Venue: TBA
Class Timings: TBA
First Meeting: TBA
Course Syllabus:
Asymptotic sequences and asymptotic expansions, solution of linear system of equations and weakly nonlinear system of equations, Regular and Singular perturbation problems, Method of multiple scales, Method of matched asymptotic expansions and boundary layer theory, WKBJ, geometric optics, and ray tracing, introduction to homogenization theory.Introduction to coding with Python, Time stepping methods for ordinary differential equations, error bounds of solutions, Fast Fourier Transform (FFT), spectral methods for solving partial differential equations, solutions of heat and wave equations, introduction to the pseudo-spectral method, dealiasing and hyperdissiaption, solution of two-dimensional Euler equation, Nonlinear Schrodinger equation, and Can-Hilliard equation.
Basic probability, Random variables, probability distribution functions and probability density functions, Moments of random variables, sequence of random variables, Law of large numbers and Central limit theorem, Ito's theorem and Stochastic calculus, Stochastic differentiation and integration, Stochastic differential equations (SDEs), analytical and numerical solution of SDE's, applications of SDE's to practical problems.
Course Evaluation: Regular homework and exams will be used to assign the course grade.
Course outcomes: After completing this course students will develop:
- A tool kit for solving differential equations where a small or a large parameter can be taken advantage of.
- Capability to numerically solve linear and nonlinear differential equations.
- Insights on stochastic calculus and stochastic differential equations with emphasis on practical applications.
- Perturbation Methods by E. J. Hinch
- Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons by Mark J. Ablowitz
- Computational Physics by Mark Newman
- Numerical Analysis by Richard L Burden and J Douglas Faires
- Spectral Methods by Claudio Canuto and Co-authors
- The Physics of Fluid Turbulence by W. D. McComb
- A First Course in Stochastic Calculus Louis-Pierre Arguin
- Teacher: Jim Thomas
Venue: TBA
Class Timings: TBA
First Meeting: TBA
Course Description: Linear-algebraic concepts underlie many theoretical ideas across mathematics. The utility of these concepts is not theoretical alone but also arises from the fact that these concepts motivate the development of highly efficient algorithms. Taking motivation from classical approximation theory and modern data science, in this course we will see how numerical algorithms and theoretical concepts go hand-in-hand to solve practical problems. A brief outline of the course is given below.
- Motivating examples: data, approximations and projections
- Floating point representation, conditioning, computational stability
- Matrix factorizations and iterative algorithms: QR, LU, Cholesky, Schur, eigenvalue
- The singular value decomposition
- Under- & over-determined systems of equations
- Advanced topics: FFT, image processing, inverse problems, optimization, randomized algorithms
Course Outcome: After completing this course, the student will:
- Possess tools and skills to develop computational solutions to practical problems as well as to understand computational complexities
- Understand various linear algebra concepts widely used in statistics, machine learning and approximation theory.
- Formulate well-defined problems using these concepts
- Develop oral and written communication skills relevant for mathematical discourse
Prerequisites: Proficiency in Python/Julia/MATLAB, linear algebra, introductory probability and statistics. Analysis concepts will be developed as needed hence undergraduate real analysis will be useful, but not necessary.
Textbooks:
- Teacher: Vishal Vasan
Venue: TBA
Class Timings: TBA
First Meeting: TBA
Course Description: The course will introduce broad topics in theoretical ecology and evolution, but selected sections will include technical results. how populations grow: logistic models and frequency dependent growth how multiple species coexist: biodiversity, predator-prey and Lotka-Volterra models how populations interact: resource competition and cooperation, consumer-resource models how ecosystems remain stable: the May bound and the ecological instability transition - random matrix, network science and geometric approaches to ecology multistability, stochasticity, and catastrophes in ecosystems statistical inference and data-driven approaches to ecology spatiotemporal structure and metacommunity ecology evolutionary processes: selection, mutation, and drift dynamics on fitness landscapes, epistasis and ruggedness
Textbooks:
Mark Kot, Elements of Mathematical Ecology (Cambridge University Press, 2012
Robert May and Angela McLean, Theoretical Ecology: Principles and Applications (Oxford University Press, 2007)
Kevin S. McCann and Gabriel Gellner, Theoretical Ecology: Concepts and Applications (Oxford University Press, 2020)
Martin Nowak, Evolutionary Dynamics: Exploring the Equations of Life (Harvard University Press, 2006)
References:
1. Statistical physics and dynamics of evolution
2. “Evolution in rapidly evolving populations”, BH Good, Harvard University (2016)
3. "Evolutionary dynamics", DS Fisher, Les Houches Course (2006)
4. "Statistical Genetics and Evolution of Quantitative Traits", RA Neher and BI Shraiman, Reviews of Modern Physics (2011)
5. "Genetic demixing and the evolution in linear stepping stone models", K. S. Korolev, Mikkel Avlund, Oskar Hallatschek, and David R. Nelson,Reviews of Modern Physics (2010)
- Teacher: Akshit Goyal