Harmonic analysis (Fourier coefficients behaviour in functional classes: theorems of De Leeuw - Katznelson - Kahane type and connected statements, operators - multipliers, Men‘shov-type correction theorems, uncertainty principals)
Geometric analysis (harmonic measure in Rn, counterexamples to its absolute continuity to the surface measure, harmonic measure for sets of high co-dimensions, analogues to Dahlberg’s theorem for the latter setting)
Dynamical systems, mathematical physics (hysteresis, operators spectrum and properties)
Minor interests include:
Bellman function, TCS (classical algorithms, Machine Learning, AI and Godel‘s incompleteness theorem), combinatorics and geometry (geodesics and Laplace operator spectrum on surfaces).
Polina Perstneva is currently studying to receive a Ph.D. from Université Paris-Saclay, under the directed of Prof. Guy David. She received her B.S. in Mathematics from Saint-Petersburg State University in 2019 and has worked on several undergrad and graduate theses:
Master thesis: “Harmonic measure in high co-dimensions” under supervision of Professor Guy David (grade 18, general diploma note Bien),
Bachelor thesis: “Theorems of de Leeuw-Katznelson-Kahane type on the two-dimensional torus” under the supervision of professor Sergey Kislyakov (diploma with distinction).
2020 — M.S. in 'Arithmetics, Analysis, Geometry Master,' Université Paris-Saclay
2019 — B.S. in Mathematics, Saint-Petersburg State University
Winner of MIPT math olympiad (2014), prize winner of OMMO (2014)
Prize winner of Saint-Petersburg stage of All-Russian physics and astronomy olympiads (2014)
Special diploma of Sakharov‘s Reading for poster, physics section, 2014
Scholarship of “Gazprom Neft” prize winner (2016 - 2019)
Fondation Math ́ematique Jaсques Hadamard scholarshiper (2019 - 2020)
Kislyakov S.V., Perstneva P.S. “Indicator functions with uniformly bounded Fourier sums and large gaps in the spectrum”, arxiv.org/abs/2006.02561, 2020,
Aleksandr Enin, Polina Perstneva, Sergey Tikhomirov. “Periodic solutions of parabolic equations with hysteresis in dimension 1”, Zap. Nauchn. Sem. POMI, Volume 489, pp 36-54, 2020 (to be translated in Journal of Mathematical Sciences).