Along with a rigorous foundation the the core areas of study, mathematics majors can also take classes in related departments at Yale. These elective classes afford students the chance to explore the applications of mathematics to Computer Science, Physics, Chemistry, Economics, and Philosophy, among other areas. (Note: not all of these classes count toward the math major.)
Computer Science
CPSC 262 (Computational Tools for Data Science): An introduction to computational tools for data science. The analysis of data using regression, classification, clustering, principal component analysis, independent component analysis, dictionary learning, topic modeling, dimension reduction, and network analysis. Optimization by gradient methods and alternating minimization. The application of high performance computing and streaming algorithms to the analysis of large data sets. You can view the syllabus for the Fall 2017 course here.
CPSC 365 (Algorithms): The study of algorithm design and complexity theory is integral to theoretical computer science, and is directly related to discrete mathematics. You can view the syllabus for the Spring 2017 course here.
CPSC 366 (Intensive Algorithms): Similar to CPSC 365, Intensive Algorithms allows students who are comfortable with proofs and interested in taking a faster-paced, more in-depth course on algorithmic design. This course is taught by Daniel Spielman, recipient of the MacArthur Fellowship. You can view the syllabus for the Spring 2018 course here.
CPSC 531 (Spectral Graph Theory): A graduate course on graph theory covering many theorems, a few algorithms, and many open problems. From the first lecture in 2009, “this course is about the eigenvalues and eigenvectors of matrices associated with graphs, and their applications. I will never forget my amazement at learning that combinatorial properties of graphs could be revealed by an examination of the eigenvalues and eigenvectors of their associated matrices. I hope to both convey my amazement to you, but then to make it feel like common sense. I’m now shocked when any important property of a graph is not revealed by its eigenvalues and eigenvectors.” Taught by Daniel Spielman. You can find the syllabus for the 2015 course here, and view the lectures given here.
Physics
PHYS 401/402 (Advanced Classical Physics): Advanced physics as the field developed from the time of Newton to the age of Einstein. Topics include mechanics, electricity and magnetism, statistical physics, and thermodynamics. The development of classical physics into a “mature” scientific discipline, an idea that was subsequently shaken to the core by the revolutionary discoveries of quantum physics and relativity. You can view the syllabus for the Fall 2017 course of 401 here, and for the Spring 2018 course of 402 here.
PHYS 410 (Classical Mechanics): An advanced treatment of mechanics, with a focus on the methods of Lagrange and Hamilton. Lectures and problems address the mechanics of particles, systems of particles, and rigid bodies, as well as free and forced oscillations. Introduction to chaos and special relativity. You can find the syllabus for the Fall 2017 course here.
PHYS 440/441 (Quantum Mechanics I/II): The first term of a two-term sequence covering principles of quantum mechanics with examples of applications to atomic physics. The solution of bound-state eigenvalue problems, free scattering states, barrier penetration, the hydrogen-atom problem, perturbation theory, transition amplitudes, scattering, and approximation techniques. You can find the syllabi for the 2017-2018 courses here and here.
PHYS 420 (Thermodynamics and Statistical Mechanics): An introduction to the laws of thermodynamics and their theoretical explanation by statistical mechanics. Applications to gases, solids, phase equilibrium, chemical equilibrium, and boson and fermion systems. You can find the syllabus for the Fall 2017 course here.
PHYS 430 (Electromagnetic Fields and Optics): Electrostatics, magnetic fields of steady currents, electromagnetic waves, and relativistic dynamics. Provides a working knowledge of electrodynamics. You can find the syllabus for the Spring 2018 course here.
PHYS 460 (Mathematical Methods of Physics): Survey of mathematical techniques useful in physics. Physical examples illustrate vector and tensor analysis, group theory, complex analysis (residue calculus, method of steepest descent), differential equations and Green’s functions, and selected advanced topics.
Chemistry
CHEM 332/333 (Physical Chemistry I/II): A comprehensive survey of modern physical and theoretical chemistry, including topics drawn from thermodynamics, chemical equilibrium, electrochemistry, and kinetics. Second semester covers topics drawn from quantum mechanics, atomic/molecular structure, spectroscopy, and statistical thermodynamics. You can find syllabi for the Fall 2017 course of 332 here, and for the Spring 2018 course of 333 here.
Economics
ECON 125 (Microeconomic Theory): An intensive treatment of consumer and producer theory, covering additional topics including choice under uncertainty, game theory, contracting under hidden actions or hidden information, externalities and public goods, and general equilibrium theory. You can find the syllabus for the Fall 2017 course here.
ECON 126 (Macroeconomic Theory): An intensive treatment of the mathematical foundations of macroeconomic modeling, and with rigorous study of additional topics.
ECON 135/136 (Probability, Statistics, and Econometrics): Foundations of mathematical statistics: probability theory, distribution theory, parameter estimation, hypothesis testing, regression, and computer programming. Second semester focuses on econometric theory and practice: problems that arise from the specification, estimation, and interpretation of models of economic behavior. Topics include classical regression and simultaneous equations models; panel data; and limited dependent variables. You can find syllabi for the Fall 2017 course of 135 here, and for the Spring 2018 course of 136 here.
ECON 350/351 (Mathematical Economics: Equilibrium and Game Theories): An introduction to general equilibrium theory and its extension to equilibria involving uncertainty and time. Discussion of the economic role of insurance and of intertemporal models, namely, the overlapping generations model and the optimal growth theory model. Second semester is an introduction to game theory and choice under uncertainty. Analysis of the role of information and uncertainty for individual choice behavior, as well as application to the decision theory under uncertainty. Analysis of strategic interaction among economic agents, leading to the theory of auctions and mechanism design. You can find syllabi for the Fall 2017 course of 350 here, and for the Spring 2018 course of 351 here.
Philosophy
PHIL 115 (First-Order Logic): An introduction to formal logic. Study of the formal deductive systems and semantics for both propositional and predicate logic.
PHIL 267 (Mathematical Logic): An introduction to the metatheory of first-order logic, up to and including the completeness theorem for the first-order calculus. Introduction to the basic concepts of set theory. This course aims to introduce the important results of the metatheory of first-order logic, up through the basic results in model theory (completeness, compactness, and Löwenheim-Skolem-Tarski theorems). You can find the syllabus for the Fall 2017 course here.
PHIL 427 (Computability and Logic): A technical exposition of Gödel’s first and second incompleteness theorems and of some of their consequences in proof theory and model theory, such as Löb’s theorem, provability logic, and nonstandard models of arithmetic. The main goal of the course is to understand technical, historical, and philosophical aspects of Gödel’s first and second incompleteness theorems, Church’s undecidability theorem, and Tarski’s undefinability theorem. Starting with the historical background for these theorems, we will learn techniques for proving these theorems. After learning different ways to characterize effective computation; Turing machine computability, register machine computability, and recursive functions,we prove these three notions are equivalent to one another. Church’s thesis will be discussed. You can find the syllabus for the Spring 2018 course here.