# PMA Graduate Courses (2019-20)

Ay 101.
Physics of Stars.
9 units (3-0-6):
first term.
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: Fuller.

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 2019-20.

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

Ay/Ph 104.
Relativistic Astrophysics.
9 units (3-0-6):
third term.
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: Instructor: Kasliwal.

Ay 105.
Optical Astronomy Instrumentation Lab.
9 units (1-5-3):
third term.
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.
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.

Ph 106 abc.
Topics in Classical Physics.
9 units (4-0-5):
first, second, third terms.
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: Weinstein, Golwala, Hutzler.

Ma 108 abc.
Classical Analysis.
9 units (3-0-6):
first, second, third terms.
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: Lazebnik, Durcik, Makarov.

Ma 109 abc.
Introduction to Geometry and Topology.
9 units (3-0-6):
first, second, third terms.
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, Gekhtman, Chen.

Ma 110 abc.
Analysis.
9 units (3-0-6):
first, second, third terms.
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: Angelopoulos, Rains, Durcik.

Ay 111 a.
Introduction to Current Astrophysics Research.
3 units:
second 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: Kirby.

Ma 111 abc.
Topics in Analysis.
9 units (3-0-6):
first, second, third terms.
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
Instructors: Cuenca, Radziwill, Lazebnik.

Ma 112 ab.
Statistics.
9 units (3-0-6):
second term.
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 2019-20.

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

Ge/Ay 117.
Bayesian Statistics and Data Analysis.
9 units (3-0-6):
second term.
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.
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 2019-20.

Ma 118.
Topics in Mathematical Logic: Geometrical Paradoxes.
9 units (3-0-6):
second term.
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 2019-20.

Ph/APh/EE/BE 118 abc.
Physics of Measurement.
9 units (3-0-6):
first, third terms.
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. Part c not offered in 2019-20.
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-divisibe 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.

CS/Ph 120.
Quantum Cryptography.
9 units (3-0-6):
first term.
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.
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: Ramakrishnan, Zhu, Graber.

Ay 121.
Radiative Processes.
9 units (3-0-6):
first term.
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: Phinney.

Ma 121 ab.
Combinatorial Analysis.
9 units (3-0-6):
second, third terms.
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: Tyomkyn, 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, Motrunich.

Ay 122 abc.
Astronomical Measurements and Instrumentation.
9 units (3-0-6):
first, second terms.
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 2019-20) concentrates on X-ray through gamma-ray techniques.
Instructors: Howard, Martin, Hallinan, Ravi.

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

Ma 123.
Classification of Simple Lie Algebras.
9 units (3-0-6):
third term.
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 2019-20.

Ay 124.
Structure and Dynamics of Galaxies.
9 units (3-0-6):
second term.
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.

Ma 124.
Elliptic Curves.
9 units (3-0-6):
second term.
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 2019-20.

Ay 125.
High-Energy Astrophysics.
9 units (3-0-6):
third term.
High-energy astrophysics, the final stages of stellar evolution; supernovae, binary stars, accretion disks, pulsars; extragalactic radio sources; active galactic nuclei; black holes.
Instructors: Kasliwal, Kulkarni.

Ma 125.
Algebraic Curves.
9 units (3-0-6):
third term.
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 2019-20.

Ph 125 abc.
Quantum Mechanics.
9 units (4-0-5):
first, second, third terms.
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.
Instructor: Wise.

Ay 126.
Interstellar and Intergalactic Medium.
9 units (3-0-6):
third term.
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.
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.
Instructor: Effros.

Ay 127.
Cosmology and Galaxy Formation.
9 units (3-0-6):
second term.
Cosmology; extragalactic distance determinations; relativistic cosmological models; galaxy formation and clustering; thermal history of the universe, microwave background; nucleosynthesis; cosmological tests.
Instructors: Hopkins, Martin.

EE/Ma/CS/IDS 127.
Error-Correcting Codes.
9 units (3-0-6):
second term.
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 (4-0-5):
first, second, third terms.
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: Motrunich, Brandao.

Ma 128.
Homological Algebra.
9 units (3-0-6):
second term.
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.
Instructor: Zhu.

Ph 129 abc.
Mathematical Methods of Physics.
9 units (4-0-5):
first, second, third terms.
Mathematical methods and their application in physics. First term covers probability and statistics in physics. Second term focuses on group theoretic methods in physics. Third term includes analytic and numerical methods for solving differential equations, integral equations, and transforms, and other applications of real analysis. The three terms can be taken independently.
Instructors: Porter, Chen, Oguri.

Ma 130 abc.
Algebraic Geometry.
9 units (3-0-6):
first, second, third terms.
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: Kivinen, Xu, Campbell.

Ge/Ay 132.
Atomic and Molecular Processes in Astronomy and Planetary Sciences.
9 units (3-0-6):
first term.
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 2019-20.
Instructor: Blake.

Ma 132 abc.
Topics in Algebraic Geometry.
9 units (3-0-6):
.
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 2019-20.

Ge/Ay 133.
The Formation and Evolution of Planetary Systems.
9 units (3-0-6):
third 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.
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. Not offered 2019-20.

Ph 135.
Introduction to Condensed Matter.
9 units (3-0-6):
first term.
This course is an introduction to condensed matter which covers electronic properties of solids, including band structures, transport, and optical properties. Ph 135 a is continued by Ph 223 ab in second and third terms.
Instructor: Refael.

EE/Ma/CS/IDS 136.
Topics in Information Theory.
9 units (3-0-6):
third term.
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 2019-20.
Instructor: Kostina.

Ph 136 abc.
Applications of Classical Physics.
9 units (3-0-6):
first, second, third terms.
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 2019-20.

