# PMA Graduate Courses (2016-17)

Ay 101.
Physics of Stars.
11 units (3-2-6):
second 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: Kirby.

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.
Instructor: Phinney.

Ay 102.
Physics of the Interstellar Medium.
9 units (3-0-6):
third 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: Hillenbrand.

Ph 103.
Atomic and Molecular Spectroscopy.
9 units (3-0-6):
second term.
Prerequisites: instructor's permission.
This course will review the basic spectroscopy of atoms and molecules, with applications to astrophysics, the terrestrial atmosphere, and the laboratory. Species to be discussed include hydrogen and simple multielectron atoms such as carbon, diatomic and polyatomic molecules, and some solids. Mechanisms and effects determining linewidths and lineshapes will be discussed for laboratory, atmospheric, and astrophysical conditions. Not offered 2016-17.

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. Not Offered 2016-2017

Ay 105.
Optical Astronomy Instrumentation Lab.
10 units (1-5-4):
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: Ph 1 abc, Ph 3, or equivalents (Ph 8 may be substituted for Ph 3).
A laboratory course focusing on practical electronic circuits, with emphasis on analog electronics. The following topics are studied: RC circuits, electrical oscillations, operational amplifiers, diodes and transistors, combining circuit elements, and computer data acquisition. The course culminates in a two-week project of the student's choosing.
Instructors: Rice, Libbrecht.

Ma 106.
Elliptic Curves.
9 units (3-0-6):
third 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 2016-17.

Ph 106 abc.
Topics in Classical Physics.
9 units (3-0-6):
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. Roughly half of the year will be devoted to mechanics, and half to electromagnetism. Topics include Lagrangian and Hamiltonian formulations of mechanics, small oscillations and normal modes, boundary-value problems, multipole expansions, and various applications of electromagnetic theory.
Instructors: Weinstein, Golwala.

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: Demirel, Schimmer, Fathizadeh.

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: Markovic, Zhang, Vafaee.

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: Makarov, Markovic, Katz.

Ay 111 a.
Introduction to Current Astrophysics Research.
3 units:
first term.
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.
Instructor: Phinney.

Ma 111 a.
Topics in Analysis.
9 units (3-0-6):
first 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, Painlev\'e transcendents and/or elliptic analogues
Instructor: Schimmer.

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 2016-17.

APh/Ph 115.
Physics of Momentum Transport in Hydrodynamic Systems.
12 units (3-0-9):
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.
Instructor: Troian.

APh/Ph/Ae 116.
Physics of Thermal and Mass Transport in Hydrodynamic Systems.
12 units (3-0-9):
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.
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. Not offered 2016-17.

Ge/Ay 117.
Statistics and Data Analysis.
9 units (3-0-6):
second term.
Prerequisites: CS 1 and instructors permission..
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.
Instructors: Kechris, Lupini.

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 2016-17.

Ph/APh/EE/BE 118 abc.
Physics of Measurement.
9 units (3-0-6):
first and second terms.
Prerequisites: Ph127, APh 105, or equivalent, or permission from instructor.
This course focuses on exploring the fundamental underpinnings of experimental measurements from the perspectives of responsivity, noise, backaction, and information. Its overarching goal is to enable students to critically evaluate real measurement systems, and to determine the ultimate fundamental 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 correlations. The first term will cover the essential fundamental underpinnings, while topics in second term will include examples from optical methods, high-frequency and fast temporal measurements, biological interfaces, signal transduction, biosensing, and measurements at the quantum limit.
Instructor: Roukes.

Ay 119.
Methods of Computational Science.
9 units (3-0-6):
third term.
Open to graduate and upper-division undergraduate students in all options. Practical computational science methods useful in disciplines dealing with large and/or complex data sets. Topics include: Scientific databases and archives; data mining and exploration; data visualization techniques; practical techniques for physical modeling, including numerical and stochastic models; data sharing over networks, Web services, computational and data grids; design and understanding of scientific computational systems and experiments, and good software practices. Not Offered 2016-2017

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.
Instructor: Vidick.

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: Graber, Rains, 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: Kirby.

Ma 121 a.
Combinatorial Analysis.
9 units (3-0-6):
third term.
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.
Instructor: Rains.

