The research interests of the members of the Analysis Group focus on partial differential equations (PDEs), harmonic analysis, and various closely related fields. Their work spans a broad array of topics, including global analysis, which investigates the behavior of differential equations on large scales or complex geometric structures, and fluid mechanics, where they study the mathematical principles governing fluid flow, often modeled by nonlinear PDEs. Another major focus is optimal transport theory, a field concerned with finding the most efficient ways to move and redistribute resources, which connects deeply to topics in PDEs, geometry, and probability theory. This theory has found applications in economics, logistics, image processing, and more, offering powerful tools to solve problems of optimization and resource allocation. The group's research also extends into the calculus of variations, where it studies the optimization of functionals, often arising in physics, geometry, and economics and leading to equations that describe minimal surfaces, optimal shapes, and energy minimization problems. Together, these varied research directions highlight the group's comprehensive approach to advancing theoretical and applied mathematics in key areas of modern analysis.
Current members:
Advisor: Cristian Gutierrez. Research interests: optimal transport and optics.
Advisor: Irina Mitrea. Research interests: partial differential equations, harmonic analysis and geometric measure theory
Advisor: Yury Grabovsky. Research interests: homogenization and analytic continuation
Webpage. Research interests: calculus of variations, homogenization and analytic continuation.
Webpage. Research interests: optimal transport, optics and partial differential equations.
Webpage. Research interests: fluid mechanics and partial differential equations.
Webpage. Research interests: geometric measure theory, harmonic analysis, partial differential equations.
Webpage. Research interests: global analysis and partial differential equations.
Webpage. Research interests: fluid mechanics and partial differential equations.
Analysis Seminar, one of the regular seminars in the department, usually meets on Mondays 2:30-3:30pm in Wachman 617. The current contact for the Analysis Seminar is Mihaela Ignatova.
Several graduate students have completed Ph.D.s under the direction of members of the Analysis Group. Interested graduate students are encouraged to take advanced topics courses in these and related areas and to attend the analysis seminar. General information about graduate study in mathematics at Temple University can be found on the graduate program website.
Selected Courses
This course covers the core areas of analysis. It focuses on the development of Lebesgue measure and integration theory, differentiation, abstract measures and integration, maximal functions, Hilbert spaces, basic functional analysis, and Hausdorff measure and dimension. Emphasis will be on exercises and problems. The course prepares students to take the Real Analysis section of the qualifying exam.
Textbook: Measure and Integral by R. Wheeden and A. Zygmund; and Selected Problems in Real Analysis by B. Makarov et al.
This is a course about fundamental concepts and results in the theory of elliptic, parabolic, and hyperbolic differential equations, preparing students for problem-solving in PDEs and their applications. Topics covered:
- Systems of first-order odes existence and uniqueness of solutions, continuous and differentiable dependence on the parameters and the initial data. The Cauchy problem for quasilinear 1st order pdes, method of characteristics, shocks.
- Solution of the Cauchy problem for nonlinear first order PDEs.
- Laplace's equation, maximum principles, Dirichlet problem, Green's function and Poisson kernel.
- Solution of the Dirichlet problem by the method of subharmonic functions. Solution by energy methods.
- Heat equation: initial boundary value problems, fundamental solution, mean value formula, maximum principle, uniqueness theorems, examples of non-uniqueness, backward heat equation. Energy methods.
- Wave equation: D'Alambert's formula, plane waves, solution by spherical means. Huygen's phenomenon, finite propagation speed. Energy methods: uniqueness, domain of dependence. Solution of the Maxwell equations.
- Fourier transform. Plancherel theorem, multiplicative properties of the Fourier transform, solution to constant coefficient equations.
- Distributions, weak derivatives, Sobolev spaces, approximation theorems by smooth functions: interior and up to the boundary. Extension of functions, traces, embedding's theorems, Rellich's lemma. Resolution of elliptic second-order equations with energy methods, Lax-Milgram, energy estimates, Fredholm alternative.
Textbook: L. C. Evans, "Partial differential Equations".
The course has variable content reflecting the research interests of members of the analysis group. For example, in Spring 2020 the topic of the course was Optimal Transport. In the Spring 2024, the course was devoted to the mathematical analysis of PDEs of fluids. These included: results on well-posedness of Euler, Navier-Stokes and related equations will be described; classical and modern approaches for long time behavior, regularity and singularity formation; Beale-Kato-Majda-type results for conservative systems and Ladyzhenskaya-Prodi-Serrin-type conditions for dissipative systems. A basic reference text is "The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations" by J. Bedrossian and V. Vicol, (Graduate Studies in Mathematics 225, AMS 2020).
This is a course covering fundamental topics of functional analysis having multiple applications in theoretical and applied mathematics areas such as harmonic analysis, PDEs, probability, optimization, economics, physics, etc. Topics covered: the Hahn-Banach theorem and its applications; the uniform boundedness principle, the closed graph, and open mapping theorems; unbounded linear operators basics; weak topologies, duality; compact operators and spectral decomposition; applications to boundary value problems; Fourier transform and distributions.
Textbook: Haim Brezis, "Functional Analysis, Sobolev spaces, and partial differential equations"