## 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.

Terms: Aut
| Units: 3

Instructors:
Basu, R. (PI)
;
He, J. (TA)

## 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.

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

Instructors:
Sommer, R. (PI)
;
Helfer, J. (TA)

## MATH 162: Philosophy of Mathematics (PHIL 162, PHIL 262)

(Graduate students register for
PHIL 262.) General survey of the philosophy of mathematics, focusing on epistemological issues. Includes survey of some basic concepts (proof, axiom, definition, number, set); mind-bending theorems about the limits of our current mathematical knowledge, such as Gödel's Incompleteness Theorems, and the independence of the continuum hypothesis from the current axioms of set theory; major philosophical accounts of mathematics: Logicism, Intuitionism, Hilbert's program, Quine's empiricism, Field's program, Structuralism; concluding with a discussion of Eugene Wigner's `The Unreasonable Effectiveness of Mathematics in the Natural Sciences'. Students won't be expected to prove theorems or complete mathematical exercises. However, includes some material of a technical nature. Prerequisite: PHIL150 or consent of instructor.

Last offered: Autumn 2015
| UG Reqs: GER:DB-Math

## MATH 163: The Greek Invention of Mathematics (CLASSICS 136)

How was mathematics invented? A survey of the main creative ideas of ancient Greek mathematics. Among the issues explored are the axiomatic system of Euclid's Elements, the origins of the calculus in Greek measurements of solids and surfaces, and Archimedes' creation of mathematical physics. We will provide proofs of ancient theorems, and also learn how such theorems are even known today thanks to the recovery of ancient manuscripts.

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

## MATH 171: Fundamental Concepts of Analysis

Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 115 but altered content and more theoretical orientation. Properties of Riemann integrals, continuous functions and convergence in metric spaces; compact metric spaces, basic point set topology. Prerequisite: 61CM or 61DM or 115 or consent of the instructor. WIM

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

Instructors:
Kerckhoff, S. (PI)
;
Zhu, X. (PI)
;
Arana Herrera, F. (TA)
...
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Instructors:
Kerckhoff, S. (PI)
;
Zhu, X. (PI)
;
Arana Herrera, F. (TA)
;
Kraushar, N. (TA)
;
Ottolini, A. (TA)

## MATH 172: Lebesgue Integration and Fourier Analysis

Similar to 205A, but for undergraduate Math majors and graduate students in other disciplines. Topics include Lebesgue measure on Euclidean space, Lebesgue integration, L^p spaces, the Fourier transform, the Hardy-Littlewood maximal function and Lebesgue differentiation. Prerequisite: 171 or consent of instructor.

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

Instructors:
Gu, Y. (PI)
;
Zou, J. (TA)

## MATH 173: Theory of Partial Differential Equations

A rigorous introduction to PDE accessible to advanced undergraduates. Elliptic, parabolic, and hyperbolic equations in many space dimensions including basic properties of solutions such as maximum principles, causality, and conservation laws. Methods include the Fourier transform as well as more classical methods. The Lebesgue integral will be used throughout, but a summary of its properties will be provided to make the course accessible to students who have not had 172 or 205A. In years when
Math 173 is not offered,
Math 220 is a recommended alternative (with similar content but a different emphasis). Prerequisite: 171 or equivalent.

Last offered: Winter 2016

## MATH 174: Calculus of Variations

An introductory course emphasizing the historical development of the theory, its connections to physics and mechanics, its independent mathematical interest, and its contacts with daily life experience. Applications to minimal surfaces and to capillary surface interfaces. Prerequisites:
Math 171 or equivalent.

Last offered: Winter 2010

## MATH 175: Elementary Functional Analysis

Linear operators on Hilbert space. Spectral theory of compact operators; applications to integral equations. Elements of Banach space theory. Prerequisite: 115 or 171.

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

Instructors:
Luk, J. (PI)
;
Wigderson, Y. (TA)

## MATH 177: Geometric Methods in the Theory of Ordinary Differential Equations

Hamiltonian systems and their geometry. First order PDE and Hamilton-Jacobi equation. Structural stability and hyperbolic dynamical systems. Completely integrable systems. Perturbation theory.

Last offered: Spring 2015

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