# PMA Undergraduate Courses (2021-22)

Ay 1.
The Evolving Universe.
9 units (3-3-3):
third term.
Introduction to modern astronomy that will illustrate the accomplishments, techniques, and scientific methodology of contemporary astronomy. The course will be organized around a set of basic questions, showing how our answers have changed in response to fresh observational discoveries. Topics to be discussed will include telescopes, stars, planets, the search for life elsewhere in the universe, supernovae, pulsars, black holes, galaxies and their active nuclei, and Big Bang cosmology. A field trip to Palomar Observatory will be organized. Not offered on a pass/fail basis.
Instructor: Hallinan.

Ma 1 abc.
Calculus of One and Several Variables and Linear Algebra.
9 units (4-0-5):
first, second, third terms.
Prerequisites: high-school algebra, trigonometry, and calculus.
Special section of Ma 1 a, 12 units (5-0-7). Review of calculus. Complex numbers, Taylor polynomials, infinite series. Comprehensive presentation of linear algebra. Derivatives of vector functions, multiple integrals, line and path integrals, theorems of Green and Stokes. Ma 1 b, c is divided into two tracks: analytic and practical. Students will be given information helping them to choose a track at the end of the fall term. There will be a special section or sections of Ma 1 a for those students who, because of their background, require more calculus than is provided in the regular Ma 1 a sequence. These students will not learn series in Ma 1 a and will be required to take Ma 1 d.
Instructors: Demiroglu, Katz, Kechris, Graber, Flach, Ni.

Ma 1 d.
Series.
4 units (2-0-2):
second term only.
Prerequisites: special section of Ma 1 a.
This is a course intended for those students in the special calculus-intensive sections of Ma 1 a who did not have complex numbers, Taylor polynomials, and infinite series during Ma 1 a. It may not be taken by students who have passed the regular Ma 1 a.
Instructor: Demiroglu.

Ph 1 abc.
Classical Mechanics and Electromagnetism.
9 units (4-0-5):
first, second, third terms.
The first year of a two-year course in introductory classical and modern physics. Topics: Newtonian mechanics in Ph 1 a; electricity and magnetism, and special relativity, in Ph 1 b, c. Emphasis on physical insight and problem solving. Ph 1 b, c is divided into two tracks: the Practical Track emphasizing practical electricity, and the Analytic Track, which teaches and uses methods of multivariable calculus. Students enrolled in the Practical Track are encouraged to take Ph 8 bc concurrently. Students will be given information helping them to choose a track at the end of fall term.
Instructors: Cheung, Hsieh, Refael, Cissé.

Ma 2/102.
Differential Equations.
9 units (4-0-5):
first term.
Prerequisites: Ma 1 abc.
The course is aimed at providing an introduction to the theory of ordinary differential equations, with a particular emphasis on equations with well known applications ranging from physics to population dynamics. The material covered includes some existence and uniqueness results, first order linear equations and systems, exact equations, linear equations with constant coefficients, series solutions, regular singular equations, Laplace transform, and methods for the study of nonlinear equations (equilibria, stability, predator-prey equations, periodic solutions and limiting cycles).
Instructors: Makarov, Angelopoulos.

Ph 2 abc.
Waves, Quantum Mechanics, and Statistical Physics.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ph 1 abc, Ma 1 abc.
An introduction to several areas of physics including applications in modern science and engineering. Topics include discrete and continuous oscillatory systems, wave mechanics, applications in telecommunications and other areas (first term); foundational quantum concepts, the quantum harmonic oscillator, the Hydrogen atom, applications in optical and semiconductor systems (second term); ensembles and statistical systems, thermodynamic laws, applications in energy technology and other areas (third term). Although best taken in sequence, the three terms can be taken independently.
Instructors: Porter, Cheung, Adhikari.

FS/Ay 3.
Freshman Seminar: Automating Discovering the Universe.
6 units (2-0-4):
third term.
Powerful new instruments enable astronomers to collect huge volumes of data on billions of objects. As a result, astronomy is changing dramatically: by the end of this decade, most astronomers will probably be analysing data collected in large surveys, and only a few will still be visiting observatories to collect their own data. The tool chest of future astronomers will involve facility with "big data", developing clever queries, algorithms (some based on machine learning) and statistics, and combining multiple databases. This course will introduce students to some of these tools. After "recovering" known objects, students will be unleashed to make their own astronomical discoveries in new data sets. Limited enrollment.
Instructor: Kulkarni.

Ma 3/103.
Introduction to Probability and Statistics.
9 units (4-0-5):
second term.
Prerequisites: Ma 1 abc.
Randomness is not anarchy-it follows mathematical laws that we can understand and use to clarify our knowledge of the universe. This course is an introduction to the main ideas of probability and statistics. The first half is devoted to the fundamental concepts of probability theory, including distributions and random variables, independence and conditional probability, expectation, the Law of Averages (Laws of Large Numbers), and "the bell curve" (Central Limit Theorem). The second half is devoted to statistical reasoning: given our observations of the world, what can we infer about the stochastic mechanisms generating our data? Major themes include estimation of parameters (e.g. maximum likelihood), hypothesis testing, confidence intervals, and regression analysis (least squares). Students will be expected to be able to carry out computer-based analyses.
Instructor: Cvitanic.

Ph 3.
Introductory Physics Laboratory.
6 units (0-3-3):
first, second, third terms.
Prerequisites: Ph 1 a or instructor's permission.
Introduction to experimental physics and data analysis, with techniques relevant to all fields that deal in quantitative data. Specific physics topics include ion trapping, harmonic motion, mechanical resonance, and precision interferometry. Broader skills covered include introductions to essential electronic equipment used in modern research labs, basic digital data acquisition and analysis, statistical interpretation of quantitative data, professional record keeping and documentation of experimental research, and an introduction to the Mathematica programming language. Only one term may be taken for credit.
Instructors: Black, Libbrecht.

FS/Ph 4.
Freshman Seminar: Astrophysics and Cosmology with Open Data.
6 units (3-0-3):
first term.
Astrophysics and cosmology are in the midst of a golden age of science-rich observations from incredibly powerful telescopes of various kinds. The data from these instruments are often freely available on the web. Anyone can do things like study x-rays from pulsars in our galaxy or gamma rays from distant galaxies using data from Swift and Fermi; discover planets eclipsing nearby stars using data from Kepler; measure the expansion of the universe using supernovae data; study the cosmic microwave background with data from Planck; find gravitational waves from binary black hole mergers using data from LIGO; and study the clustering of galaxies using Hubble data. We will explore some of these data sets and the science than can be extracted from them. A primary goal of this class is to develop skills in scientific computing and visualization-bring your laptop! Not offered 2021-22.

Ma 4/104.
Introduction to Mathematical Chaos.
9 units (3-0-6):
third term.
An introduction to the mathematics of "chaos." Period doubling universality, and related topics; interval maps, symbolic itineraries, stable/unstable manifold theorem, strange attractors, iteration of complex analytic maps, applications to multidimensional dynamics systems and real-world problems. Possibly some additional topics, such as Sarkovski's theorem, absolutely continuous invariant measures, sensitivity to initial conditions, and the horseshoe map.
Instructor: Vigneaux.

Ma 5/105 abc.
Introduction to Abstract Algebra.
9 units (3-0-6):
first, second, third terms.
Introduction to groups, rings, fields, and modules. The first term is devoted to groups and includes treatments of semidirect products and Sylow's theorem. The second term discusses rings and modules and includes a proof that principal ideal domains have unique factorization and the classification of finitely generated modules over principal ideal domains. The third term covers field theory and Galois theory, plus some special topics if time permits.
Instructors: Mantovan, Aluffi, Oswal.

Ph 5.
Analog Electronics for Physicists.
9 units (0-5-4):
first term.
Prerequisites: Ph1abc, Ma1abc, Ma2 taken concurrently.
A fast-paced laboratory course covering the design, construction, and testing of practical analog and interface circuits, with emphasis on applications of operational amplifiers. No prior experience with electronics is required. Basic linear and nonlinear elements and circuits are studied, including amplifiers, filters, oscillators and other signal conditioning circuits. Each week includes a 45 minute lecture/recitation and a 2½ hour laboratory. The course culminates in a two-week project of the student's choosing.
Instructors: Rice, Libbrecht.

Ma/CS 6/106 abc.
Introduction to Discrete Mathematics.
9 units (3-0-6):
first, seconnd, third terms.
Prerequisites: for Ma/CS 6 c, Ma/CS 6 a or Ma 5 a or instructor's permission.
First term: a survey emphasizing graph theory, algorithms, and applications of algebraic structures. Graphs: paths, trees, circuits, breadth-first and depth-first searches, colorings, matchings. Enumeration techniques; formal power series; combinatorial interpretations. Topics from coding and cryptography, including Hamming codes and RSA. Second term: directed graphs; networks; combinatorial optimization; linear programming. Permutation groups; counting nonisomorphic structures. Topics from extremal graph and set theory, and partially ordered sets. Third term: elements of computability theory and computational complexity. Discussion of the P=NP problem, syntax and semantics of propositional and first-order logic. Introduction to the Gödel completeness and incompleteness theorems.
Instructors: Conlon, Schülke, Kechris.