Ay 122 abc.
Astronomical Measurements and Instrumentation.
9 units (3-0-6):
first term (a), second term (b).
Prerequisites: Ph 106bc or equivalent.
Measurement and signal analysis techniques throughout 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 2016-17) concentrates on X-ray through gamma-ray techniques.
Instructors: (a) Kasliwal, Mawet, (b) Hallinan, Kulkarni.

Ay 123.
Structure and Evolution of Stars.
9 units (3-0-6):
first 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: Hillenbrand.

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 2016-2017.

Ay 124.
Structure and Dynamics of Galaxies.
9 units (3-0-6):
second term.
Prerequisites: Ay 21; Ph 106 or equivalent (undergraduates).
Stellar dynamics and properties of galaxies; kinematics and dynamics of our galaxy; spiral structure; stellar composition, masses, and rotation of external galaxies; star clusters; galactic evolution; binaries, groups, and clusters of galaxies.
Instructor: Hopkins.

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: Kasliwa.

Ma 125.
Algebraic Curves.
8 units (3-0-6):
.
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 2016-17.

Ph 125 abc.
Quantum Mechanics.
9 units (3-0-6):
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.
Instructors: Brandao, Cheung.

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: Kulkarni.

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 127, EE 161, EE 167, and/or EE226 should prepare the student for research in information theory, coding theory, wireless communications, and/or data compression.
Instructor: Effros.

Ay 127.
Cosmology and Galaxy Formation.
9 units (3-0-6):
second term.
Prerequisites: Ay 21; Ph 106 or equivalent (undergraduates).
Cosmology; extragalactic distance determinations; relativistic cosmological models; galaxy formation and clustering; thermal history of the universe, microwave background; nucleosynthesis; cosmological tests.
Instructor: Staff.

EE/Ma/CS 127.
Error-Correcting Codes.
9 units (3-0-6):
second term.
Prerequisites: Ma 2.
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 algebraic block codes, e.g., Hamming, BCH, Reed-Solomon (including a self-contained introduction to the theory of finite fields); and the modern theory of sparse graph codes with iterative decoding, e.g. LDPC codes, turbo codes. The students will become acquainted with encoding and decoding algorithms, design principles and performance evaluation of codes.
Instructor: Kostina.

Ph 127 abc.
Statistical Physics.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ph 12 c or equivalent, and a basic understanding of quantum and classical mechanics.
A course in the fundamental ideas and applications of classical and quantum statistical mechanics. Topics to be covered include the statistical basis of thermodynamics; ideal classical and quantum gases (Bose and Fermi); lattice vibrations and phonons; weak interaction expansions; phase transitions; and fluctuations and dynamics.
Instructors: Refael, Motrunich.

CS/EE/Ma 129 abc.
Information and Complexity.
9 units (3-0-6), first and second terms:
(1-4-4) third term.
Prerequisites: basic knowledge of probability and discrete mathematics.
A basic course in information theory and computational complexity with emphasis on fundamental concepts and tools that equip the student for research and provide a foundation for pattern recognition and learning theory. First term: what information is and what computation is; entropy, source coding, Turing machines, uncomputability. Second term: topics in information and complexity; Kolmogorov complexity, channel coding, circuit complexity, NP-completeness. Third term: theoretical and experimental projects on current research topics. Not offered 2016-17.

Ph 129 abc.
Mathematical Methods of Physics.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ph 106 abc and ACM 95/100 ab or Ma 108 abc, or equivalents.
Mathematical methods and their application in physics. First term includes analytic and numerical methods for solving differential equations, integral equations, and transforms, and other applications of real analysis. Second term covers probability and statistics in physics. Third term focuses on group theoretic methods in physics. The three terms can be taken independently.
Instructors: Porter, Chen.

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, will vary from year to year. This year, the topic will be deformation theory. Not offered 2016-17.

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; offered 2016-17.
Instructor: Blake.

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.
Instructors: Knutson, Batygin.

Ma 135 ab.
Arithmetic Geometry.
9 units (3-0-6):
first, second terms.
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 (etale cohomology, flat cohomology, motivic cohomology, or p-adic Hodge theory), curves and Abelian varieties over arithmetic schemes, moduli spaces, Diophantine geometry, algebraic cycles. Not offered 2016-17.

