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:

  • Numerical Linear Algebra by Lloyd Trefethen and David Bau
  • Analysis and Linear Algebra: The Singular Value Decomposition and Applications by James Bisgard
  • Applied Numerical Linear Algebra by J.W. Demmel
  • Matrix Computations by Gene Golub and Charles Van Loan
  • Matrix Analysis by Roger Horn, Charles Johnson

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)


Credit Score: 4