Ruoyuan Liu

I am a third year Ph.D. student at The University of Edinburgh, under the supervision of Prof. Tadahiro Oh.

Research Interests:

Nonlinear partial differential equations and (singular) stochastic PDEs. In particular, study of (deterministic and probabilistic) nonlinear dispersive PDEs, using the techniques from PDEs, harmonic analysis, and probability.

Papers:

(with E. Brun, G. Li, and Y. Zine) Global well-posedness of the one-dimensional fractional cubic nonlinear Schrödinger equations.
(with E. Brun and G. Li) Global well-posedness of the energy-critical stochastic nonlinear wave equations, preprint.
(with N. Tzvetkov and Y. Wang) Existence, uniqueness, and universality of global dynamics for the fractional hyperbolic Φ⁴₃-model.
(with A. Chapouto and G. Li) Global dynamics for the stochastic nonlinear beam equations on the four-dimensional torus, preprint.
(with A. Debussche, N. Tzvetkov, and N. Visciglia) Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space, submitted (2023). arXiv link
Local well-posedness of the periodic nonlinear Schrödinger equation with a quadratic nonlinearity u² in negative Sobolev spaces, to appear in J. Dyn. Diff. Equat. arXiv link
On the probabilistic well-posedness of the two-dimensional periodic nonlinear Schrödinger equation with the quadratic nonlinearity |u|², J. Math. Pures Appl. 171 (2023), 75-101. arXiv link
(with T. Oh) Sharp local well-posedness of the two-dimensional periodic nonlinear Schrödinger equation with a quadratic nonlinearity |u|², to appear in Math. Res. Lett. arXiv link
Global well-posedness of the two-dimensional stochastic viscous nonlinear wave equations, to appear in Stoch. Partial Differ. Equ. Anal. Comput. arXiv link
(with T. Oh) On the two-dimensional singular stochastic viscous nonlinear wave equations, C. R. Math. Acad. Sci. Paris 360 (2022), 1227-1248. arXiv link
(with G. Chasapis and T. Tkocz) Rademacher-Gaussian tail comparison for complex coefficients and related problems, Proc. Amer. Math. Soc. 150 (2022), no. 3, 1339-1349. arXiv link
(with T. Tkocz) A note on the extremal non-central sections of the cross-polytope, Adv. in Appl. Math. 118 (2020), 102031, 17pp. arXiv link

Notes:

Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation, Edinburgh, Fall 2022. Notes

Reading groups and mini-courses:

Teaching assistant for the exercise session of the mini-course "Statistical mechanics of the focusing nonlinear Schrödinger equations" (by L. Tolomeo), Summer School on Stochastic Analysis, Bernoulli Center, EPFL, Lausanne, Switzerland, 7-18 Aug. 2023.
Reading group on "Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation" (together with Y. Zine), The University of Edinburgh, Edinburgh, UK. Sept. 2022 - Aug. 2023.

Talks:

(Student talk) Summer School on Stochastic Analysis, Bernoulli Center, EPFL, Lausanne, Switzerland, 7-18 Aug. 2023.
(Poster) Summer School on Stochastic Analysis, Bernoulli Center, EPFL, Lausanne, Switzerland, 7-18 Aug. 2023.
Workshop: Random dispersive PDEs 2023, School of Mathematics, University of Birmingham, Birmingham, UK, 19-21 Jul. 2023.
Deterministic and Probabilistic Dynamics of Nonlinear Dispersive PDEs, Maxwell Institute for Mathematical Sciences, Edinburgh, UK, 14-16 Jun. 2023.
Analysis Seminar, University of Birmingham, Birmingham, UK, Feb. 2023.
Analysis Seminar, The University of Edinburgh, Edinburgh, UK, Jan. 2023.
(Poster) Trials in Wave Turbulence: from random waves to kinetic equations, Gran Sasso Science Institute, L'Aquila, Italy, 5-10 Sep. 2022.
Dispersive Day 2022 from the deterministic and stochastic perspectives, Maxwell Institute for Mathematical Sciences, Edinburgh, UK, Jun 17, 2022.

