out individually. Check-in is at the accommodation. If you arrive after 23.00 you need to pick up your keys from the Reception Centre which is at the main entrance to the Pollock Halls area. It is staffed 24 h. If you experience delays during your travels, meaning you will not arrive on the intended date, please contact the accommodation on +44 131 651 2198 to let them know.

Visas
Attendees may need to apply for a visa to enter the UK. This government website gives more information.

Travel
Information about travel to the UK and Edinburgh is available here. Lothian buses charge 1.30 GBP for a single journey and 3.20 GBP for a day ticket. Please note that the exact fare is required and no change is given.

A taxi directly from the airport will cost approximately 18.00 to 20.00 GBP to the city centre for a one-way journey.

From the airport, you can take the airport bus Airlink to the city centre. It will cost you 3.50 GBP for a single ticket, or 6.00 GBP return. More details on the airport bus Airlink is available here. The bus will take you to Waverley Bridge (next to Waverley Railway Station).

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at ICMS and George Square. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge map. The second railway station is called Haymarket and is at the West End of the city centre.

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies that may occur on your trip.

Conference Co-ordinator
Helene Frossling is the conference co-ordinator for this workshop.

Programme

Programme posted 20 May 2011. Updated 31 May. Slides added from 15 June.

Monday 6 June

08.45 - 09.45	Registration and coffee/tea
09.45 - 10.00	Introductions
10.00 - 10.40	Eli Stein (Princeton University) Cauchy integrals in for domains in C^n with minimal smoothness
10.40 - 11.10	Coffee and tea served in the Chapterhouse
11.10 - 11.50	Brian Street (University of Wisconsin-Madison) Multiparameter singular and fractional Radon transforms Link to slides
11.55 - 12.35	Jonathan Bennett (University of Birmingham) Weighted norm inequalities for a class of oscillatory integrals
12.35 - 14.00	Lunch served in the Chapterhouse
14.00 - 14.40	Keith Rogers (Instituto de Ciencias Matemáticas, Madrid) Improved bounds for Stein's square function Link to slides
14.45 - 15.30	Lillian Pierce (University of Oxford) Discrete fractional Radon transforms
15.30 - 16.00	Coffee and tea served in the Chapterhouse
16.00 - 16.40	Richard Oberlin (Louisiana State University) Variations on the Carleson-Hunt theorem Link to slides
16.45 - 17.25	Detlef Mueller (University of Kiel) Uniqueness for Schroedinger equations on two-step nilpotent groups

Tuesday 7 June

09.00 - 09.40	Michael Christ (University of California, Berkeley) Extremizers and Near-extremizers for the Radon Transform Link to slides
09.45 - 10.30	Neal Bez (University of Birmingham) Multilinear Radon-like transforms Link to slides
10.30 - 11.00	Coffee and tea served in the Chapterhouse
11.00 - 11.40	Allan Greenleaf (University of Rochester) A multi-dimensional resolution of singularities with applications Link to slides
11.45 - 12.25	Neill Lyall (University of Georgia) Polynomial patterns in subsets of the integers
12.25 - 14.00	Lunch
14.00 - 14.40	Malabika Pramanik (University of British Columbia) An uncertainty-type principle for Salem measures of fractional dimension Link to slides
14.45 - 15.25	Philip Gressman (University of Pennsylvania) Uniform sublevel set estimates and applications Link to slides
15.25 - 16.00	Coffee and tea served in the Chapterhouse
16.00 - 16.40	Christoph Thiele (University of California at Los Angeles) The twisted paraproduct and the triangular Hilbert transform
16.45 - 17.25	A.D. Ionescu (Princeton University) On the local extension of Killing vector-fields in Ricci flat manifolds Link to slides

Wednesday 8 June

09.00 - 09.40	Bill Beckner (University of Texas at Austin) Multilinear fractional Sobolev embedding
09.45 - 10.25	Dan Oberlin (Florida State University) Uniform convolution and Fourier restriction estimates with affine measures - a survey Link to slides
10.25 - 11.00	Coffee and tea served in the Chapterhouse
11.00 - 11.40	Spyros Dendrinos (University of Jyväskylä) Sharp uniform $L^p$ estimates for local Fourier restriction to curves Link to slides
11.45 - 12.25	Anthony Carbery (University of Edinburgh) The Multilinear Kakeya theorem, factorisation and algebraic topology Link to slides
	Knowledge Transfer Event
18.00-19.00	Allan Greenleaf (University of Rochester) Mathematical foundations of invisibility cloaks and transformation optics
Followed by	Wine reception