Ph 6.
Physics Laboratory.
9 units:
second term.
Prerequisites: Ph 2 a or Ph 12 a, Ma 2, Ph 3, Ph 2 b or Ph 12 b (may be taken concurrently), Ma 3 (may be taken concurrently).
A laboratory introduction to experimental physics and data analysis. Experiments use research-grade equipment and techniques to investigate topics in classical electrodynamics, resonance phenomena, waves, and other physical phenomena. Students develop critical, quantitative evaluations of the relevant physical theories; they work individually and choose which experiments to conduct. Each week includes a 30-minute individual recitation and a 3 hour laboratory.
Instructors: Rice, Politzer.

Ma 7/107.
Number Theory for Beginners.
9 units (3-0-6):
third term.
Some of the fundamental ideas, techniques, and open problems of basic number theory will be introduced. Examples will be stressed. Topics include Euclidean algorithm, primes, Diophantine equations, including an + bn = cn and a2-db2 = Â±1, constructible numbers, composition of binary quadratic forms, and congruences.
Instructor: Szumowicz.

Ph 7.
Physics Laboratory.
9 units:
third term.
Prerequisites: Ph6, Ph2b or Ph12b, Ph2c or Ph12c taken concurrently.
A laboratory course continuing the study of experimental physics introduced in Physics 6. The course introduces some of the equipment and techniques used in quantum, condensed matter, nuclear, and particle physics. The menu of experiments includes some classics which informed the development of the modern quantum theory, including electron diffraction, the Stern-Gerlach experiment, Compton scattering, and the Mössbauer Effect. The course format follows that of Physics 6: students work individually and choose which experiments to conduct, and each week includes a 30 minute individual recitation and a 3 hour laboratory.
Instructors: Rice, Politzer.

Ma 8.
Problem Solving in Calculus.
3 units (3-0-0):
first term.
Prerequisites: simultaneous registration in Ma 1 a.
A three-hour per week hands-on class for those students in Ma 1 needing extra practice in problem solving in calculus.
Instructor: Demiroglu.

Ph 8 bc.
Experiments in Electromagnetism.
3 units (0-3-0):
second, third terms.
Prerequisites: Ph 1 a.
A two-term sequence of experiments that parallel the material of Ph 1 bc. It includes measuring the force between wires with a homemade analytical balance, measuring properties of a 1,000-volt spark, and building and studying a radio-wave transmitter and receiver. The take-home experiments are constructed from a kit of tools and electronic parts. Measurements are compared to theoretical expectations.
Instructor: Spiropulu.

FS/Ph 9.
Freshman Seminar: The Science of Music.
6 units (2-0-4):
first term.
This course will focus on the physics of sound, how musical instruments make it, and how we hear it, including readings, discussions, demonstrations, and student observations using sound analysis software. In parallel we will consider what differentiates music from other sounds, and its role psychically and culturally. Students will do a final project of their choice and design, with possibilities including a book review, analysis of recordings of actual musical instruments, or instrument construction and analysis. Freshmen only; limited enrollment.
Instructor: Politzer.

Ma 10.
Oral Presentation.
3 units (2-0-1):
first term.
Open for credit to anyone. Freshmen must have instructor's permission to enroll. In this course, students will receive training and practice in presenting mathematical material before an audience. In particular, students will present material of their own choosing to other members of the class. There may also be elementary lectures from members of the mathematics faculty on topics of their own research interest.
Instructor: Mantovan.

Ph 10.
Frontiers in Physics.
3 units (2-0-1):
first term.
Open for credit to freshmen and sophomores. Weekly seminar by a member of the physics department or a visitor, to discuss his or her research at an introductory level; the other class meetings will be used to explore background material related to seminar topics and to answer questions that arise. The course will also help students find faculty sponsors for individual research projects. Graded pass/fail.
Instructor: Spiropulu.

FS/Ph 11 abc.
Freshman Seminar: Beyond Physics.
6 units (2-0-4):
second, third terms of freshman year and first term of sophomore year.
Freshmen are offered the opportunity to enroll in this class by submitting potential solutions to problems posed in the fall term. A small number of solutions will be selected as winners, granting those students permission to register. This course demonstrates how research ideas arise, are evaluated, and tested and how the ideas that survive are developed. Weekly group discussions and one-on-one meetings with faculty allow students to delve into cutting edge scientific research. Ideas from physics are used to think about a huge swath of problems ranging from how to detect life on extrasolar planets to exploring the scientific underpinnings of science fiction in Hollywood films to considering the efficiency of molecular machines. Support for summer research at Caltech between freshman and sophomore years will be automatic for students making satisfactory progress. Graded pass/fail. Freshmen only; limited enrollment.
Instructor: Phillips.

Ge/Ay 11 c.
Introduction to Earth and Planetary Sciences: Planetary Sciences.
9 units (3-0-6):
third term.
Prerequisites: Ma 1 ab, Ph 1 ab.
A broad introduction to the present state and early history of the solar system, including terrestrial planets, giant planets, moons, asteroids, comets, and rings. Earth-based observations, observations by planetary spacecraft, study of meteorites, and observations of extrasolar planets are used to constrain models of the dynamical and chemical processes of planetary systems. Although Ge 11 abcd is designed as a sequence, any one term may be taken as a standalone course. Physicists and astronomers are particularly welcome.
Instructor: Ingersoll.

Ma 11.
Mathematical Writing.
3 units (0-0-3):
third term.
Prerequisites: Freshmen must have instructor’s permission to enroll.
Students will work with the instructor and a mentor to write and revise a self-contained paper dealing with a topic in mathematics. In the first week, an introduction to some matters of style and format will be given in a classroom setting. Some help with typesetting in TeX may be available. Students are encouraged to take advantage of the Hixon Writing Center’s facilities. The mentor and the topic are to be selected in consultation with the instructor. It is expected that in most cases the paper will be in the style of a textbook or journal article, at the level of the student’s peers (mathematics students at Caltech). Fulfills the Institute scientific writing requirement. Not offered on a pass/fail basis.
Instructor: Graber.

FS/Ma 12.
Freshman Seminar: The Mathematics of Enzyme Kinetics.
6 units (2-0-4):
third term.
Prerequisites: Ma 1a, b.
Enzymes are at the heart of biochemistry. We will begin with a down to earth discussion of how, as catalysts, they are used to convert substrate to product. Then we will model their activity by using explicit equations. Under ideal conditions, their dynamics are described by a system of first order differential equations. The difficulty will be seen to stem from them being non-linear. However, under a steady state hypothesis, they reduce to a simpler equation, whose solution can describe the late time behavior. The students will apply it to some specially chosen, real examples. Not offered 2021-22.

Ph 12 abc.
Waves, Quantum Physics, and Statistical Mechanics.
9 units (4-0-5):
first, second, third terms.
Prerequisites: Ph 1 abc, Ma 1 abc, or equivalents.
A one-year course primarily for students intending further work in the physics option. Topics include classical waves; wave mechanics, interpretation of the quantum wave-function, one-dimensional bound states, scattering, and tunneling; thermodynamics, introductory kinetic theory, and quantum statistics. Chen, Filippone, Patterson.
Instructor: X.

Ma 13.
Problem Solving in Vector Calculus.
2 units (2-0-0):
second term.
Prerequisites: Concurrent registration in Ph 1b.
A two-hour per week, hands-on class for those students enrolled in Ph 1b needing extra practice with problem solving in vector calculus.
Instructor: Demiroglu.

Ma 17.
How to Solve It.
4 units (2-0-2):
first term.
There are many problems in elementary mathematics that require ingenuity for their solution. This is a seminar-type course on problem solving in areas of mathematics where little theoretical knowledge is required. Students will work on problems taken from diverse areas of mathematics; there is no prerequisite and the course is open to freshmen. May be repeated for credit. Graded pass/fail.
Instructor: Staff.

Ay 20.
Basic Astronomy and the Galaxy.
9 units (3-1-5):
first term.
Prerequisites: Ma 1 abc, Ph 1 abc or instructor's permission.
The electromagnetic spectrum and basic radiative transfer; ground and space observing techniques; "pictorial Fourier description" of astrophysical optics; Kepler's laws; exoplanets; stellar masses, distances, and motions; the birth, structure, evolution, and death of stars; the structure and dynamics of the Galaxy. Lessons will emphasize the use of order-of-magnitude calculations and scaling arguments in order to elucidate the physics of astrophysical phenomena. Short labs will introduce astronomical measurement techniques.
Instructor: Kasliwal.

