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

Instructors:
Hershkovits, O. (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.

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

Instructors:
Varolgunes, U. (PI)

## MATH 148: Algebraic Topology

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

Last offered: Winter 2019
| UG Reqs: GER:DB-Math

## MATH 151: Introduction to Probability Theory

A proof-oriented development of basic 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. nPrerequisite:
Math 52, and
Math 115 or equivalent.

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

Instructors:
Chatterjee, S. (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, WAY-FR

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.

Last offered: Spring 2019
| UG Reqs: GER:DB-Math

## MATH 155: Analytic Number Theory

Introduction to Dirichlet series and Dirichlet characters, Poisson summation, Gauss sums, analytic continuation for Dirichlet L-functions, applications to prime numbers (e.g., prime number theorem, Dirichlet's theorem). Prerequisites: Complex analysis (
Math 106 or 116),
Math 152 (or comparable familiarity with the Euclidean algorithm, multiplicative group modulo n, and quadratic reciprocity), and experience with basic analysis arguments.

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

Instructors:
Conrad, B. (PI)

## 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:
Papanicolaou, G. (PI)

## 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)

## MATH 161: Set Theory

Informal and axiomatic set theory: sets, relations, functions, and set-theoretical operations. The Zermelo-Fraenkel axiom system and the special role of the axiom of choice and its various equivalents. Well-orderings and ordinal numbers; transfinite induction and transfinite recursion. Equinumerosity and cardinal numbers; Cantor's Alephs and cardinal arithmetic. Open problems in set theory. Prerequisite: students should be comfortable doing proofs.

Last offered: Winter 2019
| UG Reqs: GER:DB-Math

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