Ph 135 abc.
Applications of Quantum Mechanics.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ph 125 abc or equivalent.
Applications of quantum mechanics to topics in contemporary physics. First term: introduction to condensed matter which covers electronic properties of solids, including band structures, transport, and optical properties. Ph 135a is continued by Ph 223 ab in second and third terms. Second term: 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. Third term: an overview of modern Quantum Optics with particular emphasis on quantum measurement science, the quantum-classical interface, quantum networks, and quantum many-body physics with atoms and photons. The course will concentrate on the essential roles of manifestly quantum (i.e., nonclassical) and entangled states of light and matter. The course covers theoretical tools for analyses of coherent light-matter interactions including the quantum master equation, and will combine examples on both theory and experiment from the current research literature. This is a one-term class aimed at advanced undergraduates as well as beginning graduate students. Terms may be taken independently.
Instructors: Yeh, Endres, Patterson.

Ph 136 abc.
Applications of Classical Physics.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ph 106 abc 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.
Instructors: Hopkins, Phinney, Vallisneri.

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.

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.
Instructor: (a) Mawet/Phinney; (b) Kirby/Sargent; (c) Hallinan/Hillenbran.

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 a.
Ordinary and Partial Differential Equations.
9 units (3-0-2):
second term.
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.
Instructor: Zhou.

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.

Ma/ACM 144 a.
Probability.
9 units (3-0-6):
first, term.
Prerequisites: For 144a, Ma 108b is strongly recommended; for 144b, 108b and 144a are prerequisite.
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.
Instructor: Tamuz.

Ma 145 a.
Representation Theory.
9 units (3-0-6):
third term.
Prerequisites: For 145a Ma 5 is a prerequisite; for 145b, Ma 108ab are strongly recommended.
The study of representations of a group by unitary operators on a Hilbert space, including finite and compact groups, and, to the extent that time allows, other groups. First term: general representation theory of finite groups. Frobenius's theory of representations of semidirect products. The Young tableaux and the representations of symmetric groups. Second term: the Peter-Weyl theorem. The classical compact groups and their representation theory. Weyl character formula.
Instructor: Din.

Ma 147 abc.
Dynamical Systems.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 108, Ma 109, or equivalent.
First term: real dynamics and ergodic theory. Second term: Hamiltonian dynamics. Third term: complex dynamics.
Instructors: Makaro, Ivrii.

Ma 148 a.
Topics in Mathematical Physics.
9 units (3-0-6):
third 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. First term: geometric theory of quantum information and quantum entanglement based on information geometry and entropy.
Instructor: Marcolli.

Ma 151 abc.
Algebraic and Differential Topology.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 109 abc or equivalent.
A basic graduate core course. Fundamental groups and covering spaces, homology and calculation of homology groups, exact sequences. Fibrations, higher homotopy groups, and exact sequences of fibrations. Bundles, Eilenberg-Maclane spaces, classifying spaces. Structure of differentiable manifolds, transversality, degree theory, De Rham cohomology, spectral sequences.
Instructors: Markovic, Vafaee, Ni.

Ma 157 abc.
Riemannian Geometry.
9 units (3-0-6):
first, second, third 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.
Instructors: Zhang, Li.

Ge/Ay 159.
Planetary Evolution and Habitability.
9 units (3-0-6):
second term.
Photochemistry of planetary atmospheres, comparative planetology, atmospheric evolution. What makes Earth habitable? Remote sensing of extrasolar planets, biosignatures. Given in alternate years; offered 2016-17.
Instructor: Yung.

Ma 160 ab.
Number Theory.
9 units (3-0-6):
first, second 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: Flach, Zavosh.

Ma 162 ab.
Topics in Number Theory.
9 units (3-0-6):
second, third 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 2016-17.

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 Experimental Physics.
Units in accordance with work accomplished:
.
Students registering for 6 or more units of Ph 172 must give a 15-minute oral presentation to the Physics Undergraduate Committee at the Physics Undergraduate Research Seminar Day. Approval of the student's research supervisor and departmental adviser must be obtained before registering. Graded pass/fail.

Ph 173.
Research in Theoretical Physics.
Units in accordance with work accomplished:
.
Students registered for 6 or more units of Ph 173 must give a 15-minute oral presentation to the Physics Undergraduate Committee at the Physics Undergraduate Research Seminar Day. Approval of the student's research supervisor and departmental adviser must be obtained before registering. Graded pass/fail.