Ma 20.
Frontiers in Mathematics.
1 unit (1-0-0):
first term.
Prerequisites: Open for credit to freshman and sophomores.
Weekly seminar by a member of the math department or a visitor, to discuss his or her research at an introductory level. The course aims to introduce students to research areas in mathematics and help them gain an understanding of the scope of the field. Graded pass/fail.
Instructor: Mantovan.

Ph 20.
Computational Physics Laboratory I.
6 units (0-6-0):
first, second, third terms.
Prerequisites: CS 1 or equivalent.
Introduction to the tools of scientific computing. Use of numerical algorithms and symbolic manipulation packages for solution of physical problems. Python for scientific programming, Mathematica for symbolic manipulation, Unix tools for software development.
Instructors: Mach, Weinstein.

Ay 21.
Galaxies and Cosmology.
9 units (3-0-6):
second term.
Prerequisites: Ma 1 abc, Ph 1 abc or instructor's permission.
Cosmological models and parameters, extragalactic distance scale, cosmological tests; constituents of the universe, dark matter, and dark energy; thermal history of the universe, cosmic nucleosynthesis, recombination, and cosmic microwave background; formation and evolution of structure in the universe; galaxy clusters, large-scale structure and its evolution; galaxies, their properties and fundamental correlations; formation and evolution of galaxies, deep surveys; star formation history of the universe; quasars and other active galactic nuclei, and their evolution; structure and evolution of the intergalactic medium; diffuse extragalactic backgrounds; the first stars, galaxies, and the reionization era.
Instructor: Djorgovski.

Ph 21.
Computational Physics Laboratory II.
6 units (0-6-0):
first, second, third terms.
Prerequisites: Ph 20 or equivalent experience with programming.
Computational tools for data analysis. Use of python for accessing scientific data from the web. Bayesian techniques. Fourier techniques. Image manipulation with python.
Instructors: Mach, Weinstein.

Ph 22.
Computational Physics Laboratory III.
6 units (0-6-0):
first, second, third terms.
Prerequisites: Ph 20 or equivalent experience with programming and numerical techniques.
Computational tools and numerical techniques. Applications to problems in classical mechanics. Numerical solution of 3-body and N-body systems. Monte Carlo integration.
Instructors: Mach, Weinstein.

Ay 30.
Introduction to Modern Research.
3 units (2-0-1):
first term.
Weekly seminar open to declared Ay majors. At the discretion of the instructor, nonmajors who have taken astronomy courses may be admitted. Course is intended for sophomores and juniors. This seminar is held in faculty homes in the evening and is designed to encourage student communication skills as they are introduced to faculty members and their research. Each week a student will review a popular-level article in astronomy for the class. Graded pass/fail.
Instructor: Kulkarni.

Ay 31.
Writing in Astronomy.
3 units (1-0-2):
third term.
This course is intended to provide practical experience in the types of writing expected of professional astronomers. Example styles include research proposals, topical reviews, professional journal manuscripts, and articles for popular magazines such as Astronomy or Sky and Telescope. Each student will adopt one of these formats in consultation with the course instructor and write an original piece. An outline and several drafts reviewed by both a faculty mentor familiar with the topic and the course instructor are required. This course is most suitable for juniors and seniors. Fulfills the Institute scientific writing requirement.
Instructor: Steidel.

Ay 43.
Reading in Astronomy and Astrophysics.
Units in accordance with work accomplished, not to exceed 3:
.
Course is intended for students with a definite independent reading plan or who attend regular (biweekly) research and literature discussion groups. Instructor's permission required. Graded pass/fail.
Instructor: Staff.

Ph 50 ab.
Caltech Physics League.
3 units (1-0-2):
first, second terms.
Prerequisites: Ph 1 abc.
This course serves as a physics club, meeting weekly to discuss and analyze real-world problems in physical sciences. A broad range of topics will be considered, such as energy production, space and atmospheric phenomena, astrophysics, nano-science, and others. Students will use basic physics knowledge to produce simplified (and perhaps speculative) models of complex natural phenomena. In addition to regular assignments, students will also compete in solving challenge problems each quarter with prizes given in recognition of the best solutions. Not offered 2021-22.

Ph 70.
Oral and Written Communication.
6 units (2-0-4):
first, second, third terms.
Provides practice and guidance in oral and written communication of material related to contemporary physics research. Students will choose a topic of interest, make presentations of this material in a variety of formats, and, through a guided process, draft and revise a technical or review article on the topic. The course is intended for senior physics majors. Fulfills the Institute scientific writing requirement.
Instructor: Hitlin.

Ph 77 abc.
Advanced Physics Laboratory.
9 units (0-5-4):
first, second, third terms.
Prerequisites: Ph 7 or instructor's permission.
Advanced preparation for laboratory research. Dual emphasis on practical skills used in modern research groups and historic experiments that illuminate important theoretical concepts. Topics include advanced signal acquisition, conditioning, and data processing, introductions to widely-used optical devices and techniques, laser-frequency stabilization, and classic experiments such as magnetic resonance, optical pumping, and doppler-free spectroscopy. Fundamentals of vacuum engineering, thin-film sample growth, and cryogenics are occasionally offered. Special topics and student-led projects are available on request.
Instructors: Black, Libbrecht.

Ay 78 abc.
Senior Thesis.
9 units:
.
Prerequisites: To register, student must obtain approval of the astronomy option representative and the prospective thesis adviser.
Previous SURF or independent study work can be useful experience. Course open to senior astronomy majors only. Research must be supervised by a faculty member. Students wishing assistance in finding an adviser and/or a topic for a senior thesis are invited to consult with the astronomy option representative. The student will work with an adviser to formulate a research project, conduct original research, present new results, and evaluate them in the context of previously published work in the field. In the first term, the student should be fully engaged in, and make significant progress on, the research project. In the second term, the research continues and an outline of the thesis itself should be reviewed with the adviser and the option representative. In the third term, research work is completed and the focus should turn to thesis writing. A written thesis of 20-100 pages must be completed and approved by the adviser and the option representative before the end of third term. The student and advisor should maintain good communication regarding the scope, content, draft due dates, and final copy of the thesis. First two terms are graded pass/fail, with grades updated at the end of the course to the appropriate letter grade for all three terms.
Instructor: Staff.

Ph 78 abc.
Senior Thesis (Experiment).
9 units:
first, second, third terms.
Prerequisites: To register for this course, the student must obtain approval of the chair of the Physics Undergraduate Committee (Ken Libbrecht).
Open only to senior physics majors. Experimental research must be supervised by a faculty member, the student's thesis adviser. Two 15-minute presentations to the Physics Undergraduate Committee are required, one near the end of the first term and one near the end of third term. The written thesis must be completed and distributed to the committee one week before the second presentation. Students wishing assistance in finding an adviser and/or a topic for a senior thesis are invited to consult with the chair of the Physics Undergraduate Committee, or any other member of this committee. A grade will not be assigned in Ph 78 untli the end of the third term. P grades will be given the first two terms, and then changed at the end of the course to the appropriate letter grade. Not offered on a pass/fail basis.

Ph 79 abc.
Senior Thesis (Theory).
9 units:
first, second, third terms.
Prerequisites: To register for this course, the student must obtain approval of the chair of the Physics Undergraduate Committee (Ken Libbrecht).
Open only to senior physics majors. Theoretical research must be supervised by a faculty member, the student's thesis adviser. Two 15-minute presentations to the Physics Undergraduate Committee are required, one near the end of the first term and one near the end of third term. The written thesis must be completed and distributed to the committee one week before the second presentation. Students wishing assistance in finding an adviser and/or a topic for a senior thesis are invited to consult with the chair of the Physics Undergraduate Committee, or any other member of this committee. A grade will not be assigned in Ph 79 until the end of the third term. P grades will be given the first two terms, and then changed at the end of the course to the appropriate letter grade. Not offered on a pass/fail basis.

Ma 92 abc.
Senior Thesis.
9 units (0-0-9):
first, second, third terms.
Prerequisites: To register, the student must obtain permission of the mathematics undergraduate representative.
Open only to senior mathematics majors who are qualified to pursue independent reading and research. This research must be supervised by a faculty member. The research must begin in the first term of the senior year and will normally follow up on an earlier SURF or independent reading project. Two short presentations to a thesis committee are required: the first at the end of the first term and the second at the midterm week of the third term. A draft of the written thesis must be completed and distributed to the committee one week before the second presentation. Graded pass/fail in the first and second terms; a letter grade will be given in the third term.

Ma 97.
Research in Mathematics.
Units to be arranged in accordance with work accomplished:
.
This course is designed to allow students to continue or expand summer research projects and to work on new projects. Students registering for more than 6 units of Ma 97 must submit a brief (no more than 3 pages) written report outlining the work completed to the undergraduate option rep at the end of the term. Approval from the research supervisor and student's adviser must be granted prior to registration. Graded pass/fail.

Ma 98.
Independent Reading.
3-6 units by arrangement:
.
Occasionally a reading course will be offered after student consultation with a potential supervisor. Topics, hours, and units by arrangement. Graded pass/fail.

