# Applied Mathematics & Computational Science (AMCS)

**AMCS 567 Mathematical and Computational Modeling of Biological Systems.**

This is an introductory course in mathematical biology. The emphasis will be on the use of mathematical and computational tools for modeling physical phenomena which arise in the study biological systems. Possible topics include random walk models of polymers, membrane elasticity, electrodiffusion and excitable systems, single-molecule kinetics, and stochastic models of biochemical networks.

Course not offered every year

Also Offered As: BE 567

Prerequisites: BE 324 and BE 350

Activity: Lecture

1 Course Unit

**AMCS 599 Independent Study**

Activity: Independent Study

1 Course Unit

**AMCS 602 Algebraic Techniques for Applied Mathematics and Computational Science, I.**

We turn to linear algebra and the structural properties of linear systems of equations relevant to their numerical solution. In this context we introduce eigenvalues and the spectral theory of matrices. Methods appropriate to the numerical solution of very large systems are discussed. We discuss modern techniques using randomized algorithms for fast matrix-vector multiplication, and fast direct solvers. Topics covered include the classical Fast Multipole Method, the interpolative decomposition, structured matrix algebra, randomized methods for low-rank approximation, and fast direct solvers for sparse matrices. These techniques are of central importance in applications of linear algebra to the numerical solution of PDE, and in Machine Learning. The theoretical content of this course is illustrated and supplemented throughout the year with substantial computational examples and assignments.

Course not offered every year

Activity: Lecture

1 Course Unit

**AMCS 603 Algebraic Techiques II**

We begin with an introduction to group theory. The emphasis is on groups as symetries and transformations of space. After an introduction to abstract groups, we turn our attention to compact Lie groups, in particular SO(3), and their representations. We explore the connections between orthogonal polynomials, classical transcendental functions and group representations. This unit is completed with a discussion of finite groups and their applications in coding theory.

Course not offered every year

Activity: Lecture

1 Course Unit

**AMCS 608 Analysis**

Complex analysis: analyticity, Cauchy theory, meromorphic functions, isolated singularities, analytic continuation, Runge's theorem, d-bar equation, Mittlag-Leffler theorem, harmonic and sub-harmonic functions, Riemann mapping theorem, Fourier transform from the analytic perspective. Introduction to Real Analysis: Weierstrass approximation, Lebesgue measure and integration Euclideanspaces, Borel measures and convergence theorems, C0 and the Riesz-Markov theorem, Lp-spaces, Fubini's Theorem.

Course not offered every year

Also Offered As: MATH 608

Prerequisite: Math 508-509

Activity: Lecture

1 Course Unit

**AMCS 609 Analysis**

Real analysis continued: general measure theory, outer measures and Cartheodoryconstruction, Hausdorff measures, Radon-Nikodym theorem, the general Fubini theorem. Functional Analysis: Hilbert space and L2-theory of the Fourier transform, normed linear spaces, convexity, the Hahn-Banach theorem, duality for Banach spaces, weak convergence, bounded linear operators, Baire category theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, compact operators, Fredholm theory, interpolation theorems, Lp-theory for the Fourier transform, functional calculus and spectral theory.

Course not offered every year

Also Offered As: MATH 609

Prerequisites: Math 608 or permission of the instructor.

Activity: Lecture

1 Course Unit

**AMCS 990 Masters Reg Tuition**

Activity: Masters Thesis

1 Course Unit

**AMCS 999 Independent Study & Research**

Activity: Independent Study

1 Course Unit