Gennady Uraltsev

RG Analysis and Partial Differential Equations


Contact

Postal address: Office:
Mathematisches Institut Endenicher Allee 60, N2.008
Universität Bonn Phone: +49 (0) 228 73-6888
Endenicher Allee 60 E-mail: guraltse (at) math.uni-bonn.de
D - 53115 Bonn

I am a currently a PhD student. My advisor is Prof. Dr. Christoph Thiele.

  • My current CV. (updated to the 01 Nov, 2016)

Preprints and Publications

My Master Thesis was done in Pisa on the topic of: Multi-parameter Singular Integrals: Product and Flag Kernels . Advisor: Prof. Fulvio Ricci. This thesis includes an overview of existing literature and a new result about stability of such kernels under non-linear changes of variables.

My Bachelor Thesis was done in Pisa on the topic of: Regularity of Minimizers of One-Dimensional Scalar Variational Problems with Lagrangians with Reduced Smoothness Conditions . Advisor: Prof. Luigi Ambrosio

Research interests

A complete description of my research is available in my research statement (updated 15 Nov, 2016).

My main research interest lies in the area of Real Harmonic Analysis and particularly in what is ften referred to as time-frequency analysis (singular integral operators, oscillatory integrals). I study boundedness properties of operators with large classes of symmetries such as translations, scaling, and modulations, where classical techniques like Calderón-Zygmund theory, wavelet analysis, Gabor frames fail to encode the relevant properties.

Time-frequency analysis ties in with an important theme of Harmonic Analysis: exploring properties of highly oscillatory integrals and associated operators, a research direction widely encouraged by Stein over the last several decades. A celebrated result by Stein and Wainger about maximally quadratically modulated Hilbert transform gave a beautiful way of combining results of Calderón-Zygmund theory, which relies on certain smoothness assumptions, with highly oscillatory of the modulated kernel. In similar spirit the following work can be generally seen as an attempt to understand how more classical results from Harmonic Analysis interplay with novel ideas from time-frequency analysis.

Variational Carleson Operator and generalizations

My PhD thesis concentrates around embedding maps and iterated outer-measure L^p norms on the time-frequency space in the context of the Variational Carleson Operator. This approach allowed to obtain previously unknown weighted bounds for the operator in a sharp range of exponents. It also seems that other open problems seem suscitible to the same techniques.

An interesting research area is that of the Non-Linear Fourier Transform . The bounds on the standard and Variational Carleson operators are still open problems in this setting. Attempting to apply Terry Lyons' theory calls for studying maximal and variational bounds on iterated Fourier inversion integral operators.

Uniform bounds for the Bilinear Hilbert Transfrom

I am currently working with Michał Warchalski on uniform bounds for the Bilinear Hilbert Transfrom. We are trying to apply the framework iterated outer measure Lp spaces the time-frequency plane. This framework seems to appropriately encode the behavior of the BHT for degenerating parameters. This problem can give insights towards a better understanding of the famous Triangular Hilbert Transfrom.

Quadratic Carleson operator and Gowers U-norms

Recent works on Gowers U-norms and on the inverse theorem suggest important operators to study in time-frequency analysis. With Pavel Zorin-Kranich we are studying a generalization of the Quadratic Carleson Operator with modulations coming from nilsequences of the Heisenberg group.

Other research interests

A good understanding of Harmonic and Time-Frequency analysis has interesting applications in Stochastic PDEs. See slides below.

Slides and Notes

  • Talk at Brown and at Yale (March, 2016): Iterated outer measure L^p spaces. slides
  • Summer School on Harmonic Analysis and Rough Paths (Sept, 2016): After Levin, Daniel, and Terry Lyons. "A signed measure on rough paths associated to a PDE of high order: results and conjectures." Revista Matemática Iberoamericana 25.3 (2009): 971-994. summary, slides.

    Several examples of rough paths and an overview of some interesting topics on frontier of Harmonic Analysis and rough path theory Slides (part 2)

  • Summer School on Sharp Inequalities in Harmonic Analysis (Sept, 2015): After Cordero-Erausquin, D., B. Nazaret, and Cédric Villani. "A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities." Advances in Mathematics 182.2 (2004): 307-332. summary, slides (part 1), slides (part 2)
  • Seminar on Blow-up for non-linear Dispersive PDEs (May, 2015): After Raffaël On the blow up phenomenon for the $L^2$ critical nonlinear Schrödinger Equation Handout: PDF,
  • Summer School on Carleson theorems and Radon type behavior (May, 2014): After Lie, Victor. "The (weak-L 2) boundedness of the quadratic Carleson operator." Geometric and Functional Analysis 19.2 (2009): 457-497 and Lie, Victor. "The polynomial Carleson operator." arXiv preprint arXiv:1105.4504 (2011). slides (part 1), slides (part 2)

Teaching