Gennady Uraltsev
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Contact
Address:
Office 593
Malott Hall – Math Department
Cornell University
212 Garden Avenue
Ithaca NY 14853
USA
Email: guraltsev@math.cornell.edu
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 timefrequency 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. Timefrequency behavior often arises when considering maximal or multilinear analogues of CalderรณnZygmund SIOs.
My PhD thesis (2016) was concerned with developing and applying outer measure Lebesgue space theory: a powerful and general functionalanalytic framework allowing one to systematically deal with with a large class of timefrequency operators.
Banach spacevalued 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 timefrequency analysis results to the context of Banach spaces. Studying multilinear 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 (multilinear Banach space geometry).
Uniform bounds
Many multilinear operators in harmonic analysis, known as "BrascampLieb" operators, depend on several geometric parameters. Many operators in timefrequency analysis happen to be singular variants of such "BrascampLieb" 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 timefrequency 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 adhoc, 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 timefrequency 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  Banachvalued 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.14431458 arXiv:1612.03028 
Uraltsev, G.  2016  Variational Carleson embeddings into the upper 3space  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.151166 arXiv:1303.1421 
Theses
PhD Thesis  TimeFrequency Analysis of the Variational Carleson Operator using outermeasure $ L^{p} $ spaces ๐ 
Master Thesis  Multiparameter Singular Integrals: Product and Flag Kernels ๐ 
Bachelor Thesis  Regularity of Minimizers of OneDimensional Scalar Variational Problems with Lagrangians with Reduced Smoothness Conditions ๐ 
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 spacesโ  Seminar speaker 
Teaching
Spring 2020
Honors Analysis I (math4130) ๐
Lebesgue Integration crash course
Crash course materials (email me for the password; please do not share nor password nor the downloaded files).
Suggested book: Stein, E. M., & Shakarchi, R. (2005). Real analysis: measure theory, integration, and Hilbert spaces. Princeton, N.J: Princeton University Press.