## 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.

Instructors:
Marberg, E. (PI)

## MATH 16: Mathematics in the Real World (STATS 90)

Introduction to non-calculus applications of mathematical ideas and principles in real-world problems. Topics include probability and counting, basic statistical concepts, geometric series. Applications include insurance, gambler's ruin, false positives in disease testing, present value of money, and mortgages. No knowledge of calculus required. Enrollment limited to students who do not have Stanford credit for a high school or college course in calculus or statistics.

| UG Reqs: GER:DB-Math

Instructors:
Poulson, J. (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.

| UG Reqs: GER:DB-Math

Instructors:
Sommer, R. (PI)

## 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.

| UG Reqs: GER:DB-Math

Instructors:
Donaldson, T. (PI)

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

(Formerly
CLASSGEN 103.) 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.

| UG Reqs: GER:DB-Hum

Instructors:
Netz, R. (PI)

## 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: 51H or 115 or consent of the instructor. WIM

| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Diaconis, P. (PI)
;
Kerckhoff, S. (PI)

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

Instructors:
Vasy, A. (PI)

## 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.

| UG Reqs: GER:DB-Math

Instructors:
Ford, A. (PI)

## 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.

Instructors:
Eliashberg, Y. (PI)

## MATH 180: Introduction to Financial Mathematics

Financial derivatives: contracts and options. Hedging and risk management. Arbitrage, interest rate, and discounted value. Geometric random walk and Brownian motion as models of risky assets. Initial boundary value problems for the heat and related partial differential equations. Self-financing replicating portfolio. Black-Scholes pricing of European options. Dividends. Implied volatility. Optimal stopping and American options. Prerequisite: 53. Corequisites: 131, 151 or
STATS 116.

| UG Reqs: GER:DB-Math

Instructors:
Menz, G. (PI)

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