Conferences and workshops attended:

Conference: Harmonic Analysis and Partial Differential Equations, Hausdorff Center for Mathematics, Bonn, Germany, 29 May - 2 Jun. 2023.
Summer School on PDEs and Randomness, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, 10-24 May. 2023.
Fourier Analysis @200, International Centre for Mathematical Sciences, Edinburgh, UK, 27 Jun - 01 Jul. 2022.
Harmonic Analysis, Stochastics and PDEs, International Centre for Mathematical Sciences, Edinburgh, UK, 20-24 Jun. 2022.
Online conference: Trends in Calculus of Variations and Partial Differential Equations, jointly organized by University of Sussex, UK, and UGent, Belgium, 18-20 May. 2022.
(Online) Unifying Concepts in PDEs with Randomness, Center de Recherches Mathématiques, Montréal, Canada, 16-27 May 2022.
Probability, Analysis and Dynamics '22, Bristol, UK, Apr 6-8, 2022.
(Online) Hamiltonian Methods and Asymptotic Dynamics, Hamiltonian Methods in Dispersive and Wave Evolution Equations, ICERM, Brown University, Providence, RI, USA, Dec. 2021.
(Online) Convex Integration and Nonlinear Partial Differential Equations, International Centre for Mathematical Sciences, Edinburgh, UK, 08-12 Nov. 2021.
(Online) Evolution Equations in Mathematical Physics: Dispersive and Hyperbolic,Tianjin University, China, Sep. 2021.
(Online) Bath Mathematical Symposium on "PDE and Randomness", University of Bath, UK, Sep 1-10, 2021.
(Online) Harmonic Analysis and Nonlinear Partial Differential Equations, Research Institute for Mathematical Sciences, Kyoto, Japan, Jun. 2021.

The University of Edinburgh, UK

Tutor for "Calculus and its Applications", Spring 2023
Tutor for "Honours Complex Variables - Skills", Spring 2023
Tutor for "Honours Analysis", Fall 2022
Tutor for "Honours Analysis - Skills", Fall 2022
Tutor for "Maths Base Semester 1", Fall 2022
Tutor for "Fundamentals of Pure Mathematics", Spring 2022
Tutor for "Maths Base Semester 2", Spring 2022

Carnegie Mellon University, US

Teaching assistant for "Multidimensional Calculus", Spring 2020
Teaching assistant for "Concepts of Mathematics", Spring 2018
Teaching assistant for "Differential and Integral Calculus", Fall 2017

Education:

Sep. 2021 - present	Ph.D. candidate, The University of Edinburgh, Edinburgh, UK
Jan. 2019 - May 2020	M.S. in Mathematical Science, Carnegie Mellon University, Pittsburgh, PA, US
Sep. 2016 - May 2020	B.S. in Mathematical Science (with University Honors and College Honors), Carnegie Mellon University, Pittsburgh, PA, US
Sep. 2016 - May 2020	B.S. in Computer Science (with University Honors), Carnegie Mellon University, Pittsburgh, PA, US

Awards:

The University of Edinburgh, UK

Aug. 2023	Sir William Darling Memorial Prize 2023 (a university-wide prize for pre-eminence in sport and academia)
Mar. 2023	Laura Wisewell travel fund

Besides mathematics, I like playing billiards games, especially snooker. I play competitively as an amateur player. Below are some of my achievements.

Snooker

Highest breaks: 136 (practice), 104 (real frame), 88 (tournament).
Runner up, BUCS-UPC Snooker Individual Championship, Woking, UK, Mar. 2023.

American pool

9th place and winner of men's sportsmanship award, ACUI Collegiate 9-Ball National Championships, Champaign, IL, US, Jun. 2019.
7th place, ACUI Collegiate 9-Ball National Championships, West Lafayette, IN, US, Jun. 2018.