• Analysis I
  ° Measure Theory, Fourier Analysis, Calculus on Normed Linear Spaces.


• Analysis II
  ° Topological Vector Spaces and Functional Analysis, Spectral Theory, Distribution (and Fourier) Theory.

(*The topics covered can differ depending on the instructor.)

Time Table & Lecture Notes:

Analysis I

° Instructor- Dr. Amrita Ghosh.

°° Lecture notes

Analysis II

° Instructor- Dr. Tuhin Ghosh.

°° Lecture notes

• Analysis III
° Distributions, Fundamental solutions, Sobolev Spaces.
° L^2 theory of Linear Elliptic equations, Stability estimates, Regularity results.
° Linear Evolution equations, Cauchy problem, Galerkin's approach, Semigroup Theory.

Time Table & Lecture Notes:

Analysis III_A

° Instructor- Dr. Tuhin Ghosh.

°° Lecture notes

Analysis III_B

° Instructor(s)- Dr. Tuhin Ghosh, Dr. Saumyajit Das, Dr. Tanvi

°° Lecture notes

• Topics in Analysis and PDEs
  

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  • Inverse Problems
    
    Mathematical inverse problems refer to a class of problems where the goal is to determine the unknown causes or parameters of a system based on observed data or measurements. A typical inverse problem is to recover the coefficients of a PDE, i.e. physical properties like conductivity, medium propagation, internal density distribution etc. from the boundary or exterior measurements.
    
    
    °Some familiar Topics: Carleman estimates, Unique Continuation, Calderon problem, Scattering problem, Gelfand problem.
    
    
    °References:
    
    
    1. Isakov, Victor, Inverse problems for Partial Differential Equations. Springer-Verlag.
    2. Eskin, Gregory, Lectures on Linear Partial Differential Equations. American Mathematical Society.
    3. Katchalov, Alexander; Kurylev, Yaroslav; Lassas, Matti; Inverse Boundary Spectral Problems. Chapman & Hall/CRC.
  
  
  • Introduction to the Mathematical Fluid Dynamics
    
    Mathematical fluid dynamics deals with the study of fluid flow using mathematical models and techniques. It provides a theoretical framework to understand and analyze the behavior of fluids in various physical systems. This involves PDEs (for example Navier-Stokes equation) which are usually derived from the principles of conservation of mass, momentum, and energy.
    
    
    °Some familar Topics: Boundary layers, Instability analysis, PDEs as gradient flows, Fluid-structure Interaction problem, Thin film flows.
    
    
    °References:
    
    
    1. Galdi, G. P., An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. Springer.
    2. Temam, Roger, Navier-Stokes equations. North-Holland Publishing House.
    3. Ockendon, H.; Ockendon, J. R., Viscous Flow. Cambridge University Press.
  
  
  • Homogenization
    
    Homogenization processes provide a macro scale approximation to a problem with heterogeneities at micro scale by suitably averaging out small scales and incorporating their effects on large scales. These effects are quantified by the so-called homogenized coefficients. A common thread is often projected in order to come up with their optimal bounds, spectral characterization etc.
    
    
    °Some familiar Topics: Asymptotic Analysis, H-limit, Compensated Compactness, H-Measure, Relative homogenization.
    
    
    °References:
    
    
    1. Tartar, Luc, The General Theory of Homogenization. Springer-Verlag.
    2. Allaire, Gregoire, Shape Optimization by the Homogenization Method. Springer-Verlag.

Inverse Scattering for the Fractional Schrödinger equation

On Fractional Borg-Levinson Problem

Adaptive Modified Weak Galerkin Method for Obstacle Problem.

Boundary Control and Calderón type Inverse Problems in Non-local heat equation.

Anisotropic Calderón Problem for a Non-Local Second Order Elliptic Operator.

Fractional anisotropic Calderón problem with external data.

Anisotropic Calderón problem of a nearly Laplace-Beltrami operator of order 2+

The Malgrange-Ehrenpreis Theorem.

Long time behaviour of evolution equations using Entropy method.

On the Homogenization of Maxwell Equations.

On Inverse Problems for Dynamical Sysytem.

• Seminars on Harmonic Analysis and PDEs

• Workshop on Harmonic Analysis: Fourier Multipliers and Related Topics

• SPIM- ODE/PDE Week

• HRI Library.

• Inverse Problems Talks (running online seminar at UC, Irvine).