## MATH 138: Celestial Mechanics

Mathematically rigorous introduction to the classical N-body problem: the motion of N particles evolving according to Newton's law. Topics include: the Kepler problem and its symmetries; other central force problems; conservation theorems; variational methods; Hamilton-Jacobi theory; the role of equilibrium points and stability; and symplectic methods. Prerequisites: 53, and 115 or 171.

Last offered: Autumn 2014
| UG Reqs: GER:DB-Math

## MATH 143: Differential Geometry

Geometry of curves and surfaces in three-space and higher dimensional manifolds. Parallel transport, curvature, and geodesics. Surfaces with constant curvature. Minimal surfaces.

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

Instructors:
Lindblad Petersen, O. (PI)
;
Fauteux-Chapleau, F. (TA)

## MATH 144: Introduction to Topology and Geometry

Point set topology, including connectedness, compactness, countability and separation axioms. The inverse and implicit function theorems. Smooth manifolds, immersions and submersions, embedding theorems. Prerequisites:
Math 61CM, or
Math 51 and familiarity with linear algebra and analysis arguments at the level of 113 and 115 respectively.

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

Instructors:
Manolescu, C. (PI)
;
Nguyen, D. (TA)

## 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.

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

## MATH 147: Differential Topology

Introduction to smooth methods in topology including tranvsersality, intersection number, fixed point theorems, as well as differential forms and integration. Prerequisites:
Math 62CM or
Math 144.

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

Instructors:
Chodosh, O. (PI)

## MATH 148: Algebraic Topology

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

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

Instructors:
Kerckhoff, S. (PI)
;
Arana Herrera, F. (TA)

## 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: Either
Math 61CM or
Math 52, and
Math 115 or equivalent.

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

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, WAY-FR

Instructors:
Soundararajan, K. (PI)
;
Miagkov, K. (TA)

## 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, WAY-FR

Instructors:
Conrad, B. (PI)

## 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.

Last offered: Spring 2020
| UG Reqs: GER:DB-Math, WAY-FR

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