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

Mathematics is a very peculiar human activity. It delivers a type of knowledge that is particularly stable, often conceived as a priori and necessary. Moreover, this knowledge is about abstract entities, which seem to have no connection to us, spatio-temporal creatures, and yet it plays a crucial role in our scientific endeavors. Many philosophical questions emerge naturally: What is the nature of mathematical objects? How can we learn anything about them? Where does the stability of mathematics comes from? What is the significance of results showing the limits of such knowledge, such as Gödel's incompleteness theorem? The first part of the course will survey traditional approaches to philosophy of mathematics ("the big Isms") and consider the viability of their answers to some of the previous questions: logicism, intuitionism, Hilbert's program, empiricism, fictionalism, and structuralism. The second part will focus on philosophical issues emerging from the actual practice of mathematics. We will tackle questions such as: Why do mathematicians re-prove the same theorems? What is the role of visualization in mathematics? How can mathematical knowledge be effective in natural science? To conclude, we will explore the aesthetic dimension of mathematics, focusing on mathematical beauty. Prerequisite: PHIL150 or consent of instructor.

Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math

Instructors:
De Toffoli, S. (PI)
;
Thompson, D. (TA)

## 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.

Terms: Aut
| Units: 3-5
| UG Reqs: GER:DB-Hum, WAY-A-II

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: 61CM or 61DM or 115 or consent of the instructor. WIM

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

## 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:
Hershkovits, O. (PI)

## 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.

Terms: Win
| Units: 3

Instructors:
Chatterjee, S. (PI)
;
Bates, E. (TA)

## 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:
Galkowski, J. (PI)
;
De Groote, C. (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.

Terms: Spr
| Units: 3

Instructors:
Eliashberg, Y. (PI)
;
Fauteux-Chapleau, F. (TA)

## MATH 193: Polya Problem Solving Seminar

Topics in mathematics and problem solving strategies with an eye towards the Putnam Competition. Topics may include parity, the pigeonhole principle, number theory, recurrence, generating functions, and probability. Students present solutions to the class. Open to anyone with an interest in mathematics.

Terms: Aut
| Units: 1
| Repeatable
5 times
(up to 5 units total)

Instructors:
Soundararajan, K. (PI)
;
Vakil, R. (PI)

## MATH 197: Senior Honors Thesis

Honors math major working on senior honors thesis under an approved advisor carries out research and reading. Satisfactory written account of progress achieved during term must be submitted to advisor before term ends. May be repeated 3 times for a max of 9 units. Contact department student services specialist to enroll.

Terms: Aut, Win, Spr, Sum
| Units: 1-6
| Repeatable
3 times
(up to 9 units total)

Instructors:
Bump, D. (PI)
;
Eliashberg, Y. (PI)
;
Fox, J. (PI)
;
Luk, J. (PI)
;
Reingold, O. (PI)
;
Ryzhik, L. (PI)
;
Savarese, S. (PI)
;
Vakil, R. (PI)