Thursday 9 June

09.00 - 09.40	Antonio Cordoba (Universidad Autonoma de Madrid) Singular integrals in fluid mechanics Link to slides
09.45 - 10.25	Sanghyuk Lee (Seoul National University) On pointwise convergence of solutions to the Schrödinger equations
10.25 - 11.00	Coffee and tea served in the Chapterhouse
11.00 - 11.40	Akos Magyar (University of British Columbia) Discrete singular radon transforms on step-2 nilpotent groups
11.45 - 14.00	Lunch
14.00 - 14.40	Michael Greenblatt (University of Illinois at Chicago) A resolution of singularities algorithm for local fields of characteristic zero and some applications Link to slides
14.45 - 15.25	Alexander Iosevich ( University of Rochester) Sobolev bounds for generalized Radon transforms and application to geometric measure theory Link to slides
15.25 - 16.00	Coffee and tea served in the Chapterhouse
16.00 - 16.40	Ana Vargas (Universidad Autonoma de Madrid) About the Born approximation in potential inverse scattering Link to slides
16.45 - 17.25	Michael Cowling (Universiy of New South Wales) An oscillatory integral related to random matrices
19.00	Workshop dinner

Friday 10 June

09.45 - 10.25	Izabella Laba (University of British Columbia) Arithmetic progressions in sumsets and $L^p$-almost periodicity Link to slides
10.25 - 11.00	Coffee and tea served in the Chapterhouse
11.00 - 11.40	Jong-Guk Bak (Pohang University of Science and Technology) A weak type Fourier restriction estimate for some degenerate curves
11.45 - 12.25	Alexander Volberg (Michigan State University) The sharp estimate of Calder'on--Zygmund operators on metric spaces with geometric doubling

Presentations:

