## MATH 145: Algebraic Geometry

An introduction to the methods and concepts of algebraic geometry. The point of view and content will vary over time, but include: affine varieties, Hilbert basis theorem and Nullstellensatz, projective varieties, algebraic curves. Required: 120. Strongly recommended: additional mathematical maturity via further basic background with fields, point-set topology, or manifolds.

Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Solis, P. (PI)
;
Qian, L. (TA)

## MATH 146: Analysis on Manifolds

Differentiable manifolds, tangent space, submanifolds, implicit function theorem, differential forms, vector and tensor fields. Frobenius' theorem, DeRham theory. Prerequisite: 62CM or 52 and familiarity with linear algebra and analysis arguments at the level of 113 and 115 respectively.

Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Wieczorek, W. (PI)

## MATH 147: Differential Topology

Smooth manifolds, transversality, Sards' theorem, embeddings, degree of a map, Borsuk-Ulam theorem, Hopf degree theorem, Jordan curve theorem. Prerequisite: 115 or 171.

Last offered: Spring 2018
| UG Reqs: GER:DB-Math, WAY-FR

## MATH 148: Algebraic Topology

Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots. Prerequisite: 109 or 120.

Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Kerckhoff, S. (PI)

## MATH 151: Introduction to Probability Theory

Counting; axioms of probability; conditioning and independence; expectation and variance; discrete and continuous random variables and distributions; joint distributions and dependence; central limit theorem and laws of large numbers. Prerequisite: 52 or consent of instructor.

Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Kwan, M. (PI)

## MATH 152: Elementary Theory of Numbers

Euclid's algorithm, fundamental theorems on divisibility; prime numbers; congruence of numbers; theorems of Fermat, Euler, Wilson; congruences of first and higher degrees; quadratic residues; introduction to the theory of binary quadratic forms; quadratic reciprocity; partitions.

Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Tsai, C. (PI)

## MATH 154: Algebraic Number Theory

Properties of number fields and Dedekind domains, quadratic and cyclotomic fields, applications to some classical Diophantine equations. Prerequisites: 120 and 121, especially modules over principal ideal domains and Galois theory of finite fields.

Terms: Spr
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Tsai, C. (PI)
;
Qian, L. (TA)

## MATH 155: Analytic Number Theory

Topics in analytic number theory such as the distribution of prime numbers, the prime number theorem, twin primes and Goldbach's conjecture, the theory of quadratic forms, Dirichlet's class number formula, Dirichlet's theorem on primes in arithmetic progressions, and the fifteen theorem. Prerequisite: 152, or familiarity with the Euclidean algorithm, congruences, residue classes and reduced residue classes, primitive roots, and quadratic reciprocity.

Last offered: Spring 2018
| UG Reqs: GER:DB-Math

## MATH 158: Basic Probability and Stochastic Processes with Engineering Applications (CME 298)

Calculus of random variables and their distributions with applications. Review of limit theorems of probability and their application to statistical estimation and basic Monte Carlo methods. Introduction to Markov chains, random walks, Brownian motion and basic stochastic differential equations with emphasis on applications from economics, physics and engineering, such as filtering and control. Prerequisites: exposure to basic probability.

Terms: Spr
| Units: 3

Instructors:
Cook, N. (PI)
;
Etter, P. (TA)

## MATH 159: Discrete Probabilistic Methods

Modern discrete probabilistic methods suitable for analyzing discrete structures of the type arising in number theory, graph theory, combinatorics, computer science, information theory and molecular sequence analysis. Prerequisite:
STATS 116/
MATH 151 or equivalent. Typically in alternating years.

Terms: Spr
| Units: 3
| UG Reqs: WAY-FR

Instructors:
Kwan, M. (PI)
;
Nuti, P. (TA)

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