CNS/Bi/Ph/CS/NB 187.
Neural Computation.
9 units (3-0-6):
first term.
Prerequisites: familiarity with digital circuits, probability theory, linear algebra, and differential equations. Programming will be required.
This course investigates computation by neurons. Of primary concern are models of neural computation and their neurological substrate, as well as the physics of collective computation. Thus, neurobiology is used as a motivating factor to introduce the relevant algorithms. Topics include rate-code neural networks, their differential equations, and equivalent circuits; stochastic models and their energy functions; associative memory; supervised and unsupervised learning; development; spike-based computing; single-cell computation; error and noise tolerance.
Instructor: Perona.

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 2016-17.

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.
Instructor: Staff.

Ay/Ge 198.
Special Topics in the Planetary Sciences.
9 units (3-0-6):
third term.
Topic for 2015-16 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. Not Offered 2016-17.

Ph 199.
Frontiers of Fundamental Physics.
9 units (3-0-6):
third term.
Prerequisites: Ph 125 abc, Ph 106 abc, 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 2016-2017.

Ph 201.
Candidacy Physics Fitness.
9 units (3-0-6):
third term.
The course will review problem solving techniques and physics applications from the undergraduate physics college curriculum. In particular, we will touch on the main topics covered in the written candidacy exam: classical mechanics, electromagnetism, statistical mechanics and quantum physics, optics, basic mathematical methods of physics, and the physical origin of everyday phenomena.
Instructors: Refael, Endres.

Ph 205 abc.
Relativistic Quantum Field Theory.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ph 125.
Topics: the Dirac equation, second quantization, quantum electrodynamics, scattering theory, Feynman diagrams, non-Abelian gauge theories, Higgs symmetry-breaking, the Weinberg-Salam model, and renormalization.
Instructor: Wise.

Ay 211.
Contemporary Extragalactic Astronomy.
9 units (3-0-6):
third term.
Prerequisites: Ay 123, Ay 124, and Ay 127.
Topics in extragalactic astronomy and cosmology, including observational probes of dark matter and dark energy; cosmological backgrounds and primordial element abundances; galaxy formation and evolution, including assembly histories, feedback and environmental effects; physics of the intergalactic medium; the role of active galactic nuclei; galactic structure and stellar populations; future facilities and their likely impact in the field. Not offered 2016-17.

SS/Ma 214.
Mathematical Finance.
9 units (3-0-6):
second term.
A course on pricing financial derivatives, risk management, and optimal portfolio selection using mathematical models. Students will be introduced to methods of Stochastic, Ito Calculus for models driven by Brownian motion. Models with jumps will also be discussed.
Instructor: Cvitanic.

Ay 215.
Seminar in Theoretical Astrophysics.
9 units (3-0-6):
second term.
Course for graduate students and seniors in astronomy and planetary science. Students will be required to lead some discussions. Topic will be selected based on student interest. Not offered 2016-17.

Ph 217.
Introduction to the Standard Model.
9 units (3-0-6):
first term.
Prerequisites: Ph 205 abc and Ph 236 abc, or equivalent.
An introduction to elementary particle physics and cosmology. Students should have at least some background in quantum field theory and general relativity. The standard model of weak and strong interactions is developed, along with predictions for Higgs physics and flavor physics. Some conjectures for physics beyond the standard model are introduced: for example, low-energy supersymmetry and warped extra dimensions. Not offered 2016-17.

Ay 218.
Extrasolar Planets.
9 units (3-0-6):
third term.
Not offered 2016-17.

Ay 219.
Elements in the Universe and Galactic Chemical Evolution.
9 units (3-0-6):
second term.
Prerequisites: Ay 121, 123, 124, 126.
Survey of the formation of the elements in the universe as a function of cosmic time. Review of the determination of abundances in stars, meteorites, H II regions, and in interstellar and intergalactic gas. Overview of models of galactic chemical evolution. Participants will measure elemental abundances from the Keck spectrum of a star and construct their own numerical chemical evolution models. Not offered 2016-17.

Ph/CS 219 abc.
Quantum Computation.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ph 129 abc or equivalent.
The theory of quantum information and quantum computation. Overview of classical information theory, compression of quantum information, transmission of quantum information through noisy channels, quantum error-correcting codes, quantum cryptography and teleportation. Overview of classical complexity theory, quantum complexity, efficient quantum algorithms, fault-tolerant quantum computation, physical implementations of quantum computation.
Instructors: Kitaev, Preskill.