Presentation Details
Bak, Jong-Guk
A weak type Fourier restriction estimate for some degenerate curves
View Abstract Hide Abstract
We will discuss how to obtain a weak type endpoint estimate for the restriction of the Fourier transform to some classes of degenerate curves. The estimates are for the affine arclength measure and are uniform in a suitable sense.
Bennett, Jonathan
Weighted norm inequalities for a class of oscillatory integrals
View Abstract Hide Abstract
We present some general weighted $L2$ norm inequalities for oscillatory convolution operators on the line with phases of finite type. This is joint work with Sam Harrison.
Carbery, Anthony
The Multilinear Kakeya theorem, factorisation and algebraic topology
View Abstract Hide Abstract
We discuss some developments arising out of the Multilinear Kakeya theorem which concern factorisation of functions and convex optimisation, and also (in relation to Guth's proof) some aspects of algebraic topology.
Christ, Michael
Extremizers and Near-extremizers for the Radon Transform
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The Radon transform satisfies various inequalities with respect to Lebesgue space norms. One of these inequalities has connections with several other topics, including a certain convolution operator and a multilinear inequality involving determinants. It enjoys an extraordinarily large group of symmetries. We discuss inverse questions about functions which exactly or nearly extremize this inequality. In particular, (all) extremizers have recently been identified. In this story, a leading role is played by considerations of symmetry. A remarkable happenstance is the existence of equivalent formulations for three different operators; each incarnation reveals its own facet of the full symmetry group. We will outline the steps which lead to this identification. Along the way we will touch briefly on weighted norm inequalities, a nonlinear and nonlocal Euler-Lagrange equation, rearrangement inequalities, and the Hardy-Littlewood-Sobolev inequality. All would be for nought, were it not for the timely appearance of a bonus symmetry.
Cordoba, Antonio
Singular integrals in fluid mechanics.
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Some new estimates for singular integrals having applications to well-possedness and blow-up, in finite time, in several models relevant to fluid mechanics.
Cowling, Michael
An oscillatory integral related to random matrices
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It is known that, if $X$ is a uniformly distributed $d times d$ orthogonal matrix, then the powers $X^n$ of $X$ are no longer uniformly distributed, but are concentrated near the central orthogonal matrices. One can try to understand the distribution of these powers in various ways; the Fourier analytic approach leads one to want to compute Fourier transforms of rather singular functions on the maximal torus of the orthogonal group. This gives rise to some interesting singular oscillatory integrals. I will explain what I know about powers of random matrices and about the singular integrals that arise in this context.
Dendrinos, Spyros
Sharp uniform $L^p$ estimates for local Fourier restriction to curves
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We present sharp estimates, with respect to the affine arclength measure, for the restriction of the Fourier transform to a class of curves in $R^d$ that includes curves of finite type. This measure possesses certain invariance and mitigation properties which are important in establishing uniform results.
Greenblatt, Michael
A resolution of singularities algorithm for local fields of characteristic zero and some applications
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A resolution of singularities algorithm for general local fields of characteristic zero is described, generalizing the speaker's earlier algorithm for functions on R^n. The p-adic case leads to new estimates for exponential sums as well as some number-theoretic applications.
Greenleaf, Allan
A multi-dimensional resolution of singularities with applications
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We give a fairly explicit algorithm for resolution of singularities of real analytic functions in dimensions greater than or equal to three, and use it to compute critical integrability indices. This is joint work with Tristan Collins and Malabika Pramanik.
Gressman, Philip
Uniform sublevel set estimates and applications
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We will discuss several recent results concerning the growth rates of sublevel sets and related estimates for the norms of certain geometric averaging operators. These results share the common feature that they exhibit a natural uniformity (of a sort similar to the works of Carbery, Christ, and Wright; Carbery and Wright; and Phong, Stein, and Sturm) as well as affine invariance. These results have applications to oscillatory integrals and restriction estimates.
Iosevich, Alexander
Sobolev bounds for generalized Radon transforms and application to geometric measure theory
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We shall see how Sobolev bounds for generalized Radon transforms can be used to recover a fractal generalization of the regular value theorem, a result on the distribution of lattice points near convex domains and related problems.
Laba, Izabella
Arithmetic progressions in sumsets and $L^p$-almost periodicity
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We prove results about the $L^p$-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in $L^p$, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,...,N} of sizes $alpha N$ and $beta N$ then $A+B$ contains an arithmetic progression of length at least about $exp(c (alpha beta log N)^{1/2})$. Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least about $exp(c (alpha log N/(log(beta^{-1}))^3)^{1/2})$. (Joint work with Ernie Croot and Olof Sisask.)
Lee, Sanghyuk
On pointwise convergence of solutions to the Schr"odinger equations
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A classical problem regarding pointwise behavior of the free Schrödinger equation is to determine the optimal exponent s for which lim_{t → 0} e^{it Δ}f(x)=f(x) a.e. x in R^d, whenever f ∈ H^s(R^d). This problem was settled for d=1 but still remains open for the higher dimensions even though some progress has been made for d=2. In this talk we consider pointwise convergence problem for Schrödinger equations with quadratic potentials, which include the hermite Schrödinger equation. This problem is closely related to pointwise convergence along variable curves. We will show that these problems are more or less equivalent to that of the free Schrödinger equation and also discuss equivalence of related time-space estimates. Part of this talk is based on a joint work with Keith Rogers.
Lyall, Neill
Polynomial patterns in subsets of the integers
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It is a striking and elegant fact (proved independently by Furstenberg and Sarkozy) that any subset of the integers of positive upper density necessarily contains two distinct elements whose difference is given by a perfect square. We will present a new proof of this result and also discuss a number of variations, extensions and generalizations.
Magyar, Akos
Discrete singular radon transforms on step-2 nilpotent groups
View Abstract Hide Abstract
Download abstract: http://www.icms.org.uk/downloads/harmonic/Magyar.pdf
Muller, Detlef
Uniqueness for Schroedinger equations on two-step nilpotent groups
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In a recent preprint, S. Ben Said and S. Thangavelu have proved a uniqueness theorem for solutions to the Schroedinger equation on the Heisenberg group. In joint work with J. Ludwig, we extend this result to arbitrary 2-step nilpotent Lie groups, by means of an alternative approach.
Oberlin, Dan
Uniform convolution and Fourier restriction estimates with affine measures - a survey
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This talk will be a survey of some convolution and Fourier restriction estimates which are uniform over certain classes of curves or surfaces.
Oberlin, Richard
Variations on the Carleson-Hunt theorem
View Abstract Hide Abstract
We discuss recent results related to the pointwise convergence of Fourier integrals.
Pramanik, Malabika
An uncertainty-type principle for Salem measures of fractional dimension
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e prove that probability measures supported on sets of Hausdorff dimension < 1 in the real line cannot simultaneously satisfy the optimal ball condition and the optimal Fourier decay condition subject to dimension, even though there are many examples of Salem measures where one of the two holds. This provides a negative answer to a question of Mitsis and of Mattila. This is joint work with Karsten Chipeniuk, Anne de Roton and Izabella Laba.
Rogers, Keith
Improved bounds for Stein's square function
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We prove a weighted norm inequality for the maximal Bochner-Riesz operator and the associated square-function. Connections with space-time regularity estimates for the Schrödinger and wave equations will also be highlighted. This is joint work with Sanghyuk Lee and Andreas Seeger.
Street, Brian
Multiparameter singular and fractional Radon transforms
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The purpose of this talk is to discuss the $L^p$ regularity of operators of the form [fmapsto psi(x) int f(gamma(t,x)) K(t) dt] where $gamma(t,x)$ is a $C^infty$ function satisfying $gamma(0,x)equiv x$, $psi$ is a $C^infty$ cutoff function supported near $x=0$, and $K(t)$ is a ``multiparameter'' fractional kernel supported near $t=0$. For instance, we discuss the case when $K$ is a so-called product kernel, or a fractional kernel which is slightly better than a product kernel. In the ``single-parameter'' case we discuss the case when $K(t)$ is a Cader'on-Zygmund kernel or a fractional kernel which is close to a Calder'on-Zygmund kernel. When $K$ is a Calder'on-Zygmund kernel, our results generalize some of the result of Christ, Nagel, Stein, and Wainger. This talk is partially based on joint work with E. M. Stein.
Vargas, Ana
About the Born Approximation in Potential Inverse Scattering
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This is a joint work with J.A.Barceló, D. Faraco and A. Ruiz. We analize the regularity information of the potential in the Schodinger equation contained in its Born approximation for different sets of data (full, backscattering). In particular, we treat the case of non compactly supported potentials in 2D and reconstruction of classical continuity in Hölder clases.
Volberg, Alexander
The sharp estimate of Calder'on--Zygmund operators on metric spaces with geometric doubling.
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We give a short and simple polynomial estimate of the norm of weighted dyadic shift on metric space with geometric doubling, which is linear in the norm of the weight. Combined with the existence of special probability space of dyadic lattices built in recent note of Reznikov--Volberg, and with decomposition of Calder'on--Zygmund operators to dyadic shifts from Hyt"onen's paper, we will be able to have a linear (in the norm of weight) estimate of an arbitrary Calder'on--Zygmund operator on a metric space with geometric doubling.

