## Mathematics

- APC 199/MAT 199: Math AliveMathematics has profoundly changed our world, from the way we communicate with each other and listen to music, to banking and computers. This course is designed for those without college mathematics who want to understand the mathematical concepts behind important modern applications. The course consists of individual modules, each focusing on a particular application (e.g. compression, animation and using statistics to explain, or hide, facts). The emphasis is on ideas, not on sophisticated mathematical techniques, but there will be substantial problem-set requirements. Students will learn by doing simple examples.
- APC 350/MAT 322: Introduction to Differential EquationsThis course will introduce the basic theory, models and techniques for ordinary and partial differential equations. Emphasis will be placed on the connection with other disciplines of science and engineering. We will try to strike a balance between the theoretical (e.g. existence and uniqueness issues, qualitative properties) and the more practical issues such as analytical and numerical approximations.
- MAE 305/MAT 391/EGR 305/CBE 305: Mathematics in Engineering IA treatment of the theory and applications of ordinary differential equations with an introduction to partial differential equations. The objective is to provide the student with an ability to solve problems in this field.
- MAE 306/MAT 392: Mathematics in Engineering IIThis course covers a range of fundamental mathematical techniques and methods that can be employed to solve problems in contemporary engineering and the applied sciences. Topics include complex analysis and integral transforms.
- MAT 100: Calculus FoundationsIntroduction to limits and derivatives as preparation for further courses in calculus. Fundamental functions (polynomials, rational functions, exponential, logarithmic, trigonometric) and their graphs will be also reviewed. Other topics include tangent and normal lines, linearization, computing area and rates of change. The emphasis will be on learning to think independently and creatively in the mathematical setting.
- MAT 103: Calculus IFirst semester of calculus. Topics include limits, continuity, the derivative, basic differentiation formulas and applications (curve-sketching, optimization, related rates), definite and indefinite integrals, the fundamental theorem of calculus.
- MAT 104: Calculus IIContinuation of MAT 103. Topics include techniques of integration, arclength, area, volume, convergence of series and improper integrals, L'Hopital's rule, power series and Taylor's theorem, introduction to differential equations and complex numbers.
- MAT 175: Mathematics for Economics/Life SciencesSurvey of topics from multivariable calculus as preparation for future course work in economics or life sciences. Topics include basic techniques of integration, average value, vectors, partial derivatives, gradient, optimization of multivariable functions, and constrained optimization with Lagrange multipliers.
- MAT 201: Multivariable CalculusVectors in the plane and in space, vector functions and motion, surfaces, coordinate systems, functions of two or three variables and their derivatives, maxima and minima and applications, double and triple integrals, vector fields and Stokes's theorem.
- MAT 202: Linear Algebra with ApplicationsCompanion course to MAT 201. Matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvectors and their applications to quadratic forms and dynamical systems.
- MAT 204: Advanced Linear Algebra with ApplicationsCompanion course to MAT203. Linear systems of equations, linear independence and dimension, linear transforms, determinants, (real and complex) eigenvectors and eigenvalues, orthogonality, spectral theorem, singular value decomposition, Jordan forms, other topics as time permits. More abstract than MAT202 but more concrete than MAT217. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or MAT215 or equivalent.
- MAT 215: Single Variable Analysis with an Introduction to ProofsAn introduction to the mathematical discipline of analysis, to prepare for higher-level course work in the department. Topics include rigorous epsilon-delta treatment of limits, convergence, and uniform convergence of sequences and series. Continuity, uniform continuity, and differentiability of functions. The Heine-Borel Theorem. The Riemann integral, conditions for integrability of functions and term by term differentiation and integration of series of functions, Taylor's Theorem.
- MAT 217: Honors Linear AlgebraA rigorous course in linear algebra with an emphasis on proof rather than applications. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms.
- MAT 218: Multivariable Analysis and Linear Algebra IIContinuation of the rigorous introduction to analysis in MAT 216
- MAT 325: Analysis I: Fourier Series and Partial Differential EquationsBasic facts about Fourier Series, Fourier Transformations, and applications to the classical partial differential equations will be covered. Also Finite Fourier Series, Dirichlet Characters, and applications to properties of primes.
