MATH 110: Number Theory for Cryptography
Number theory and its applications to modern cryptography. Topics include: congruences, primality testing and factorization, public key cryptography, and elliptic curves, emphasizing algorithms. Includes an introduction to proof-writing. This course develops math background useful in
CS 255. WIM. Prerequisite:
Math 51
Terms: Aut
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
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; dual space and dual basis; eigenvectors and eigenvalues; diagonalization. Includes an introduction to proof-writing. (
Math 104 offers a more application-oriented treatment.) Prerequisites:
Math 51
Terms: Aut, Win, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Malinnikova, E. (PI)
;
Swaminathan, M. (PI)
;
Vondrak, J. (PI)
...
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Instructors:
Malinnikova, E. (PI)
;
Swaminathan, M. (PI)
;
Vondrak, J. (PI)
;
Amar, S. (TA)
;
Niu, J. (TA)
MATH 115: Functions of a Real Variable
The development of 1-dimensional real analysis (the logical framework for why calculus works): sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Includes introduction to proof-writing. Prerequisite:
Math 51 or
Math 56.
Terms: Aut, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 116: Complex Analysis
Holomorphic and analytic functions, power series, Cauchy integral and Cauchy integral formula, meromorphic functions and differential forms, calculus of residues and applications, analytic continuation, conformal mappings, Riemann mapping theorem, Laurent series and conformal classification of annuli, harmonic functions and Dirichlet problem, introduction to Riemann surfaces, theory of elliptic functions and integrals. (
Math 106 offers a less theoretical treatment). Prerequisites: 51,52 and 171, or 61cm and 62cm.
Terms: Aut
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Eliashberg, Y. (PI)
;
Chen, Y. (TA)
MATH 117: Advanced Complex Analysis
Review of holomorphic and meromorphic 1-forms, product development, Gamma-function and Riemann zeta-function, Fourier series and integrals, Fourier and Laplace transforms, differential geometric and analytic approaches to Riemann surfaces and conformal mappings, introduction to hyperbolic geometry, Laplace and d-bar equations and their solvability, the Uniformization Theorem, divisors and line bundles, Riemann-Roch theorem, Abel Jacobi theory. Prerequisite:
Math 116.
Terms: Win
| Units: 4
| UG Reqs: WAY-FR
Instructors:
Eliashberg, Y. (PI)
;
Fushida-Hardy, S. (TA)
MATH 118: Mathematics of Computation
Notions of analysis and algorithms central to modern scientific computing: continuous and discrete Fourier expansions, the fast Fourier transform, orthogonal polynomials, interpolation, quadrature, numerical differentiation, analysis and discretization of initial-value and boundary-value ODE, finite and spectral elements. Prerequisites:
MATH 51 and 53.
Terms: Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Cortinovis, A. (PI)
MATH 120: Groups and Rings
Recommended for Mathematics majors and required of honors Mathematics majors. A more advanced treatment of group theory than in
Math 109, also including ring theory. 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 course. Prerequisite:
Math 51 and some prior proof-writing experience.
Terms: Aut, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
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.
Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Bump, D. (PI)
;
Lopez, A. (TA)
MATH 122: Modules and Group Representations
Modules over PID. Tensor products over fields. Group representations and group rings. Maschke's theorem and character theory. Character tables, construction of representations. Prerequisite:
Math 113 and 120.
Terms: Spr
| Units: 4
| UG Reqs: WAY-FR
Instructors:
Taylor, R. (PI)
MATH 131P: Partial Differential Equations
An introduction to techniques for solving PDE's. Topics include physical examples (such as the heat equation, wave equation, and Laplace's equation in 2 and 3 dimensions) and separation of variables with various coordinate systems to relate them to Sturm-Liouville problems using Fourier, Bessel, and Legendre series. Prerequisite:
Math 53.
Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
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