Ay 101.
Physics of Stars.
9 units (3-0-6):
first term.
Prerequisites: Ay 20 is recommended.
Physics of stellar interiors and atmospheres. Properties of stars, stellar spectra, radiative transfer, line formation. Stellar structure, stellar evolution. Nucleosynthesis in stars. Stellar oscillations.
Instructor: Mawet.

Ph 101.
Order-of-Magnitude Physics.
9 units (3-0-6):
third term.
Emphasis will be on using basic physics to understand complicated systems. Examples will be selected from properties of materials, geophysics, weather, planetary science, astrophysics, cosmology, biomechanics, etc. Offered in alternate years. Not offered 2021-22.

Ay 102.
Physics of the Interstellar Medium.
9 units (3-0-6):
second term.
Prerequisites: Ay 20 is recommended.
An introduction to observations of the inter-stellar medium and relevant physical processes. The structure and hydrodynamic evolution of ionized hydrogen regions associated with massive stars and supernovae, thermal balance in neutral and ionized phases, star formation and global models for the interstellar medium.
Instructor: Steidel.

Ay/Ph 104.
Relativistic Astrophysics.
9 units (3-0-6):
third term.
Prerequisites: Ph 1, Ph 2 ab.
This course is designed primarily for junior and senior undergraduates in astrophysics and physics. It covers the physics of black holes and neutron stars, including accretion, particle acceleration and gravitational waves, as well as their observable consequences: (neutron stars) pulsars, magnetars, X-ray binaries, gamma-ray bursts; (black holes) X-ray transients, tidal disruption and quasars/active galaxies and sources of gravitational waves.
Instructor: Kasliwal.

Ay 105.
Optical Astronomy Instrumentation Lab.
9 units (1-5-3):
third term.
Prerequisites: Ay 20.
An opportunity for astronomy and physics undergraduates (juniors and seniors) to gain firsthand experience with the basic instrumentation tools of modern optical and infrared astronomy. The 10 weekly lab experiments include radiometry measurements, geometrical optics, polarization, optical aberrations, spectroscopy, CCD characterization, vacuum and cryogenic technology, infrared detector technology, adaptive optics (wavefront sensors, deformable mirrors, closed loop control) and a coronography tuturial.
Instructor: Mawet.

Ph 105.
Analog Electronics for Physicists.
9 units:
first term.
Prerequisites: Ph1abc, Ma2, or equivalent.
A laboratory course intended for graduate students, it covers the design, construction, and testing of simple, practical analog and interface circuits useful for signal conditioning and experiment control in the laboratory. No prior experience with electronics is required. Students will use operational amplifiers, analog multipliers, diodes, bipolar transistors, and passive circuit elements. Each week includes a 45 minute lecture/recitation and a 2½ hour laboratory. The course culminates in a two-week project of the student's choosing.
Instructors: Rice, Libbrecht.

Ph 106 abc.
Topics in Classical Physics.
9 units (4-0-5):
first, second, third terms.
Prerequisites: Ph 2 ab or Ph 12 abc, Ma 2.
An intermediate course in the application of basic principles of classical physics to a wide variety of subjects. Ph106a will be devoted to mechanics, including Lagrangian and Hamiltonian formulations of mechanics, small oscillations and normal modes, central forces, and rigid-body motion. Ph106b will be devoted to fundamentals of electrostatics, magnetostatics, and electrodynamics, including boundary-value problems, multipole expansions, electromagnetic waves, and radiation. It will also cover special relativity. Ph106c will cover advanced topics in electromagnetism and an introduction to classical optics.
Instructors: Fuller, Golwala.

Ay/Ge 107.
Introduction to Astronomical Observation.
9 units (1-1-7):
first term.
Prerequisites: CS1 or equivalent coding experience recommended.
This hands-on, project-based course covers the design, proposal, and execution of astronomical observations, the basics of data reduction and analysis, and interacting with astronomical survey catalogs. In the first module, students will learn to use small, portable telescopes and find and image objects of interest using finder charts. In the second module, students will use Palomar Observatory to propose and execute their own research projects focused on astrophysical or planetary topics. In the third module, students will query and work with data from on-line archives and catalogs. The scope of the course includes imaging and spectroscopic observational techniques at optical and infrared wavelengths. The format centers on projects and practical skills but also includes a lecture and problem set component to establish the theoretical underpinnings of the practical work. The course meets once a week in the evening, and there are 1-2 required field trips to Palomar Observatory.
Instructors: Hillenbrand, de Kleer.

Ma 108 abc.
Classical Analysis.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 1 or equivalent, or instructor’s permission. May be taken concurrently with Ma 109.
First term: structure of the real numbers, topology of metric spaces, a rigorous approach to differentiation in R^n. Second term: brief introduction to ordinary differential equations; Lebesgue integration and an introduction to Fourier analysis. Third term: the theory of functions of one complex variable.
Instructors: Dunn, Wu, Karpukhin.

Ma 109 abc.
Introduction to Geometry and Topology.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 2 or equivalent, and Ma 108 must be taken previously or concurrently.
First term: aspects of point set topology, and an introduction to geometric and algebraic methods in topology. Second term: the differential geometry of curves and surfaces in two- and three-dimensional Euclidean space. Third term: an introduction to differentiable manifolds. Transversality, differential forms, and further related topics.
Instructors: Marcolli, Karpukhin, Smillie.

Ma 110 abc.
Analysis.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 108 or previous exposure to metric space topology, Lebesgue measure.
First term: integration theory and basic real analysis: topological spaces, Hilbert space basics, Fejer's theorem, measure theory, measures as functionals, product measures, L^p -spaces, Baire category, Hahn- Banach theorem, Alaoglu's theorem, Krein-Millman theorem, countably normed spaces, tempered distributions and the Fourier transform. Second term: basic complex analysis: analytic functions, conformal maps and fractional linear transformations, idea of Riemann surfaces, elementary and some special functions, infinite sums and products, entire and meromorphic functions, elliptic functions. Third term: harmonic analysis; operator theory. Harmonic analysis: maximal functions and the Hardy-Littlewood maximal theorem, the maximal and Birkoff ergodic theorems, harmonic and subharmonic functions, theory of H^p -spaces and boundary values of analytic functions. Operator theory: compact operators, trace and determinant on a Hilbert space, orthogonal polynomials, the spectral theorem for bounded operators. If time allows, the theory of commutative Banach algebras.
Instructors: Karpukhin, Rains, Radziwill.

Ay 111 ab.
Introduction to Current Astrophysics Research.
1 unit (1-0-0):
first, second, third terms.
This course is intended primarily for first-year Ay graduate students, although participation is open and encouraged. Students are required to attend seminar-style lectures given by astrophysics faculty members, describing their research, to attend the weekly astronomy colloquia, and to follow these with additional readings on the subject. At the end of each term, students are required to summarize in oral or written form (at the discretion of the instructor), one of the covered subjects that is of most interest to them.
Instructors: Djorgovski, Howard, Hopkins.

Ma 111 abc.
Topics in Analysis.
9 units (3-0-6):
third term.
Prerequisites: Ma 110 or instructor's permission.
This course will discuss advanced topics in analysis, which vary from year to year. Topics from previous years include potential theory, bounded analytic functions in the unit disk, probabilistic and combinatorial methods in analysis, operator theory, C*-algebras, functional analysis. The third term will cover special functions: gamma functions, hypergeometric functions, beta/Selberg integrals and $q$-analogues. Time permitting: orthogonal polynomials, Painleve transcendents and/or elliptic analogues. Part a (first term) and part b (second term) will not be offered 2021-2022.
Instructor: Katz.

APh/Ph 112 ab.
Noise and Stochastic Resonance.
9 units (3-0-6):
second, third terms.
Prerequisites: Ph 12 abc, ACM 95/100 ab and Ph 106 abc, equivalent background, or instructor's permission.
The presence of noise in experimental systems is often regarded as a nuisance since it diminishes the signal to noise ratio thereby obfuscating weak signals or patterns. From a theoretical perspective, noise is also problematic since its influence cannot be elicited from deterministic equations but requires stochastic-based modeling which incorporates various types of noise and correlation functions. In general, extraction of embedded information requires that a threshold be overcome in order to outweigh concealment by noise. However, even below threshold, it has been demonstrated in numerous systems that external forcing coupled with noise can actually boost very weak signatures beyond threshold by a phenomenon known as stochastic resonance. Although it was originally demonstrated in nonlinear systems, more recent studies have revealed this phenomenon can occur in linear systems subject, for example, to color-based noise. Techniques for optimizing stochastic resonance are now revolutionizing modeling and measurement theory in many fields ranging from nonlinear optics and electrical systems to condensed matter physics, neurophysiology, hydrodynamics, climate research and even finance. This course will be conducted in survey and seminar style and is expected to appeal to theorists and experimentalists alike. Review of the current literature will be complimented by background readings and lectures on statistical physics and stochastic processes as needed.
Instructor: Troian.