- MAT 330: Complex Analysis with ApplicationsThe theory of functions of one complex variable, covering analyticity, contour integration, residues, power series expansions, and conformal mapping. The goal in the course is to give adequate treatment of the basic theory and also demonstrate the use of complex analysis as a tool for solving problems.
- MAT 346: Algebra IILocal Fields and the Galois theory of Local Fields.
- MAT 355: Introduction to Differential GeometryIntroduction to geometry of surfaces. Surfaces in Euclidean space: first fundamental form, second fundamental form, geodesics, Gauss curvature, Gauss-Bonnet Theorem. Minimal surfaces in the Euclidean space.
- MAT 375/COS 342: Introduction to Graph TheoryThe fundamental theorems and algorithms of graph theory. Topics include: connectivity, matchings, graph coloring, planarity, the four-color theorem, extremal problems, network flows, and related algorithms.
- MAT 378: Theory of GamesGames in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications.
- MAT 419: Topics in Number Theory: The Arithmetic of Quadratic FormsAn introduction to the arithmetic and geometry of quadratic forms. Topics will include lattices, class number, local-global principle, composition, and other connections to algebraic number theory. Applications to the representation of integers by quadratic forms will be discussed, including various generalizations of Lagrange's Theorem which states that every positive integer is the sum of four squares.
- MAT 425: Analysis III: Integration Theory and Hilbert SpacesThe theory of Lebesgue integration in n-dimensional space. Differentiation theory. Hilbert space theory and applications to Fourier Transforms, and partial differential equations. Introduction to fractals. This course is the third semester of a four-semester sequence, but may be taken independently of the other semesters.
- MAT 427: Ordinary Differential EquationsThis course offers an introduction to the study of ordinary differential equations. Topics include: Explicit solutions of some linear and non-linear equations (using tools that span: separation of variables, integrating factors, Greens functions, and Laplace transform methods); Series solutions of ODEs with analytic coefficients and with regular singular points; Fundamental existence and uniqueness theorems (Peano, Picard-Lindelof, and Osgood); Introduction to dynamical systems (Poincare-Bendixon theorem); Stability of equilibrium points and periodic orbits.
- MAT 478: Topics In Combinatorics: The Probabilistic MethodThis course covers probabilistic methods in combinatorics and their applications in theoretical computer science. The topics include linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, geometry, VC-dimension and possibly more as time permits.
- MAT 515: Topics in Number Theory and Related Analysis: Spectral Theory of Locally Uniform GeometriesWe introduce and discuss problems connected with prescribing the differential operator spectra of locally uniform geometries, primarily Euclidean hyperbolic and regular graphs. Various applications are given. Most of the tools from number theory, analysis, geometry, combinatorics and group theory are introduced as needed.
- MAT 517: Topics in Arithmetic Geometry: Diophantine Geometry and Modular FormsWe cover three topics: The geometric Bombieri-Lang conjecture and explain a proof of the geometric Bombieri-Lang conjecture for ramified covers of abelian varieties by J. Xie and X. Yuan; Faltings heights of abelian varieties and explain a proof of the Northcott property of Faltings heights in an isogeny class of abelian variety by M. Kisin and L. Mocz; and Kroneck limit formula for SL(3) and explain a proof of this formula and an application to the Stark conjecture for cubic fields by N. Bergeron, P. Charollois, and L. Garcia.
- MAT 519: Topics in Number Theory: Sieves and Algebraic Number TheoryWe discuss recent developments in arithmetic statistics. Topics include the study of squarefree values of polynomials, Galois groups of random polynomials, field extensions with given Galois group, sizes and structure of class groups, Selmer groups, Tate-Shafarevich groups, and related problems involving sieve theory and algebraic number theory.
- MAT 527: Topics in Differential Equations: Parabolic PDEThis is an introductory course to Parabolic PDE. We start with a brief review of 2nd order elliptic PDE, then discuss basic theory about the heat equation on the Euclidean space, including fundamental solution, Schauder and Lp estimates, maximal principle, construction and estimate of the heat kernel, energy methods. We then discuss heat equation on manifolds, Harnack principle of Li-Yau. As applications, we plan to cover a derivation of the log-Sobolev inequality on the Euclidean space via parabolic estimates, and the connectionof the inequality to Perelman's W-functional.