Ph/APh 223 ab.
Advanced Condensed-Matter Physics.
9 units (3-0-6):
second, third terms.
Prerequisites: Ph 125 or equivalent, or instructor's permission.
Advanced topics in condensed-matter physics, with emphasis on the effects of interactions, symmetry, and topology in many-body systems. Ph/Aph 223a covers second quantization, Hartree-Fock theory of the electron gas, Mott insulators and quantum magnetism, bosonization, quantum Hall effects, and symmetry protected topological phases such as topological insulators. Ph/APh 223b will continue with BCS theory of superconductivity, Ginzburg-Landau theory, elements of unconventional and topological superconductors, theory of superfluidity, Bose-Hubbard model and bosonic Mott insulators, and some aspects of quantum systems with randomness.
Instructors: Alicea, Chen.

Ph 229 abc.
Advanced Mathematical Methods of Physics.
9 units (3-0-6):
first, second terms only.
Prerequisites: Ph 129 abc or equivalent.
Advanced topics in geometry and topology that are widely used in modern theoretical physics. Emphasis will be on understanding and applications more than on rigor and proofs. First term will cover basic concepts in topology and manifold theory. Second term will include Riemannian geometry, fiber bundles, characteristic classes, and index theorems. Third term will include anomalies in gauge-field theories and the theory of Riemann surfaces, with emphasis on applications to string theory.
Instructor: Ooguri.

Ph 230 abc.
Elementary Particle Theory.
9 units (3-0-6):
second, third terms.
Prerequisites: Ph 205 abc or equivalent.
Advanced methods in quantum field theory. First term: introduction to supersymmetry, including the minimal supersymmetric extension of the standard model, supersymmetric grand unified theories, extended supersymmetry, supergravity, and supersymmetric theories in higher dimensions. Second and third terms: nonperturbative phenomena in non-Abelian gauge field theories, including quark confinement, chiral symmetry breaking, anomalies, instantons, the 1/N expansion, lattice gauge theories, and topological solitons. Not offered 2016-17.

Ph 236 abc.
Relativity.
9 units (3-0-6):
first, second terms.
Prerequisites: a mastery of special relativity at the level of Goldstein's.
Classical Mechanics , or of Jackson's Classical Electrodynamics. A systematic exposition of Einstein's general theory of relativity and its applications to gravitational waves, black holes, relativistic stars, causal structure of space-time, cosmology and brane worlds. Not offered 2016-2017.

Ph 237.
Gravitational Waves.
9 units (3-0-6):
third term.
Prerequisites: Ph 236 a.
The theory and astrophysical phenomenology of gravitational-wave sources (black holes, neutron stars, compact binaries, early-universe phenomena, etc.). Gravitational-wave detectors (LIGO, LISA, and others), and data analysis. Not offered 2016-17.

Ph 242 ab.
Physics Seminar.
3 units (2-0-1):
first, second terms.
Topics in physics emphasizing current research at Caltech. One two-hour meeting per week. Speakers will be chosen from both faculty and students. Registration restricted to first-year graduate students in physics; exceptions only with permission of instructor. Graded pass/fail.
Instructors: Stone, Refael.

Ph 250 abc.
Introduction to String Theory.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ph 205 or equivalent.
The first two terms will focus largely on the bosonic string. Topics covered will include conformal invariance and construction of string scattering amplitudes, the origins of gauge interactions and gravity from string theory, T-duality, and D-branes. The third term will cover perturbative aspects of superstrings, supergravity, various BPS branes, and string dualities.
Instructors: Gukov, Kapustin.

Ma 290.
Reading.
Hours and units by arrangement:
.
Occasionally, advanced work is given through a reading course under the direction of an instructor.

Ph 300.
Thesis Research.
Units in accordance with work accomplished:
.
Ph 300 is elected in place of Ph 172 or Ph 173 when the student has progressed to the point where research leads directly toward the thesis for the degree of Doctor of Philosophy. Approval of the student's research supervisor and department adviser or registration representative must be obtained before registering. Graded pass/fail.

Ma 390.
Research.
Units by arrangement:
.

### 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.