Ma 112 ab.
Statistics.
9 units (3-0-6):
second term.
Prerequisites: Ma 2 a probability and statistics or equivalent.
The first term covers general methods of testing hypotheses and constructing confidence sets, including regression analysis, analysis of variance, and nonparametric methods. The second term covers permutation methods and the bootstrap, point estimation, Bayes methods, and multistage sampling. Not offered 2021-22.

APh/Ph 115.
Physics of Momentum Transport in Hydrodynamic Systems.
9 units (3-0-6):
second term.
Prerequisites: ACM 95 or equivalent.
Contemporary research in many areas of physics requires some knowledge of the principles governing hydrodynamic phenomena such as nonlinear wave propagation, symmetry breaking in pattern forming systems, phase transitions in fluids, Langevin dynamics, micro- and optofluidic control, and biological transport at low Reynolds number. This course offers students of pure and applied physics a self-contained treatment of the fundamentals of momentum transport in hydrodynamic systems. Mathematical techniques will include formalized dimensional analysis and rescaling, asymptotic analysis to identify dominant force balances, similitude, self-similarity and perturbation analysis for examining unidirectional and Stokes flow, pulsatile flows, capillary phenomena, spreading films, oscillatory flows, and linearly unstable flows leading to pattern formation. Students must have working knowledge of vector calculus, ODEs, PDEs, complex variables and basic tensor analysis. Advanced solution methods will be taught in class as needed. Not offered 2021-2022.
Instructor: Troian.

APh/Ph/Ae 116.
Physics of Thermal and Mass Transport in Hydrodynamic Systems.
9 units (3-0-6):
third term.
Prerequisites: ACM 95 or equivalent and APh/Ph 115 or equivalent.
Contemporary research in many areas of physics requires some knowledge of how momentum transport in fluids couples to diffusive phenomena driven by thermal or concentration gradients. This course will first examine processes driven purely by diffusion and progress toward description of systems governed by steady and unsteady convection-diffusion and reaction-diffusion. Topics will include Fickian dynamics, thermal transfer in Peltier devices, Lifshitz-Slyozov growth during phase separation, thermocouple measurements of oscillatory fields, reaction-diffusion phenomena in biophysical systems, buoyancy driven flows, and boundary layer formation. Students must have working knowledge of vector calculus, ODEs, PDEs, complex variables and basic tensor analysis. Advanced solution methods such as singular perturbation, Sturm-Liouville and Green's function analysis will be taught in class as needed. Not offered 2021-2022.
Instructor: Troian.

Ma 116 abc.
Mathematical Logic and Axiomatic Set Theory.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 5 or equivalent, or instructor's permission.
First term: Introduction to first-order logic and model theory. The Godel Completeness Theorem and the Completeness Theorem. Definability, elementary equivalence, complete theories, categoricity. The Skolem-Lowenheim Theorems. The back and forth method and Ehrenfeucht-Fraisse games. Farisse theory. Elimination of quantifiers, applications to algebra and further related topics if time permits. Second and third terms: Axiomatic set theory, ordinals and cardinals, the Axiom of Choice and the Continuum Hypothesis. Models of set theory, independence and consistency results. Topics in descriptive set theory, combinatorial set theory and large cardinals.
Instructor: Vidnyanszky.

Ge/Ay 117.
Bayesian Statistics and Data Analysis.
9 units (3-0-6):
second term.
Prerequisites: CS1 or equivalent.
In modern fields of planetary science and astronomy, vast quantities of data are often available to researchers. The challenge is converting this information into meaningful knowledge about the universe. The primary focus of this course is the development of a broad and general tool set that can be applied to the student's own research. We will use case studies from the astrophysical and planetary science literature as our guide as we learn about common pitfalls, explore strategies for data analysis, understand how to select the best model for the task at hand, and learn the importance of properly quantifying and reporting the level of confidence in one's conclusions.
Instructor: Knutson.

Ma/CS 117 abc.
Computability Theory.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 5 or equivalent, or instructor's permission.
Various approaches to computability theory, e.g., Turing machines, recursive functions, Markov algorithms; proof of their equivalence. Church's thesis. Theory of computable functions and effectively enumerable sets. Decision problems. Undecidable problems: word problems for groups, solvability of Diophantine equations (Hilbert's 10th problem). Relations with mathematical logic and the Gödel incompleteness theorems. Decidable problems, from number theory, algebra, combinatorics, and logic. Complexity of decision procedures. Inherently complex problems of exponential and superexponential difficulty. Feasible (polynomial time) computations. Polynomial deterministic vs. nondeterministic algorithms, NP-complete problems and the P = NP question. Not offered 2021-22.

Ma 118.
Topics in Mathematical Logic: Geometrical Paradoxes.
9 units (3-0-6):
second term.
Prerequisites: Ma 5 or equivalent, or instructor's permission.
This course will provide an introduction to the striking paradoxes that challenge our geometrical intuition. Topics to be discussed include geometrical transformations, especially rigid motions; free groups; amenable groups; group actions; equidecomposability and invariant measures; Tarski's theorem; the role of the axiom of choice; old and new paradoxes, including the Banach-Tarski paradox, the Laczkovich paradox (solving the Tarski circle-squaring problem), and the Dougherty-Foreman paradox (the solution of the Marczewski problem). Not offered 2021-22.

Ph/APh/EE/BE 118 ab.
Physics of Measurement.
9 units (3-0-6):
second, third terms.
Prerequisites: Ph 127, APh 105, or equivalent, or permission from instructor.
This course explores the fundamental underpinnings of experimental measurements from the perspectives of coupling, responsivity, noise, backaction, and information. Its overarching goal is to enable students to develop intuition about, and to critically evaluate, a diversity of real measurement systems - and to provide a framework for estimating the ultimate and practical limits to information that can be extracted from them. Topics will include physical signal transduction and responsivity, fundamental noise processes, modulation, frequency conversion, synchronous detection, signal-sampling techniques, digitization, signal transforms, spectral analyses, and correlation methods. The first term will cover the essential fundamental underpinnings, while topics in second term will focus their application to high frequency, microwave, and fast time-domain measurements where distributed approaches become imperative. The second term (in alternate years) may focus on topics that include either measurements at the quantum limit, biosensing and biological interfaces, of functional brain imaging. Not offered 2021-22.
Instructor: Roukes.

Ay 119.
Astroinformatics.
6 units (3-0-3):
third term.
This class is an introduction to the data science skills from the applied computer science, statistics, and information technology, that are needed for a modern research in any data-intensive field, but with a special focus on the astronomical applications. Open to graduate and upper-division on undergraduate students in all options. The topics covered include design of data systems, regression techniques, supervised and unsupervised machine learning, databases, Bayesian statistics, high performance computing, software carpentry, deep learning, and visualization. The class will feature real-world examples from cutting-edge projects in which the instructors are involved.
Instructors: Djorgovski, Graham, Mahabal, Duev.

CS/Ph 120.
Quantum Cryptography.
9 units (3-0-6):
first term.
Prerequisites: Ma 1b, Ph 2b or Ph 12b, CS 21, CS 38 or equivalent recommended (or instructor's permission).
This course is an introduction to quantum cryptography: how to use quantum effects, such as quantum entanglement and uncertainty, to implement cryptographic tasks with levels of security that are impossible to achieve classically. The course covers the fundamental ideas of quantum information that form the basis for quantum cryptography, such as entanglement and quantifying quantum knowledge. We will introduce the security definition for quantum key distribution and see protocols and proofs of security for this task. We will also discuss the basics of device-independent quantum cryptography as well as other cryptographic tasks and protocols, such as bit commitment or position-based cryptography. Not offered 2021-2022

Ma 120 abc.
Abstract Algebra.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 5 or equivalent or instructor's permission.
This course will discuss advanced topics in algebra. Among them: an introduction to commutative algebra and homological algebra, infinite Galois theory, Kummer theory, Brauer groups, semisimiple algebras, Weddburn theorems, Jacobson radicals, representation theory of finite groups.
Instructors: Szumowicz, Zhu, Flach.

Ay 121.
Radiative Processes.
9 units (3-0-6):
first term.
Prerequisites: Ph106bc, Ph 125 or equivalent (undergraduates).
The interaction of radiation with matter: radiative transfer, emission, and absorption. Compton processes, coherent emission processes, synchrotron radiation, collisional excitation, spectroscopy of atoms and molecules.
Instructor: Ravi.

Ma 121 ab.
Combinatorial Analysis.
9 units (3-0-6):
first, third terms.
Prerequisites: Ma 5.
A survey of modern combinatorial mathematics, starting with an introduction to graph theory and extremal problems. Flows in networks with combinatorial applications. Counting, recursion, and generating functions. Theory of partitions. (0, 1)-matrices. Partially ordered sets. Latin squares, finite geometries, combinatorial designs, and codes. Algebraic graph theory, graph embedding, and coloring.
Instructors: Schülke, Conlon.

