The research carried out by the members of the Analysis Group, primarily focused on partial differential equations and related areas, covers a wide range of topics, from classical aspects of PDEs, to microlocal and global analysis, analysis on graphs, spectral theory, nonlinear equations, and complex analysis in several variables.

Shif Berhanu's research interests lie in the areas of partial differential equations and complex analysis. He has worked on the qualitative and boundary properties of complex vector fields. His current interests include Vekua-type equations and the analytic regularity of solutions of fully nonlinear first order partial differential equations

Cristian Gutierrez's research interests are in the areas of partial differential equations and harmonic analysis. He has worked on weighted norm inequalities, singular integrals, Hardy spaces, Gaussian harmonic analysis, parabolic equations, subelliptic equations, and nonlinear elliptic equations. His current research interests include geometric optics and nonlinear PDEs of Monge-Ampere type.

Gerardo Mendoza's main area of research is linear partial differential equations. He has done work in microlocal analysis, solvability, complex analysis, and analysis on manifolds with singularities. His current interests lie in global analysis, principally on manifolds with singularities, spectral theory of partial differential operators on such manifolds, and various aspects of elliptic differential complexes ranging from local to global.

Irina Mitrea's research interests are in the areas of harmonic analysis, partial differential equations, geometric measure theory, and scattering theory. Her work has been motivated by the study of well-posedness of boundary value problems for elliptic equations and systems of second and higher order in domains with irregular boundaries. Mitrea's current research interests include analysis on quasi-metric spaces and applications of harmonic analysis tools to several complex variables.

Isaac Pesenson is interested in Shannon sampling, wavelets and frames on manifolds, Heisenberg-type groups and quantum and combinatorial graphs. He is currently working on developing time-frequency analysis on manifolds and graphs and its applications to CMB, high-dimensional data dimension reduction, image processing, visualization and learning theory.

The *Analysis* group is active in a variety of
research areas including:

- microlocal analysis
- complex analysis in several variables, including analysis on CR manifolds
- nonlinear differential equations
- differential complexes, elliptic or not
- spectral theory on manifolds with singularities, including quantum graphs
- harmonic analysis

Several graduate students have completed Ph.D.s in analysis in recent years. Interested graduate students are encouraged to take advanced topics courses in these and related areas and to attend the above listed weekly seminars.

General information about graduate study in mathematics at Temple, including course descriptions, can be found on the graduate program website.

A partial differential equation (PDE) is an equation involving functions and their partial derivatives, and since many natural laws can be expressed in terms of rates of changes, PDEs appear and have applications in an enormous number of questions. For example, PDEs describe the propagation of sound and heat, the motion of fluids, the behavior of electric and magnetic fields, and the behavior of financial markets. In the first semester the course will focus in the study of three second order equations that contain the ideas and the germ of generality to study more general PDEs: the Laplace equation, the heat equation, and the wave equation. We will also cover first order equations. The solutions of these equations have different qualitative and quantitative properties and their study is essential to understand elliptic, parabolic and hyperbolic equations. The emphasis will be on ideas and techniques presented in a way that can be use later to deal with more difficult situations such us nonlinear equations. These extensions will be the subject of the second semester.**Math 8141-8142: Partial Differential Equations I-II.**

The course will be useful for students in analysis, applied mathematics, and engineering.

Textbooks:

-Partial Differential Equations, by L. C. Evans, Graduate Texts in Mathematics vol. 19, American Mathematical Society, 1998, ISBN: 0-8218-0772-2.

-Elliptic Partial Differential Equations of Second Order, by D. Gilbarg and N. S. Trudinger, Springer, ISBN: 9783540411604.

-Instructor's notes.

Prerequisites: Basic concepts of real analysis; knowledge of Lebesgue integration is useful but not required.

The topic this semester is**Math 9400: Topics in Analysis.***Applied Harmonic Analysis*. The course covers the following: The $L_{2}$-Fourier transform. The short-time Fourier transform. The Discrete Fourier transform. The Fast Fourier transform; time-frequency analysis and the uncertainty principle; Paley-Wiener theory of band-limited functions. Shannon sampling of band-limited functions; discrete and continuous wavelet transform and multiresolution analysis; existence and structure of Gabor frames in $L_{2}(R^{n})$; Zak transform methods; general Hilbert frames; the Heisenberg group, its Schrödinger representation and their role in time-frequency analysis; applications to pseudodifferential operators; and applications to signal processing.

Prerequisites. At least one of the following : Real Analysis, 8042, or Functional Analysis, 9042.

Textbook: - Grochenig, Karlheinz, Foundations of time-frequency analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA, 2001. xvi+359 pp. ISBN: 0-8176-4022-3

This is a first course in Functional Analysis. The course covers: linear spaces; normed spaces; Banach spaces. Basic examples. Hilbert spaces and orthonormal bases. Fejer's Theorem and Fourier series in L**Math 9041: Functional Analysis.**_{2}[-pi, pi]. Bounded linear operators in Banach spaces. Banach- Steinhaus and Banach theorems. Linear functionals, dual spaces and the Hahn-Banach theorem. Examples of linear functionals in basic function spaces. Compact operators. Self-adjoint bounded operators and their spectral decomposition. Unbounded self-adjoint and symmetric operators and their spectral decomposition. Basics of distributions and distributional Fourier transform. The scale of Sobolev spaces H^{s}(**R**^{n}), -infty<s<infty, with applications to elliptic differential operators with constant coefficients.

Prerequisites: Real Analysis 8042.

Text:

- Lusternic & Sobolev, Elements of Functional Analysis.

- W. Rudin, Functional Analysis.

This is a basic course in the analysis of functions of several complex variables. Beginning with Hartogs' theorem on separate analyticity, the Cauchy representation formula, Hartogs' phenomenon we will build up to domains of holomorphy, pseudoconvexity, and the L**Math 9051: Several Complex Variables.**^{2}theory as developed by Hörmander. The theory of holomorphic functions of several variables is very different from the theory in one variable.

The course requires the sequences Math 8041-8042 (Real Analysis) and Math 8051-8052 (Functions of a Complex Variable) or their equivalent, or permission of instructor.

References:

- L. Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd edition, North- Holland Publishing Co., Amsterdam, 1990 - R. O. Wells, Differential Analysis on Complex Manifolds, 3rd edition, GTM 65, Springer, New York, 2008.

- R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, GTM 108, Springer, New York, 1986.