Ge/Ay 137.
Planetary Physics.
9 units (3-0-6):
second term.
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.
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 19-20]: 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):
second and third terms.
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: Chen, Endres, Faraon, Hsieh, Minnich, Painter.

Ph 139.
Introduction to High Energy Physics.
9 units (3-0-6):
third term.
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: Patterson.

Ma/ACM/IDS 140 ab.
Probability.
9 units (3-0-6):
second, third terms.
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: Tamuz, Ouimet.

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: Mawet, Kasliwal, Fuller.

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.
Ordinary and Partial Differential Equations.
9 units (3-0-6):
third term.
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: 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.

Ma 145 abc.
Topics in Representation Theory.
9 units (3-0-6):
second, third terms.
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. Part a not offered in 2019-20.
Instructors: Kivinen, Cuenca.

Ma 147 abc.
Dynamical Systems.
9 units (3-0-6):
first, second terms.
First term: real dynamics and ergodic theory. Second term: Hamiltonian dynamics. Third term: complex dynamics. Part c not offered in 2019-20.
Instructors: Radziwill, Lazebnik.

Ma 148 ab.
Topics in Mathematical Physics.
9 units (3-0-6):
second and third terms.
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.
Instructors: Parikh, Makarov.

Ma 151 abc.
Algebraic and Differential Topology.
9 units (3-0-6):
first, second, third terms.
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, Ni, Chen.

Ma 157 abc.
Riemannian Geometry.
9 units (3-0-6):
second, third terms.
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 2019-20.
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. Given in alternate years; not offered 2019-20.
Instructor: Yung.

Ma 160 abc.
Number Theory.
9 units (3-0-6):
first, second terms.
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. Part c not offered in 2019-20.
Instructors: Burungale, Campbell.

Ma 162 ab.
Topics in Number Theory.
9 units (3-0-6):
first, second term.
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.
Instructors: Frank, Burungale.

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.
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):
first term.
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. Not offered 2019-20.
Instructor: Perona.

Ay 190.
Computational Astrophysics.
9 units (3-0-6):
second term.
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 2019-20.

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: Chen, Kechris, Smillie, Zhu, Durcik, Shikhelman, Angelopoulos, Frank, Wang, Tyomkyn, Xu, Isett, Demirel-Frank.

Ay/Ge 198.
Special Topics in the Planetary Sciences.
9 units (3-0-6):
third term.
Topic for 2019-20 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.
Instructors: Howard, Mawet.

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.
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 2019-20.

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.
Instructor: Endres.

Ph 205 abc.
Relativistic Quantum Field Theory.
9 units (3-0-6):
first, second, third terms.
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.
Instructors: Gukov, Kapustin.

Ay 211.
Contemporary Extragalactic Astronomy.
9 units (3-0-6):
third term.
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 2019-20.

Ay 215.
Seminar in Theoretical Astrophysics.
9 units (3-0-6):
second term.
Course for graduate students and seniors in astronomy. Topic for 2019-20 will be compact binaries containing white dwarfs, neutron stars and black holes. Formation, mass transfer, accretion, X-ray and pulsar binaries, magnetic and wind interactions, mergers, gravitational waves. Students will be required to lead some discussions; homework will consist exclusively of reading and working through selected papers in preparation for discussions.
Instructors: Fuller, Phinney.

Ph 217.
Introduction to the Standard Model.
9 units (3-0-6):
first term.
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 2019-20.

Ay 218.
Extrasolar Planets.
9 units (3-0-6):
third term.
Not offered 2019-20.

Ay 219.
Elements in the Universe and Galactic Chemical Evolution.
9 units (3-0-6):
second term.
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 2019-20.

Ph/CS 219 abc.
Quantum Computation.
9 units (3-0-6):
first, second terms.
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. Part c not offered in 2019-20.
Instructors: Preskill, Brandao.

Ph/APh 223 ab.
Advanced Condensed-Matter Physics.
9 units (3-0-6):
second, third terms.
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, Kitaev.

Ph 229 ab.
Advanced Mathematical Methods of Physics.
9 units (3-0-6):
first term.
A course on conformal field theory and the conformal bootstrap. Students should have some background in quantum field theory. Topics will include the renormalization group, phase transitions, universality, scale vs. conformal invariance, conformal symmetry, operator product expansion, state-operator correspondence, conformal blocks, the bootstrap equations, bootstrap in d=2 dimensions, numerical bootstrap methods in d>2, analytical bootstrap methods, introduction to AdS/CFT. Possible additional topics (time permitting) include superconformal field theories, entanglement entropy, monotonicity theorems, and conformal perturbation theory.
Instructor: Kapustin.

Ph 230 ab.
Elementary Particle Theory.
9 units (3-0-6):
first term.
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. Only offered in fall quarter in the 2019-20 academic year
Instructor: Zurek.

Ph 236 abc.
General Relativity.
9 units (3-0-6):
first, second terms.
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. Offered in alternate years. Part c not offered in 2019-20.
Instructors: Chen, Teukolsky.

Ph 237.
Gravitational Radiation.
9 units (3-0-6):
third term.
Special topics in Gravitational-wave Detection. Physics of interferometers, limits of measurement, coherent quantum feedback, noise, data analysis.
Instructor: Adhikari.

Ph 242 ab.
Physics Seminar.
4 units (2-0-2):
first, second terms.
An introduction to independent research, including training in relevant professional skills and discussion of current Caltech research areas with Caltech faculty, postdocs, and students. One meeting per week plus student projects. Registration restricted to first-year graduate students in physics.
Instructor: Patterson.

Ph 250 abc.
Introduction to String Theory.
9 units (3-0-6):
first, second, third terms.
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. Not offered 2019-20.

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 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:
.