Ph 121 abc.
Computational Physics Lab.
6 units (0-6-0):
first, second, third terms.
Many of the recent advances in physics are attributed to progress in computational power. In the advanced computational lab, students will hone their computational skills bu working through projects inspired by junior level classes (such as classical mechanics and E, statistical mechanics, quantum mechanics and quantum many-body physics). This course will primarily be in Python and Mathematica. This course is offered pass/fail.
Instructors: Simmons-Duffin, Refael.

Ay 122 abc.
Astronomical Measurements and Instrumentation.
9 units (3-0-6):
first, second terms.
Prerequisites: Ph 106bc or equivalent.
Measurement and signal analysis techniques througout the electromagnetic spectrum. Courses may include lab work and field trips to Caltech observatories. Ay 122a concentrates on infrared, optical, and ultraviolet techniques: telescopes, optics, detectors, photometry, spectroscopy, active/adaptive optics, coronography. Imaging devices and image processing. Ay 122b concentrates on radio through submillimeter techniques: antennae, receivers, mixers, and amplifiers. Interferometers and aperture synthesis arrays. Signal analysis techniques and probability and statistics, as relevant to astronomical measurement. Ay 122c (not offered 2021-22) concentrates on X-ray through gamma-ray techniques.
Instructors: Martin, Howard, Hallinan, Ravi.

Ay 123.
Structure and Evolution of Stars.
9 units (3-0-6):
second term.
Prerequisites: Ay 101; Ph 125 or equivalent (undergraduates).
Thermodynamics, equation of state, convection, opacity, radiative transfer, stellar atmospheres, nuclear reactions, and stellar models. Evolution of low- and high-mass stars, supernovae, and binary stars.
Instructor: Fuller.

Ma 123.
Classification of Simple Lie Algebras.
9 units (3-0-6):
third term.
Prerequisites: Ma 5 or equivalent.
This course is an introduction to Lie algebras and the classification of the simple Lie algebras over the complex numbers. This will include Lie's theorem, Engel's theorem, the solvable radical, and the Cartan Killing trace form. The classification of simple Lie algebras proceeds in terms of the associated reflection groups and a classification of them in terms of their Dynkin diagrams. Not offered 2021-22.

Ay 124.
Structure and Evolution of Galaxies.
9 units (3-0-6):
second term.
Prerequisites: Ay 21; Ph 106 or equivalent (undergraduates).
Stellar dynamics and properties of galaxies; instabilities; spiral and barred galaxies; tidal dynamics and galaxy mergers; stellar composition, masses, kinematics, and structure of galaxies; galactic archeology; galactic star formation; feedback from stars and super-massive black holes; circum-galactic medium.
Instructor: Hopkins.

Ma 124.
Elliptic Curves.
9 units (3-0-6):
second term.
Prerequisites: Ma 5 or equivalent.
The ubiquitous elliptic curves will be analyzed from elementary, geometric, and arithmetic points of view. Possible topics are the group structure via the chord-and-tangent method, the Nagel-Lutz procedure for finding division points, Mordell's theorem on the finite generation of rational points, points over finite fields through a special case treated by Gauss, Lenstra's factoring algorithm, integral points. Other topics may include diophantine approximation and complex multiplication. Not offered 2021-22.

Ay 125.
High-Energy Astrophysics.
9 units (3-0-6):
third term.
Prerequisites: Ph 106 and Ph 125 or equivalent (undergraduates).
High-energy astrophysics, the final stages of stellar evolution; supernovae, binary stars, accretion disks, pulsars; extragalactic radio sources; active galactic nuclei; black holes.
Instructor: Phinney.

Ma 125.
Algebraic Curves.
9 units (3-0-6):
third term.
Prerequisites: Ma 5.
An elementary introduction to the theory of algebraic curves. Topics to be covered will include affine and projective curves, smoothness and singularities, function fields, linear series, and the Riemann-Roch theorem. Possible additional topics would include Riemann surfaces, branched coverings and monodromy, arithmetic questions, introduction to moduli of curves. Not offered 2021-22.

Ph 125 abc.
Quantum Mechanics.
9 units (4-0-5):
first, second, third terms.
Prerequisites: Ma 2 ab, Ph 12 abc or Ph 2 ab, or equivalents.
A one-year course in quantum mechanics and its applications, for students who have completed Ph 12 or Ph 2. Wave mechanics in 3-D, scattering theory, Hilbert spaces, matrix mechanics, angular momentum, symmetries, spin-1/2 systems, approximation methods, identical particles, and selected topics in atomic, solid-state, nuclear, and particle physics. Chen.
Instructors: Wise, Porter, Y.

Ay 126.
Interstellar and Intergalactic Medium.
9 units (3-0-6):
third term.
Prerequisites: Ay 102 (undergraduates).
Physical processes in the interstellar medium. Ionization, thermal and dynamic balance of interstellar medium, molecular clouds, hydrodynamics, magnetic fields, H II regions, supernova remnants, star formation, global structure of interstellar medium.
Instructor: Hillenbrand.

EE/Ma/CS 126 ab.
Information Theory.
9 units (3-0-6):
first, second terms.
Prerequisites: Ma 3.
Shannon's mathematical theory of communication, 1948-present. Entropy, relative entropy, and mutual information for discrete and continuous random variables. Shannon's source and channel coding theorems. Mathematical models for information sources and communication channels, including memoryless, Markov, ergodic, and Gaussian. Calculation of capacity and rate-distortion functions. Universal source codes. Side information in source coding and communications. Network information theory, including multiuser data compression, multiple access channels, broadcast channels, and multiterminal networks. Discussion of philosophical and practical implications of the theory. This course, when combined with EE 112, EE/Ma/CS/IDS 127, EE/CS 161, and EE/CS/IDS 167, should prepare the student for research in information theory, coding theory, wireless communications, and/or data compression. EE/Ma/CS 126a offered 2021-2022; EE/Ma/CS 126b not offered 2021-2022.
Instructor: Effros.

Ay 127.
Astrophysical Cosmology.
9 units (3-0-6):
first term.
Prerequisites: Ay 21; Ph 106 or equivalent (undergraduates).
Cosmology; extragalactic distance determinations; relativistic cosmological models; thermal history of the universe; nucleosynthesis; microwave background fluctuations; large-scale structure; inter-galactic medium; cosmological tests; galaxy formation and clustering.
Instructor: Martin.

EE/Ma/CS/IDS 127.
Error-Correcting Codes.
9 units (3-0-6):
third term.
Prerequisites: EE 55 or Ma 3.
This course develops from first principles the theory and practical implementation of the most important techniques for combating errors in digital transmission or storage systems. Topics include highly symmetric linear codes, such as Hamming, Reed-Muller, and Polar codes; algebraic block codes, e.g., BCH, Reed-Solomon (including a self-contained introduction to the theory of finite fields); and sparse graph codes with iterative decoding, i.e., LDPC code and turbo codes. Students will become acquainted with encoding and decoding algorithms, design principles and performance evaluation of codes.
Instructor: Kostina.

Ph 127 ab.
Statistical Physics of Interacting Systems, Phases, and Phase Transitions.
9 units (4-0-5):
first, second terms.
Prerequisites: Ph 12 c or equivalent; quantum mechanics at the level of Ph 125 ab is required for Ph 127 b; may be taken concurrently.
An advanced course in statistical physics that focuses on systems of interacting particles. Part a will cover interacting gases and spin models of magnetism, phase transitions and broken symmetries, classical field theories, and renormalization group approach to collective phenomena. Part b will introduce the path-integral based quantum to classical statistical mechanics mapping, as well as dualities and topological-defects descriptions, with applications to magnets, superfluids, and gauge field theories.
Instructor: Motrunich.

Ma 128.
Homological Algebra.
9 units (3-0-6):
second term.
Prerequisites: Math 120 abc or instructor's permission.
This course introduces standard concepts and techniques in homological algebra. Topics will include Abelian and additive categories; Chain complexes, homotopies and the homotopy category; Derived functors; Yoneda extension and its ring structure; Homological dimension and Koszul complexe; Spectral sequences; Triangulated categories, and the derived category. Not offered 2021-22.

Ph 129 abc.
Mathematical Methods of Physics.
9 units (4-0-5):
first, second terms.
Prerequisites: Ma 2 and Ph 2 abc, or equivalent.
Mathematical methods and their application in physics. First term focuses on group theoretic methods in physics. Second term includes analytic methods such as complex analysis, differential equations, integral equations and transforms, and other applications of real analysis. Third term covers probability and statistics in physics. Each part may be taken independently. Part c not offered 2021-2022. Chen, Chatziioannou.
Instructor: X.

Ma 130 abc.
Algebraic Geometry.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 120 (or Ma 5 plus additional reading).
Plane curves, rational functions, affine and projective varieties, products, local properties, birational maps, divisors, differentials, intersection numbers, schemes, sheaves, general varieties, vector bundles, coherent sheaves, curves and surfaces.
Instructors: Oswal, Yu.

