# Research

The research at CAM can be broadly classified into three broad topics, Differential equations and related areas, Probability theory and related fields and computational sciences. Differential equations arise in various contexts. Most of the models arising in engineering sciences or in natural sciences are in terms of differential equations. Differential equations also play a big role in other branches of mathematics like geometry. The mathematics involved in the study of these equations vary quite a bit from one to the other. Mathematicians at CAM work on developing mathematical tools to analyze various partial differential equations.

Probability is another important branch of applied mathematics and at CAM research is carried out in stochastic partial differential equations, theory of Gaussian processes and their relationships with geometry, Ergodic Theory and Dynamical Systems, Symbolic Dynamics and related areas of Statistical Physics and Probability, Random polynomials, etc.

CAM also does research in computational sciences including Geophysical fluid dynamics, Scientific computing which include developing efficient algorithms and also applications to various other fields like monsoon.

Ongoing research at CAM can be classified into following areas.

**Partial Differential Equations and Related Areas**

- Qualitative and quantitative properties of solutions of differential equations
- Elliptic and Parabolic PDE
- Hyperbolic Equations and Conservation Laws
- Euler and Navier-Stokes equations
- Problems in calculus of variations
- Optimal transportation problems
- Inverse problems
- Geometry and analysis
- Control theory problems of fluid flows

**Probability Theory and Related Areas**

- Probability theory
- Stochastic differential equations
- Random Geometry, point processes
- Ergodic Theory and dynamical systems
- Rough path theory, stochastic analysis

**Complex Analysis and Related Areas**

- Potential theory
- Complex analysis
- Several complex variables

**Computational Science**

- Scientific computing and machine learning
- Numerical Analysis of differential equations
- Computational fluid dynamics
- Geophysical fluid flows