- MAT 528: Topics in Nonlinear Analysis: Topics in General RelativityThis course will concern advanced topics in general relativity, including the mathematics of gravitational collapse, the dynamics of rapidly rotating and extremal black holes and the structure of spacetime singularities.
- MAT 531: Introduction to Riemann SurfacesThis course is an introduction to the theory of compact Riemann surfaces, including some basic properties of the topology of surfaces, differential forms and the basic existence theorems, the general uniformization theorem, and the Riemann-Roch theorem and some of its consequences.
- MAT 547: Topics in Algebraic Geometry: Boundedness of VarietiesIn the last a few decades, results on boundedness plays a central role in the progress of higher dimensional geometry. This course aims to give a comprehensive treatment of the topic. We cover results on the Fujita Conjecture, uniform bounds on birational maps for general type varieties, boundedness of log general type pairs/ACC Conjecture, and boundedness of complements/BAB Conjecture etc.
- MAT 549: Topics in Algebra: Equidistribution via \ell-adic cohomologyThis course pursues the study of equidistribution questions using the ideas and techniques of \ell-adic cohomology.
- MAT 559: Topics in Geometry: Lipschitz Extension, Reverse Isoperimetry and RoundingSuppose that we are given a Lipschitz function f from a subset S of a metric space X to a metric space Y. Can we extend f to a Y-valued Lipschitz function that is defined on all of X? This depends on geometric properties of X,Y,S,f, and it is typically impossible, but for over a century a range of creative methods were devised to prove such extension theorems in many settings. The course describes results on extending Lipschitz functions, starting from the classical and arriving to current research. This includes links to questions on how one can reverse the isoperimetric inequality and ways to round continuous space to a discrete subset.
- MAT 560: Algebraic TopologyThe aim of the course is to study some of the basic algebraic techniques in Topology, such as homology groups, cohomology groups and homotopy groups of topological spaces.
- MAT 569: Topics in Topology: Topology of 4-manifoldsThe aim of this course is to give an introduction to smooth 4-manifolds. Topics include elliptic surfaces, symplectic 4-manifolds, the symplectic sum, the knot surgery construction of Fintushel and Stern, Seiberg-Witten invariants, and exotic structures on simply-connected 4-manifolds with small Euler characteristics.
- MAT 577: Topics in Combinatorics: The Probabilistic MethodProbabilistic methods in Combinatorics and their applications in theoretical Computer Science. The topics include linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, geometry, the VC dimension of a range space and its applications, and possibly more as time permits.
- MAT 586/APC 511/MOL 511/QCB 513: Computational Methods in Cryo-Electron MicroscopyThis course focuses on computational methods in cryo-EM, including three-dimensional ab-initio modelling, structure refinement, resolving structural variability of heterogeneous populations, particle picking, model validation, and resolution determination. Special emphasis is given to methods that play a significant role in many other data science applications. These comprise of key elements of statistical inference, image processing, optimization, and dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.
- MAT 589: Topics in Probability, Statistics and Dynamics: Modern Discrete Probability TheoryThe aim of this course is to survey some of the fundamentals of modern discrete probability, particularly random processes on networks and discrete structures. The course introduces topics including concentration of measure, random graphs, percolation, convergence of Markov chains and other topics depending on student interest.
- MAT 595/PHY 508: Topics in Mathematical Physics: Mathematical Aspects of Condensed Matter PhysicsThe course discusses rigorous results in quantum mechanics, relevant for condensed matter physics. Topics to be covered include: Effect of disorder on quantum dynamics, quantum transport and linear response theory, topological phases of matter, quantum Hall effect and topological classification of insulators (via K-theory or otherwise).
- ORF 309/EGR 309/MAT 380: Probability and Stochastic SystemsAn introduction to probability and its applications. Topics include: basic principles of probability; Lifetimes and reliability, Poisson processes; random walks; Brownian motion; branching processes; Markov chains.