S5B1 - Graduate Seminar on Advanced Topics in PDE (summer term 2015)

Lectures:

Friday, 14 (c.t.) - 16, Seminar room 0.011

Course Description:

We will discuss in this seminar various topics in Analysis and PDE. The seminar may lead to research topics in the area that can lead to a Master or PhD thesis. Students interested in participating should contact one of the organizers.

A first organizatorial meeting will be Friday April 10, 2015.

Talks (updated periodically):

• April 17, 2015: Diogo Oliveira e Silva (University of Bonn)

Title: Restriction, Kakeya, Decoupling: an invitation.

Abstract: This is the first talk of a semester-long seminar geared towards an understanding of Bourgain-Demeter's recent breakthrough, "The proof of the l^2 decoupling conjecture". After a brief introduction to the restriction and Kakeya problems, I will focus on their multilinear counterparts and try to illustrate the various connections in play. I will state the decoupling inequality and, time permitting, will list several applications to analysis, incidence geometry and number theory. This talk is intended as a non-technical survey; as such, many topics will be discussed but few details will be provided.

A list of relevant literature (which could be used in the preparation of future talks) follows:

A. Restriction/Kakeya:
• Wolff, "Lectures in Harmonic Analysis" (Chapters 7 and 10)
• Wolff, "Recent Work Connected with the Kakeya Problem"
• Tao, "Recent progress on the Restriction conjecture"

B. Multilinear Restriction/Kakeya and Applications:
• Bennett-Carbery-Tao, "On the multilinear restriction and Kakeya conjectures"
• Guth, "The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture"
• Carbery-Valdimarsson, "The endpoint multilinear Kakeya theorem via the Borsuk-Ulam theorem"
• Guth, "A short proof of the multilinear Kakeya inequality"
• Bourgain-Guth, "Bounds on oscillatory integral operators based on multilinear estimates"

C. Decoupling and Applications:
• Bourgain-Demeter, "The proof of the l^2 decoupling conjecture"
• Bourgain-Demeter, "Decoupling for curves and hypersurfaces with nonzero Gaussian curvature"
• Bourgain-Demeter, "Decoupling for surfaces in R^4"
• Bourgain, "Decoupling inequalities and some mean-value theorems"
• Bourgain, "Decoupling, exponential sums and the Riemann zeta function"

• April 24, 2015: Shaoming Guo (University of Bonn)

Title: Bi- and Tri-linear restriction estimates in R^3.

Abstract: I will present the paper "A sharp bilinear restriction estimate for paraboloids" by Tao. I will also discuss the trilinear restriction estimate by Bennett, Carbery and Tao in the paper "On the multilinear restriction and Kakeya conjectures".

• May 8, 2015: Shaoming Guo (University of Bonn)

Title: Improving linear restriction estimates by multilinear restriction estimates.

Abstract: I will present part of the paper Bounds on oscillatory integral operators based on multilinear estimates'' by Bourgain and Guth. In this paper, the authors managed to apply the multilinear restriction estimates to obtain certain improvement on the linear restriction estimates.

• May 15, 2015: Annegret Burtscher (University of Bonn)

Title: Geodesic incompleteness in general relativity.

Abstract: The general theory of relativity describes the effect of gravitation in terms of the geometry of spacetimes via the Einstein equations. In the 1950s the initial value formulation and local existence of solutions to the Einstein equations were established. As of yet the global structure of solutions is much less understood, in general, singularities seem unavoidable. The Penrose singularity theorems give some glimpse of this singular nature by relating geodesic incompleteness to the existence of trapped surfaces. In this talk we will see how such trapped surfaces can form during evolution from regular initial data, illustrated for spherically symmetric solutions of the Einstein-Euler equations.

• June 5, 2015: Mariusz Mirek (University of Bonn)

Title: A local T(b) theorem for perfect multilinear Calder\'{o}n--Zygmund operators

Abstract: We will discuss a multilinear local T(b) theorem that differs from previously considered multilinear local T(b) theorems in using exclusively general testing functions b as opposed to a mix of general testing functions and indicator functions. The main new feature is a set of relations between the various testing functions b that to our knowledge has not been observed in the literature and is necessitated by our approach. For simplicity we restrict our attention to the perfect dyadic model. This is a joint work with Christoph Thiele.

• June 19, 2015: Anna Kosiorek

Title: A short proof of the multilinear Kakeya Inequality.

Abstract: I will present a proof of a slightly weaker version of multilinear Kakeya inequality, a geometric estimate about the overlap pattern of cylindrical tubes in R^n pointing in different directions. The talk will be based on a recent paper by Larry Guth.

• June 26, 2015: Stefan Steinerberger (Yale University)

Title: Nonlinear phase unwrapping of a function.

Abstract: One way of getting Fourier series is to take a holomorphic function, create a root in the origin by translation, factor it out and repeat. A nonlinear analogue would be to remove all roots inside the unit disk and repeat - this gives rise to an unwinding series with many nice properties. This series has been independently discovered by several people and is easy to compute and very useful. We provide the first proof of convergence in suitable spaces. This is joint work with Raphy Coifman.

• July 3, 2015: Shaoming Guo and Diogo Oliveira e Silva (University of Bonn)

Title: Proof of the \ell^2 decoupling conjecture.

Abstract: We will focus on the very recent paper "The proof of the \ell^2 Decoupling Conjecture" by J. Bourgain and C. Demeter (Ann. of Math. 2015). The proof relies on certain geometric/analytic techniques developed in Bourgain-Guth's "Bounds on oscillatory integral operators based on multilinear estimates" (Geom. Funct. Anal. 2011). In the first half of this talk, Shaoming will recall some of these techniques. After the break, Diogo will specialize to the two-dimensional situation and run the induction-on-scales argument to finish the proof.

• July 10, 2015: Blazej Wrobel (Universita di Milano)

Title: Approaching bilinear multipliers via a functional calculus

Abstract: Bilinear multipliers for the Fourier transform may be defined in terms of a joint functional calculus for partial derivatives. Using this observation we propose a spectral generalization of the theory of bilinear multipliers outside of the Fourier transform framework. We focus on Coifman-Meyer type multiplier theorems and their relations with fractional Leibniz rules. Examples admitted by our theory include bilinear multipliers for: the discrete Laplacian, the general Dunkl Laplacian, and the Jacobi operator. The talk is mostly based on work in progress.

• July 17, 2015: Master's seminar day