## MATH 109: Applied Group Theory

Applications of the theory of groups. Topics: elements of group theory, groups of symmetries, matrix groups, group actions, and applications to combinatorics and computing. Applications: rotational symmetry groups, the study of the Platonic solids, crystallographic groups and their applications in chemistry and physics. Honors math majors and students who intend to do graduate work in mathematics should take 120. WIM.

| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Li, Z. (PI)

## MATH 110: Applied Number Theory and Field Theory

Number theory and its applications to modern cryptography. Topics: congruences, finite fields, primality testing and factorization, public key cryptography, error correcting codes, and elliptic curves, emphasizing algorithms. WIM.

| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Wilson, J. (PI)

## MATH 111: Computational Commutative Algebra

Introduction to the theory of commutative rings, ideals, and modules. Systems of polynomial equations in several variables from the algorithmic viewpoint. Groebner bases, Buchberger's algorithm, elimination theory. Applications to algebraic geometry and to geometric problems.

| UG Reqs: GER:DB-Math

## MATH 113: Linear Algebra and Matrix Theory

Algebraic properties of matrices and their interpretation in geometric terms. The relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations. Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; eigenvectors and eigenvalues; diagonalization. (
Math 104 offers a more application-oriented treatment.)

| UG Reqs: GER:DB-Math, WAY-FR

## MATH 114: Introduction to Scientific Computing (CME 108)

Introduction to Scientific Computing Numerical computation for mathematical, computational, physical sciences and engineering: error analysis, floating-point arithmetic, nonlinear equations, numerical solution of systems of algebraic equations, banded matrices, least squares, unconstrained optimization, polynomial interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations, truncation error, numerical stability for time dependent problems and stiffness. Implementation of numerical methods in MATLAB programming assignments. Prerequisites:
MATH 51, 52, 53; prior programming experience (MATLAB or other language at level of
CS 106A or higher).nGraduate students should take it for 3 units and undergraduate students should take it for 4 units.

| UG Reqs: GER:DB-EngrAppSci, WAY-AQR, WAY-FR

Instructors:
Sing Long Collao, C. (PI)
;
Ying, L. (PI)

## MATH 115: Functions of a Real Variable

The development of real analysis in Euclidean space: sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Honors math majors and students who intend to do graduate work in mathematics should take 171. Prerequisite: 51.

| UG Reqs: GER:DB-Math

Instructors:
Lin, Y. (PI)
;
Miller, J. (PI)

## MATH 116: Complex Analysis

Analytic functions, Cauchy integral formula, power series and Laurent series, calculus of residues and applications, conformal mapping, analytic continuation, introduction to Riemann surfaces, Fourier series and integrals. (
Math 106 offers a less theoretical treatment.) Prerequisites: 52, and 115 or 171.

| UG Reqs: GER:DB-Math

Instructors:
Kerckhoff, S. (PI)

## MATH 120: Groups and Rings

Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 109 but altered content and more theoretical orientation. Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals; polynomial rings over a field; PID and non-PID. Unique factorization domains. WIM.

| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Bump, D. (PI)
;
Vakil, R. (PI)

## MATH 121: Galois Theory

Field of fractions, splitting fields, separability, finite fields. Galois groups, Galois correspondence, examples and applications. Prerequisite:
Math 120 and (also recommended) 113.

| UG Reqs: GER:DB-Math

Instructors:
Soundararajan, K. (PI)

## MATH 122: Modules and Group Representations

Modules over PID. Tensor algebra. Group representations and group rings. Maschke's theorem and character theory. Character tables, construction of representations. Prerequisite:
Math 120. Also recommended: 113.

Instructors:
Wilson, J. (PI)

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