# Applied Math & Computational Science (AMCS)

AMCS 510 Complex Analysis

Complex numbers, DeMoivre's theorem, complex valued functions of a complex variable, the derivative, analytic functions, the Cauchy-Riemann equations, complex integration, Cauchy's integral theorem, residues, computation of definite integrals by residues, and elementary conformal mapping.

Taught by: Staff.

One-term course offered either term

Also Offered As: MATH 410

Prerequisite: MATH 240

Activity: Lecture

1.0 Course Unit

Topics will include: Vector spaces, Basis and dimension, quotients; Linear maps and matrices; Determinants, Dual spaces and maps; Invariant subspaces, Cononical forms; Scalar products: Euclidean, unitary and symplectic spaces; Orthogonal and Unitary operators; Tensor products and polylinear maps; Symmetric and skew-symmetric tensors and exterior algebra.

Also Offered As: MATH 314, MATH 514

Prerequisite: MATH 240

Activity: Lecture

1.0 Course Unit

AMCS 520 Ordinary Differential Equations

After a rapid review of the basic techniques for solving equations, the course will discuss one or more of the following topics: stability of linear and nonlinear systems, boundary value problems and orthogonal functions, numerical techniques, Laplace transform methods.

One-term course offered either term

Also Offered As: MATH 420

Prerequisite: MATH 240

Activity: Lecture

1.0 Course Unit

AMCS 525 Partial Dif Equations

Method of separation of variables will be applied to solve the wave, heat, and Laplace equations. In addition, one or more of the following topics will be covered: qualitative properties of solutions of various equations (characteristics, maximum principles, uniqueness theorems), Laplace and Fourier transform methods, and approximation techniques.

Course usually offered in fall term

Also Offered As: MATH 425

Prerequisite: MATH 240

Activity: Lecture

1.0 Course Unit

AMCS 567 Mathematical Computation Methods for Modeling Biological Systems

This course will cover topics in systems biology at the molecular/cellular scale. The emphasis will be on quantitative aspects of molecular biology, with possible subjects including probabilistic aspects of DNA replication, transcription, translation, as well as gene regulatory networks and signaling. The class will involve analyzing and simulating models of biological behavior using MATLAB. Prereqisite: Graduate standing or permission of the instructor.

One-term course offered either term

Also Offered As: BE 567, GCB 567

Activity: Lecture

1.0 Course Unit

AMCS 584 The Mathmatics of Medical Imaging and Measurement

The last several decades have seen major revolutions in both medical and non-medical and imaging technologies. Underlying all of these advances are sophisticated mathematical tools to model the measurement process and reconstruct images. This course begins with an introduction of the mathematical models and then proceeds to discuss the integral transforms that underlie these models: the Fourier transform, the Radon transform and the Laplace transform. We discuss how each of these transforms is inverted, both in theory and in practice. Along the way we study interpolation, sampling, approximation theory, filtering and noise analysis. This course assumes a thorough knowledge of linear algebra and a knowledge of analysis at the undergraduate level (Math 314 and Math 360 and Math 361, or Math 508 and Math 509).

Taught by: Staff.

Course not offered every year

Also Offered As: BE 584, MATH 584

Prerequisites: MATH 314, 360, 361 or 508, 509

Activity: Lecture

1.0 Course Unit

AMCS 599 Independent Study

Activity: Independent Study

1.0 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

1.0 Course Unit

AMCS 603 Algebraic Techniques for Applied Mathematics and Computational Science, 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

1.0 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.

One-term course offered either term

Also Offered As: MATH 608

Prerequisites: Math 508, 509

Activity: Lecture

1.0 Course Unit

AMCS 609 Analysis

Real analysis: general measure theory, outer measures and Cartheodory construction, Hausdorff measures, Radon-Nikodym theorem, Fubini's theorem, Hilbert space and L2-theory of the Fourier transform. Functional analysis: 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.

One-term course offered either term

Also Offered As: MATH 609

Prerequisite: MATH 608

Activity: Lecture

1.0 Course Unit

AMCS 610 Functional Analysis

Convexity and the Hahn Banach Theorem. Hilbert Spaces, Banach Spaces, and examples: Sobolev spaces, Holder spaces. The uniform bounded principle, Baire category theorem, bounded operators, open mapping theorem, closed graph theorem and applications. The concepts of duality and dual spaces. The Riesz theory of compact operators and Fredholm theory. Functional calculus and elementary Spectral Theory. Interpolation theorems. Applications to partial differential equations and approximation theory.

Also Offered As: MATH 610

Prerequisite: MATH 608 OR MATH 609

Activity: Lecture

1.0 Course Unit

AMCS 999 Independent Study & Research

Study under the direction of a faculty member.

Activity: Independent Study

1.0 Course Unit