Ge/Ay 132.
Atomic and Molecular Processes in Astronomy and Planetary Sciences.
9 units (3-0-6):
first term.
Prerequisites: instructor's permission.
Fundamental aspects of atomic and molecular spectra that enable one to infer physical conditions in astronomical, planetary, and terrestrial environments. Topics will include the structure and spectra of atoms, molecules, and solids; transition probabilities; photoionization and recombination; collisional processes; gas-phase chemical reactions; and isotopic fractionation. Each topic will be illustrated with applications in astronomy and planetary sciences, ranging from planetary atmospheres and dense interstellar clouds to the early universe. Given in alternate years; not offered 2021-2022.
Instructor: Blake.

Ma 132 abc.
Topics in Algebraic Geometry.
9 units (3-0-6):
.
Prerequisites: Ma 130 or instructor's permission.
This course will cover advanced topics in algebraic geometry that will vary from year to year. Topics will be listed on the math option website prior to the start of classes. Previous topics have included geometric invariant theory, moduli of curves, logarithmic geometry, Hodge theory, and toric varieties. This course can be repeated for credit. Not offered 2021-22.

Ge/Ay 133.
The Formation and Evolution of Planetary Systems.
9 units (3-0-6):
first term.
Review current theoretical ideas and observations pertaining to the formation and evolution of planetary systems. Topics to be covered include low-mass star formation, the protoplanetary disk, accretion and condensation in the solar nebula, the formation of gas giants, meteorites, the outer solar system, giant impacts, extrasolar planetary systems.
Instructor: Batygin.

Ma 135 ab.
Arithmetic Geometry.
9 units (3-0-6):
first term.
Prerequisites: Ma 130.
The course deals with aspects of algebraic geometry that have been found useful for number theoretic applications. Topics will be chosen from the following: general cohomology theories (étale cohomology, flat cohomology, motivic cohomology, or p-adic Hodge theory), curves and Abelian varieties over arithmetic schemes, moduli spaces, Diophantine geometry, algebraic cycles.
Instructor: Flach.

Ph 135.
Introduction to Condensed Matter.
9 units (3-0-6):
first term.
Prerequisites: Ph 125 ab or equivalent or instructor's permission.
This course is an introduction to condensed matter which covers electronic properties of solids, including band structures, and transport. In addition, the course will introduce topological band-structure effects, covering Berry phase, the Thouless pump, and topological insulators. Ph 135 is continued by Ph/APh 223 ab in the winter and spring terms.
Instructor: Refael.

EE/Ma/CS/IDS 136.
Topics in Information Theory.
9 units (3-0-6):
third term.
Prerequisites: Ma 3 or ACM/EE/IDS 116 or CMS 117 or Ma/ACM/IDS 140a.
This class introduces information measures such as entropy, information divergence, mutual information, information density from a probabilistic point of view, and discusses the relations of those quantities to problems in data compression and transmission, statistical inference, language modeling, game theory and control. Topics include information projection, data processing inequalities, sufficient statistics, hypothesis testing, single-shot approach in information theory, large deviations. Not offered 2021-2022.
Instructor: Kostina.

Ph 136 abc.
Applications of Classical Physics.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ph 106 ab or equivalent.
Applications of classical physics to topics of interest in contemporary "macroscopic'' physics. Continuum physics and classical field theory; elasticity and hydrodynamics; plasma physics; magnetohydrodynamics; thermodynamics and statistical mechanics; gravitation theory, including general relativity and cosmology; modern optics. Content will vary from year to year, depending on the instructor. An attempt will be made to organize the material so that the terms may be taken independently. Ph 136a will focus on thermodynamics, statistical mechanics, random processes, and optics. Ph136b will focus on fluid dynamics, MHD, turbulence, and plasma physics. Ph 136c will cover an introduction to general relativity. Offered in alternate years. Not offered 2021-22.

Ge/Ay 137.
Planetary Physics.
9 units (3-0-6):
second term.
Prerequisites: Ph 106 abc, ACM 95/100 ab.
A quantitative review of dynamical processes that characterize long-term evolution of planetary systems. An understanding of orbit-orbit resonances, spin-orbit resonances, secular exchange of angular momentum and the onset of chaos will be developed within the framework of Hamiltonian perturbation theory. Additionally, dissipative effects associated with tidal and planet-disk interactions will be considered.
Instructor: Batygin.

Ph/APh 137 abc.
Atoms and Photons.
9 units (3-0-6):
first, second terms.
Prerequisites: Ph 125 ab or equivalent, or instructor's permission.
This course will provide an introduction to the interaction of atomic systems with photons. The main emphasis is on laying the foundation for understanding current research that utilizes cold atoms and molecules as well as quantized light fields. First term: resonance phenomena, atomic/molecular structure, and the semi-classical interaction of atoms/molecules with static and oscillating electromagnetic fields. Techniques such as laser cooling/trapping, coherent manipulation and control of atomic systems. Second term: quantization of light fields, quantized light matter interaction, open system dynamics, entanglement, master equations, quantum jump formalism. Applications to cavity QED, optical lattices, and Rydberg arrays. Third term [not offered 2021-2022]: Topics in contemporary research. Possible areas include introduction to ultracold atoms, atomic clocks, searches for fundamental symmetry violations, synthetic quantum matter, and solid state quantum optics platforms. The emphasis will be on reading primary and contemporary literature to understand ongoing experiments.
Instructors: Hutzler, Endres.

APh/Ph 138 ab.
Quantum Hardware and Techniques.
9 units (3-0-6):
third term, a and b offered in alternating years.
Prerequisites: Ph 125abc or Ph 127ab or Ph137ab or instructor's permission.
This class covers multiple quantum technology platforms and related theoretical techniques, and will provide students with broad knowledge in quantum science and engineering. It will be split into three-week modules covering: applications of near-term quantum computers, superconducting qubits, trapped atoms and ions, topological quantum matter, solid state quantum bits, tensor-product states.
Instructors: Faraon, Minnich.

Ph 139.
Introduction to Elementary Particle Physics.
9 units (3-0-6):
second term.
Prerequisites: Ph 125 ab or equivalent, or instructor's permission.
This course provides an introduction to particle physics which includes Standard Model, Feynman diagrams, matrix elements, electroweak theory, QCD, gauge theories, the Higgs mechanism, neutrino mixing, astro-particle physics/cosmology, accelerators, experimental techniques, important historical and recent results, physics beyond the Standard Model, and major open questions in the field.
Instructor: Weinstein.

Ma/ACM/IDS 140 ab.
Probability.
9 units (3-0-6):
second, third terms.
Prerequisites: For 140 a, Ma 108 b is strongly recommended.
Overview of measure theory. Random walks and the Strong law of large numbers via the theory of martingales and Markov chains. Characteristic functions and the central limit theorem. Poisson process and Brownian motion. Topics in statistics.
Instructors: Vigneaux, Tamuz.

Ay 141 abc.
Research Conference in Astronomy.
3 units (1-0-2):
first, second, third terms.
Oral reports on current research in astronomy, providing students an opportunity for practice in the organization and presentation of technical material. A minimum of two presentations will be expected from each student each year. In addition, students are encouraged to participate in a public-level representation of the same material for posting to an outreach website. This course fulfills the option communication requirement and is required of all astronomy graduate students who have passed their preliminary exams. It is also recommended for astronomy seniors. Graded pass/fail.
Instructors: Hallinan, Phinney, Martin.

Ay 142.
Research in Astronomy and Astrophysics.
Units in accordance with work accomplished:
.
The student should consult a member of the department and have a definite program of research outlined. Approval by the student's adviser must be obtained before registering. 36 units of Ay 142 or Ay 143 required for candidacy for graduate students. Graded pass/fail.

Ma/ACM 142 ab.
Ordinary and Partial Differential Equations.
9 units (3-0-6):
second, third terms.
Prerequisites: Ma 108; Ma 109 is desirable.
The mathematical theory of ordinary and partial differential equations, including a discussion of elliptic regularity, maximal principles, solubility of equations. The method of characteristics.
Instructors: Angelopoulos, Isett.

Ay 143.
Reading and Independent Study.
Units in accordance with work accomplished:
.
The student should consult a member of the department and have a definite program of reading and independent study outlined. Approval by the student's adviser must be obtained before registering. 36 units of Ay 142 or Ay 143 required for candidacy for graduate students. Graded pass/fail.

Ay 144.
Independent Writing in Astronomy.
3 units (0-0-3):
offered every term.
Prerequisites: Ay 142.
This course is intended to be taken by students conducting minor study in the Ay option, subsequent to a term of Ay 142 (Research in Astronomy and Astrophysics), or by students who have completed a SURF with an astronomy faculty member and are writing it up for publication. Students should sign up in the section of the faculty member who supervised the research project. Course requirements are (at minimum) bi-weekly meetings with the research adviser and preparation of a 5-20 page write-up of the work in the style of one of the major journals, such as ApJ/AJ or Science/Nature. This course is required as part of the Ay minor.
Instructor: Staff.

