The Harmonic Analysis Working Group meets weekly during most semesters, and
usually spends several weeks focusing on a topic in analysis of
interest to its members, and the discussion is led by a member of
staff, a visitor, a post-doc, or a research student. The level of
discussion varies to suit the needs of the members. Occasionally, we
have held postgraduate courses, which are also listed below.
Harmonic Analysis Working Group 2009-2010
This year we are currently studying, in Semester 1, recent
spectacular advances in Kakeya problems spurred by the introduction by
Dvir of the polynomial method. In Semester 2 we shall study a recent
preprint of Laba and Pramanik on maximal functions associated to
fractal sets. Meetings are on Tuesdays in 4312 JCMB from 4--6 p.m.
Harmonic Analysis Working Group 2007-2008
This year we are currently studying some issues in analytic
number theory which are informed by a harmonic analysis
perspective. Jim Wright is leading this, which in the first
semester is taking place at 1 p.m. in 4312 JCMB.
Harmonic Analysis Working Group 2006-2007
This year we studied:
- A paper by S. Cuccagna, "Estimates for averaging operators along curves with two-sided k-fold singularities" (NL)
- A paper by Bennett and Seeger, "The Fourier extension
operator on large spheres and related oscillatory integrals"
- Restriction of the Fourier transform to curves (JW)
- The Bellman Function technique (AV)
Harmonic Analysis Working Group 2005-2006
During the autumn semester of this year, we studied the
multilinear Kakeya maximal function and the multilinear
restriction problem (AC).
Harmonic Analysis Working Group 2004-2005
This year we studied:
- Brascamp--Lieb inequalities (AC)
- Carleson's theorem (JW)
- Work of Erdogan on distance set problems (JB)
- The Newton Diagram and oscillatory integrals (JW)
- Fourier Integral and Oscillatory Integral Operators (NL)
Harmonic Analysis Working Group 2003-2004
This year we began by reading the recent paper of Terry Tao on sharp linear and bilinear restriction theorems for the paraboloid in Euclidean space. After this we studied some papers on null forms and understand their relation with bilinear restriction theorems. Also, J. Wright gave a series of lectures on Fourier Analysis on compact groups.
Harmonic Analysis Working Group 2002-2003
This year we covered the following topics in the first part of the academic year:
- Restriction and Affine curvature -- a theorem of Sjolin (SD)
- Work of Mockenhaupt and Tao on restriction over finite fields (AC)
- Some other aspects of harmonic analysis over finite fields (AC)
- Weil's Riemann Hypotheis and exponential sums (JW)
- Kakeya over finite fields (AC)
- Multiparameter Hilbert transforms associated to polynomials (SP)
Analysis Working Group 2001-2002
This year we covered the following topics:
- Interpolation (JRW, IRG, RB)
- Directional Maximal Functions (LW, MaA.A)
- Von Neumann's Inequality (Vasiliki Lioudaki)
- Von Neumann Algebras (AMS)
- BMO and H^1 (TAG)
- Semigroups of Operators (TAG)
- The Cotlar - Stein lemma and almost orthogonality (AC)
- Two-parameter isoperimetric inequalities (JW)
- Hyperbolic PDE: Techniques of Klainerman (NB)
- Affine dimension (AC)
- Aspects of Multiplier Theory (VO)
Harmonic Analysis Postgraduate Course 2001-2002
This year we also had a postgraduate course in the first part of the academic year in which we covered the following topics:
- The Hardy-Littlewood Maximal Function
- Singular Integrals
- Littlewood-Paley Theory
- Fourier Multipliers
- The Role of Dilations
- Sobolev Embedding
- Curvature and Oscillatory Integrals
Analysis Working Group 2000-2001
This year we covered:
- Weighted inequalities.
- Arzela-Ascoli Theorem and applications.
- Kakeya sets.
- Weighted inequalities for the extension operator for the Fourier transform.
- Schatten-von Neumann C_p classes.
- Probabalistic ideas in analysis.
- Null forms.
Analysis Postgraduate Course 1999-2000
During the autumn term we discussed some of the rudiments of real analysis.
- Measure and integration
- Decomposition of measures and the Radon Nikodym theorem
- Construction of measures via outer measure
- Riesz representation theorems for measures
- Haar measure on locally compact abelian topological groups
- Basic theory of the Fourier transform
- Sobolev spaces
These topics were designed to be of interest to first year postgraduate students, and first year p/g's in analysis were expected to attend. The next term we moved on to a selection of less standard themes. We roughly followed the presentation in Royden's book, with Rudin and Folland's books being equally suitable. Members of the working group took it in turns to make presentations on the different topics. Once these topics were covered, we moved to a selection from:
- Sublevel set and oscillatory integral estimates
- The restriction phenomenon in nonlinear PDE
- Fractal Geometry and Hausdorff Measures
- Introduction to microlocal analysis
- Cotlar-Stein lemma and almost orthogonality