Participants

Name	Institution
Bailey, Andrew	University of Birmingham
Bak, Jong-Guk	Pohang University of Science and Technology
Beckner, W	University of Texas at Austin
Bennett, Jonathan	University of Birmingham
Bez, Neal	University of Birmingham
Candy, Timothy	University of Edinburgh
Carbery, Anthony	University of Edinburgh
Christ, Michael	University of California, Berkeley
Chung, Francis	University of Chicago
Cordoba, Antonio	Universidad Autonoma, Madrid
Cowling, Michael	University of New South Wales
Dendrinos, Spyros	University of Jyväskylä
Greenblatt, Michael	University of Illinois at Chicago
Greenleaf, Allan	University of Rochester
Gressman, Philip	University of Pennsylvania
Harris, Stephen	University of Edinburgh
Harrison, Samuel	University of Birmingham
Heywood, Philip	University of Edinburgh
Iliopoulou, Marina	University of Edinburgh
Ionescu, A.D.	Princeton University
Iosevich, Alexander	University of Rochester
Kinnear, George	University of Edinburgh
Laba, Izabella	University of British Columbia
LaVictoire, Patrick	University of Wisconsin
Lawrie, Andrew	University of Chicago
Lee, Sanghyuk	Seoul National University
Lichtin, Ben	*
Lyall, Neill	University of Georgia
Magyar, Akos	University of British Columbia
Mas, Albert	Universitat Autonoma de Barcelona
Muller, Detlef	University of Kiel
Nagel, Alexander	University of Wisconsin-Madison
Oberlin, Dan	Florida State University
Oberlin, Richard	Louisiana State University
Oliveira e Silva, Diogo	University of California, Berkeley
Pierce, Lillian	University of Oxford
Pott, Sandra	University of Lund
Pramanik, Malabika	University of British Columbia,
Rice, Alex	University of Georgia
Rodriguez, Salvador	Heriot-Watt University
Rogers, Keith	Instituto de Ciencias Matemáticas, Madrid
Rudnev, Misha	University of Bristol
Rule, David	Heriot-Watt University
Seeger, Andreas	University of Wisconsin-Madison
Staubach, Wulf	Heriot-Watt University
Stein, Elias	Princeton University
Street, Brian	University of Wisconsin-Madison
Taylor, Krystal	University of Rochester
Thiele, Christoph	University of California, Los Angeles
Vargas, Ana	Universidad Autonoma, Madrid
Volberg, Alexander	Michigan State University
Wright, Jim	University of Edinburgh