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 Gennady Uraltsev HC Wang Assistant Professor Cornell University – Mathematics

 PhD BIGS – University of Bonn Advisor: Prof. Dr. Christoph Thiele

### Contact

Office 593
Malott Hall – Math Department
Cornell University
212 Garden Avenue
Ithaca NY 14853
USA

## Research Interests

 Harmonic Analysis Partial Differential Equation Stochastic PDEs

My research is concerned with Harmonic Analysis and the theory of singular integral operators. My main area of work is time-frequency analysis, initiated by Carleson with his celebrated result on the pointwise convergence of Fourier series for $$L^{2}$$ functions. Since then the field has been significantly developed and many deep and surprising connections have been found with Ergodic Theory, Additive Combinatorics, and, crucially, with the study of dispersive PDEs and SDEs. On the other hand, many fundamental questions in the area remain open and some are beyond the reach of currently developed techniques. Time-frequency behavior often arises when considering maximal or multilinear analogues of Calderón-Zygmund SIOs.

My PhD thesis (2016) was concerned with developing and applying outer measure Lebesgue space theory: a powerful and general functional-analytic framework allowing one to systematically deal with with a large class of time-frequency operators.

### Banach space-valued harmonic analysis

Most results about singular integral operators were originally formulated for functions valued in $$\C$$. Many PDE applications however require analogous results for functions valued in Banach spaces. I am in particular interested in extending time-frequency analysis results to the context of Banach spaces. Studying multi-linear operators valued in Banach spaces can also provide interesting insights about the intrinsic geometry of different Banach spaces on their own and when related to others (multi-linear Banach space geometry).

### Uniform bounds

Many multilinear operators in harmonic analysis, known as "Brascamp-Lieb" operators, depend on several geometric parameters. Many operators in time-frequency analysis happen to be singular variants of such "Brascamp-Lieb" inequalities. While many bounds are known in specific cases, a more uniform theory is incomplete. Trying to prove "uniform bounds" for such operators i.e. bounds that are independent of a specific geometric parameter configuration elucidates how time-frequency techniques interact with more classical harmonic analysis results.

### Outer Lebesgue spaces

Outer Lebesgue spaces have found great use in formalizing, cleaning up, and streamlining quite complex, and sometimes ad-hoc, arguments in time frequency analysis proof by providing a consistent functional analytic framework. Many abstract questions (duality, interpolation etc.) about outer Lebesgue spaces remain open.

### Stochastic PDEs: renormalization and paracontrolled calculus

Evolution PDEs with random driving noise or random initial data present new challenges. Parabolic equations have been investigated at length using both analytic and algebraic techniques; my main interest is related to dispersive equations. Techniques recently developed for dealing with stochastic dispersive PDEs seem to have interesting potential applications even in deterministic settings in time-frequency analysis e.g. understanding bounds for the Nonlinear Fourier Transform (aka scattering transform).

## Publications

 Uraltsev, G. 2019 Uniform Bounds for the bilinear Hilbert transform in the Banach range draft, in preparation Amenta, A. and Uraltsev, G. 2019 The bilinear Hilbert transform in UMD spaces arXiv preprint arXiv:1909.06416, submitted Amenta, A. and Uraltsev, G. 2019 Banach-valued modulation invariant Carleson embeddings and outer-$L^ p$ spaces: the Walsh case arXiv preprint arXiv:1905.08681, submitted Di Plinio, F., Do, Y.Q. and Uraltsev, G.N. 2018 Positive Sparse Domination of Variational Carleson Operators Annali della Scuola Normale Superiore di Pisa. Classe di scienze, 18(4), pp.1443-1458 arXiv:1612.03028 Uraltsev, G. 2016 Variational Carleson embeddings into the upper 3-space arXiv preprint arXiv:1610.07657, submitted C. Mantegazza , G. Mascellani, and G. Uraltsev 2014 On the distributional Hessian of the distance function Pacific Journal of Mathematics 270.1: pp.151-166 arXiv:1303.1421

### Theses

 PhD Thesis Time-Frequency Analysis of the Variational Carleson Operator using outer-measure $Lp$ spaces 🔗 Master Thesis Multi-parameter Singular Integrals: Product and Flag Kernels 🔗 Bachelor Thesis Regularity of Minimizers of One-Dimensional Scalar Variational Problems with Lagrangians with Reduced Smoothness Conditions 🔗

## Talks and Conferences

### Planned

Dates Where Title
2020/04/04 – 2020/04/05 AMS Spring Central Sectional Meeting, Purdue University TBA Special Session on Harmonic Analysis speaker

### Past

#### 2019

Dates Where Title
2019/10/07 Washington University, St. Louis Uniform Bounds for the Bilinear Hilbert Transform Seminar speaker

#### 2018

Dates Where Title
2018/07 ICM 2018 Satellite Conference Harmonic Analysis
2018/10 NEAM 2018, New Paltz   Short talk speaker

#### 2017

Dates Where Title
2017/12 St. Petersburg Chebychev Laboratory Outer measure spaces in Time Frequency Analysis Minicourse speaker
2017/12 St. Petersburg Department of Steklov Institute of Mathematics Uniform Bounds for the Bilinear Hilbert Transform Seminar speaker
2017/10 NEAM 2017 Uniform Bounds for the Bilinear Hilbert Transform Short talk speaker
2017/06 Math Department, Delft University ariational Carleson andbeyond using embedding maps and iterated outer measure spaces” Seminar speaker

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