Ma 145 abc.
Topics in Representation Theory.
9 units (3-0-6):
second term.
Prerequisites: Ma 5.
This course will discuss the study of representations of a group (or related algebra) by linear transformations of a vector space. Topics will vary from year to year, and may include modular representation theory (representations of finite groups in finite characteristic), complex representations of specific families of groups (esp. the symmetric group) and unitary representations (and structure theory) of compact groups. Not offered in 2021-22.

Ma 147 abc.
Dynamical Systems.
9 units (3-0-6):
first term.
Prerequisites: Ma 108, Ma 109, or equivalent.
First term: real dynamics and ergodic theory. Second term: Hamiltonian dynamics. Third term: complex dynamics. Part b and c not offered in 2021-2022.
Instructor: Wu.

Ma 148 ab.
Topics in Mathematical Physics.
9 units (3-0-6):
first term.
This course covers a range of topics in mathematical physics. The content will vary from year to year. Topics covered will include some of the following: Lagrangian and Hamiltonian formalism of classical mechanics; mathematical aspects of quantum mechanics: Schroedinger equation, spectral theory of unbounded operators, representation theoretic aspects; partial differential equations of mathematical physics (wave, heat, Maxwell, etc.); rigorous results in classical and/or quantum statistical mechanics; mathematical aspects of quantum field theory; general relativity for mathematicians. Geometric theory of quantum information and quantum entanglement based on information geometry and entropy. Part b not offered in 2021-22.
Instructor: Marcolli.

Ma 151 abc.
Algebraic and Differential Topology.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 109 abc or equivalent.
Part a: Homology Theory. Homology and calculation of homology groups, exact sequences, cohomology rings, Poincaré duality. Part b: Homotopy Theory and K-theory. Fibrations, higher homotopy groups, and exact sequences of fibrations. Fiber bundles, Eilenberg-MacLane spaces, classifying spaces. K-theory, generalized cohomology theory, Bott periodicity. Part c: Characteristic classes. Stiefel-Whitney classes, Chern classes, Pontryagin classes, cobordism theory, Chern-Weil theory.
Instructors: Mazel-Gee, Ni.

Ma 157 abc.
Riemannian Geometry.
9 units (3-0-6):
first, second terms.
Prerequisites: Ma 151 or equivalent, or instructor's permission.
Part a: basic Riemannian geometry: geometry of Riemannian manifolds, connections, curvature, Bianchi identities, completeness, geodesics, exponential map, Gauss's lemma, Jacobi fields, Lie groups, principal bundles, and characteristic classes. Part b: basic topics may vary from year to year and may include elements of Morse theory and the calculus of variations, locally symmetric spaces, special geometry, comparison theorems, relation between curvature and topology, metric functionals and flows, geometry in low dimensions. Part c not offered in 2021-22.
Instructor: Smillie.

Ge/Ay 159.
Astrobiology.
9 units (3-0-6):
second term.
We approach the age-old questions "Why are we here?" and "Are we alone?" by covering topics in cosmology, the origins of life, planetary habitability, the detection of biosignatures, the search for extraterrestrial intelligence, and humanity's future in space. Specific topics include: the emergence of life at hydrothermal vents; the habitable zone and the Gaia hypothesis; the search for ancient habitable environments on Mars; icy satellites like Europa, Enceladus, and Titan as astrobiological prospects; and the hunt for atmospheric biosignatures on exoplanets. Offered 2021-2022.
Instructor: Yung.

Ma 160 abc.
Number Theory.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 5.
In this course, the basic structures and results of algebraic number theory will be systematically introduced. Topics covered will include the theory of ideals/divisors in Dedekind domains, Dirichlet unit theorem and the class group, p-adic fields, ramification, Abelian extensions of local and global fields.
Instructors: Dunn, Burungale Radziwill.

Ma 162 ab.
Topics in Number Theory.
9 units (3-0-6):
first, second terms.
Prerequisites: Ma 160.
The course will discuss in detail some advanced topics in number theory, selected from the following: Galois representations, elliptic curves, modular forms, L-functions, special values, automorphic representations, p-adic theories, theta functions, regulators. Not offered 2021-22.

Ph 171.
Reading and Independent Study.
Units in accordance with work accomplished:
.
Occasionally, advanced work involving reading, special problems, or independent study is carried out under the supervision of an instructor. Approval of the instructor and of the student's departmental adviser must be obtained before registering. The instructor will complete a student evaluation at the end of the term. Graded pass/fail.

Ph 172.
Research in Physics.
Units in accordance with work accomplished:
.
Undergraduate students registering for 6 or more units of Ph 172 must provide a brief written summary of their work, not to exceed 3 pages, to the option rep at the end of the term. Approval of the student's research supervisor and departmental adviser must be obtained before registering. Graded pass/fail.

Ph 177.
Advanced Experimental Physics.
9 units (0-4-5):
second, third terms.
Prerequisites: Ph 6, Ph 106 a, Ph 125 a or equivalents.
A one-term laboratory course which will require students to design, assemble, calibrate, and use an apparatus to conduct a nontrivial experiment involving quantum optics or other current research area of physics. Students will work as part of a small team to reproduce the results of a published research paper. Each team will be guided by an instructor who will meet weekly with the students; the students are each expected to spend an average of 4 hours/week in the laboratory and the remainder for study and design. Enrollment is limited. Permission of the instructors required.
Instructors: Rice, Hutzler.

CNS/Bi/Ph/CS/NB 187.
Neural Computation.
9 units (3-0-6):
third term.
Prerequisites: introductory neuroscience (Bi 150 or equivalent); mathematical methods (Bi 195 or equivalent); scientific programming.
This course aims at a quantitative understanding of how the nervous system computes. The goal is to link phenomena across scales from membrane proteins to cells, circuits, brain systems, and behavior. We will learn how to formulate these connections in terms of mathematical models, how to test these models experimentally, and how to interpret experimental data quantitatively. The concepts will be developed with motivation from some of the fascinating phenomena of animal behavior, such as: aerobatic control of insect flight, precise localization of sounds, sensing of single photons, reliable navigation and homing, rapid decision-making during escape, one-shot learning, and large-capacity recognition memory.
Instructors: Meister, Rutishauser.

Ay 190.
Computational Astrophysics.
9 units (3-0-6):
second term.
Prerequisites: Ph 20-22 (undergraduates).
Introduction to essential numerical analysis and computational methods in astrophysics and astrophysical data analysis. Basic numerical methods and techniques; N-body simulations; fluid dynamics (SPH/grid-based); MHD; radiation transport; reaction networks; data analysis methods; numerical relativity. Not offered 2021-2022.

Ma 191 abc.
Selected Topics in Mathematics.
9 units (3-0-6):
first, second, third terms.
Each term we expect to give between 0 and 6 (most often 2-3) topics courses in advanced mathematics covering an area of current research interest. These courses will be given as sections of 191. Students may register for this course multiple times even for multiple sections in a single term. The topics and instructors for each term and course descriptions will be listed on the math option website each term prior to the start of registration for that term.
Instructors: Burungale, Katz, Kechris, Zhu, Szumowicz, Dunn, Hutchcroft, Karpukhin, Schülke.

Ay/Ge 198.
Special Topics in the Planetary Sciences.
6 units (2-0-4):
third term.
Topic for 2021-2022 is Extrasolar Planets. Thousands of planets have been identified in orbit around other stars. Astronomers are now embarking on understanding the statistics of extrasolar planet populations and characterizing individual systems in detail, namely star-planet, planet-planet and planet-disk dynamical interactions, physical parameters of planets and their composition, weather phenomena, etc. Direct and indirect detection techniques are now completing the big picture of extra-solar planetary systems in all of their natural diversity. The seminar-style course will review the state of the art in exoplanet science, take up case studies, detail current and future instrument needs, and anticipate findings.
Instructor: Howard.

Ph 198.
Special Topics in Physics.
Units in accordance with work accomplished:
.
Topics will vary year to year and may include hands-on laboratory work, team projects and a survey of modern physics research.
Instructor: Staff.

Ph 199.
Frontiers of Fundamental Physics.
9 units (3-0-6):
third term.
Prerequisites: Ph 125 ab, Ph 106 ab, or equivalent.
This course will explore the frontiers of research in particle physics and cosmology, focusing on the physics at the Large Hadron Collider. Topics include the Standard Model of particle physics in light of the discovery of the Higgs boson, work towards the characterization and measurements of the new particle's quantum properties, its implications on physics beyond the standard model, and its connection with the standard model of cosmology focusing on the dark matter challenge. The course is geared toward seniors and first-year graduate students who are not in particle physics, although students in particle physics are welcome to attend. Not offered 2021-2022.

### Please Note

The online version of the Caltech Catalog is provided as a convenience; however, the printed version is the only authoritative source of information about course offerings, option requirements, graduation requirements, and other important topics.