Topics in Modern Hyperbolic PDEs
The goal of this course is to prepare students to enter research in hyperbolic PDE, nonlinear dispersive equations, and mathematical general relativity.
The course should also be useful to students working in applied areas where nonlinear evolution, multiscale behavior, and wave propagation play a central role. Techniques such as Fourier analysis, wave-packet decomposition, dispersive estimates, and concentration compactness have important connections with problems in fluid dynamics, nonlinear optics, imaging, signal processing, geophysics, and other areas involving complex wave phenomena and scale interactions.
We will study both conservative evolution equations, including the wave, Klein–Gordon, and Schrödinger equations, and dissipative models such as the heat equation.
A central theme will be concentration compactness and profile decomposition, with emphasis on how these methods capture loss of compactness in large-data problems. We will develop the necessary analytic tools, including Sobolev and Strichartz estimates, and study their connection with Fourier restriction, the Kakeya problem, and wave-packet/tube decompositions in nonlinear dispersive PDE.
The final part of the course will focus on applications to low-regularity Einstein equations, including the phenomenon that weak limits of vacuum solutions may carry a nontrivial effective stress-energy tensor generated by high-frequency gravitational oscillations. The course is intended for graduate students and advanced undergraduates with basic preparation in ODEs, functional analysis, and PDE.
The course should also be useful to students working in applied areas where nonlinear evolution, multiscale behavior, and wave propagation play a central role. Techniques such as Fourier analysis, wave-packet decomposition, dispersive estimates, and concentration compactness have important connections with problems in fluid dynamics, nonlinear optics, imaging, signal processing, geophysics, and other areas involving complex wave phenomena and scale interactions.
We will study both conservative evolution equations, including the wave, Klein–Gordon, and Schrödinger equations, and dissipative models such as the heat equation.
A central theme will be concentration compactness and profile decomposition, with emphasis on how these methods capture loss of compactness in large-data problems. We will develop the necessary analytic tools, including Sobolev and Strichartz estimates, and study their connection with Fourier restriction, the Kakeya problem, and wave-packet/tube decompositions in nonlinear dispersive PDE.
The final part of the course will focus on applications to low-regularity Einstein equations, including the phenomenon that weak limits of vacuum solutions may carry a nontrivial effective stress-energy tensor generated by high-frequency gravitational oscillations. The course is intended for graduate students and advanced undergraduates with basic preparation in ODEs, functional analysis, and PDE.
Lecturer
Date
16th September ~ 18th December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Friday | 13:30 - 15:05 | Shuimo-LG26 | - | - | - |
Prerequisite
The course is intended for graduate students and advanced undergraduates in mathematics or physics at Quizhen College. Basic ODE, PDE (taught as basic courses) and functional analysis
Syllabus
Tentative Syllabus
The course is expected to cover the following topics (may be slightly modified based on the demand of the students):
1. Ordinary Differential Equations and Dynamical Systems
* Qualitative theory of ODEs and dynamical systems
* Lyapunov functions and stability
* Hartman–Grobman theorem
* Stable and unstable manifold theorems
2. Nonlinear Wave and Dispersive Equations
* The Cauchy problem for nonlinear wave equations
* Local and long-time behavior of solutions
* Littlewood–Paley theory and frequency localization
* Dispersive and Strichartz estimates
3. Fourier Restriction and Kakeya-Type Methods
* Fourier restriction theory
* Wave-packet and tube decompositions
* Connections with the Kakeya problem
* Applications to nonlinear dispersive equations
4. Concentration Compactness
* Concentration–compactness principles and profile decompositions
* Applications to large-data dispersive PDE
5. Applications to the Einstein Equations
* Hyperbolic structure of the Einstein equations
* Low-regularity and weak-limit problems
* Concentration compactness and high-frequency phenomena in mathematical general relativity
The course is expected to cover the following topics (may be slightly modified based on the demand of the students):
1. Ordinary Differential Equations and Dynamical Systems
* Qualitative theory of ODEs and dynamical systems
* Lyapunov functions and stability
* Hartman–Grobman theorem
* Stable and unstable manifold theorems
2. Nonlinear Wave and Dispersive Equations
* The Cauchy problem for nonlinear wave equations
* Local and long-time behavior of solutions
* Littlewood–Paley theory and frequency localization
* Dispersive and Strichartz estimates
3. Fourier Restriction and Kakeya-Type Methods
* Fourier restriction theory
* Wave-packet and tube decompositions
* Connections with the Kakeya problem
* Applications to nonlinear dispersive equations
4. Concentration Compactness
* Concentration–compactness principles and profile decompositions
* Applications to large-data dispersive PDE
5. Applications to the Einstein Equations
* Hyperbolic structure of the Einstein equations
* Low-regularity and weak-limit problems
* Concentration compactness and high-frequency phenomena in mathematical general relativity
Reference
References and Course Materials:
The course will not follow a single textbook. Detailed lecture notes will be provided throughout the semester, particularly since a substantial portion of the material reflects recent developments and ongoing research of the instructor and Prof. Shing-Tung Yau, for which no standard textbook treatment is currently available.
For foundational material and complementary reading, the following references are recommended:
* C. D. Sogge, Lectures on Nonlinear Wave Equations.
* Qian Wang, Lectures on Nonlinear Wave Equations.
* C. E. Kenig and F. Merle, “Global well-posedness, scattering and blow-up for the energy-critical focusing nonlinear wave equation,” Acta Mathematica, 201 (2008), no. 2, 147–212.
* L. C. Evans, Partial Differential Equations, for general background and standard PDE material.
Additional references and research papers will be assigned as appropriate during the course.
The course will not follow a single textbook. Detailed lecture notes will be provided throughout the semester, particularly since a substantial portion of the material reflects recent developments and ongoing research of the instructor and Prof. Shing-Tung Yau, for which no standard textbook treatment is currently available.
For foundational material and complementary reading, the following references are recommended:
* C. D. Sogge, Lectures on Nonlinear Wave Equations.
* Qian Wang, Lectures on Nonlinear Wave Equations.
* C. E. Kenig and F. Merle, “Global well-posedness, scattering and blow-up for the energy-critical focusing nonlinear wave equation,” Acta Mathematica, 201 (2008), no. 2, 147–212.
* L. C. Evans, Partial Differential Equations, for general background and standard PDE material.
Additional references and research papers will be assigned as appropriate during the course.
Audience
Advanced Undergraduate
, Graduate
Video Public
No
Notes Public
No
Language
English
Lecturer Intro
Puskar Mondal is currently an assistant professor in BIMSA. He's been a fellow and lecturer at Harvard CMSA of Mathematics (his mentor is Prof. Shing-Tung Yau), and was a Ph.D. student at Yale University, under the supervision of Prof. Vincent Moncrief.