Mathematics 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.

This 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.

Analytic Combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the scientific analysis of algorithms in computer science and for the study of scientific models in many other disciplines. This course combines motivation for the study of the field with an introduction to underlying techniques, by covering as applications the analysis of numerous fundamental algorithms from computer science. The second half of the course introduces Analytic Combinatorics, starting from basic principles.

Computational complexity theory is a mathematical discipline that studies efficient computation. The course covers some of the truly beautiful ideas of modern complexity theory such as: approaches to the famous P vs NP question and why they are stuck; complexity classes and their relationship; circuit lower bounds; proof systems such as zero knowledge proofs, interactive proofs and probabilistically checkable proofs; hardness of approximation; de-randomization and the hardness vs randomness paradigm; quantum computing.

A 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.

This 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 algebraic equations, numerical integration, analytical and numerical solution of ordinary and partial differential equations, harmonic functions and conformal maps, and time-series data. The course synthesizes descriptive observations, mathematical theories, numerical methods, and engineering consequences.

Introduction 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.

First 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.

Continuation 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.

Survey 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.

Vectors 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.

Companion 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.

Companion 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.

An 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.

A 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.

Continuation of the rigorous introduction to analysis in MAT 216

Basic 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.

The 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.

Local Fields and the Galois theory of Local Fields.

Introduction 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.

The 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.

Games in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications.

An 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.

This 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.

This course will cover the standard material in a first course on commutative algebra. Topics include: ideals in and modules over commutative rings, localization, primary decomposition, integral dependence, Noetherian rings and chain conditions, discrete valuation rings and Dedekind domains, completion; and dimension theory.

An introduction to representation theory of Lie groups and semisimple Lie algebras.

This 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.

This course covers the basics of the spectral theory of automorphic forms, including the analytic continuation of Eisenstein Series, the trace formula (in some basic cases) and the general Ramanujan Conjectures. Applications to number theory are highlighted.

This course focuses on Sen's theory in the study of p-adic Galois representations. We also discuss several generalizations, for instance: 1. Fontaine's study of B_dR-representations, 2. a relative version of Sen's theory.

This course will cover some recent applications of the arithmetic of automorphic forms to questions about special values of L-functions. Particular attention will be given to L-functions of elliptic curves and modular forms.

Basic introductory course to modern methods of analysis. The possible topics may include Lp spaces, Banach spaces, uniform boundedness principle, closed graph theorem, locally convex spaces, distributions, Fourier transform, Riesz interpolation theorem, Hardy-Littlewood maximal function, Calderon-Zygmund theory, oscillatory integrals, almost orthogonality, Sobolev spaces, restriction theorems, spectral theory of compact operators, applications to partial differential equations.

We discuss long time behavior of solutions of nonlinear evolution equations modeling fluids and plasmas. Topics include linear and nonlinear stability, invariant manifolds, singularity formation.

This course coverd the following topics: 1. Fast introduction to mathematical General Relativity. 2. Kerr family 3. Wave Equation in Minkowski space. Boundedness and Decay. Vectorfield Method. 4. Wave Equation on Black Holes. Boundedness and Decay. 5. Introduction to the problem of the nonlinear stability of Kerr.

The course studies the Muskat equation, which governs the evolution of the interface between two fluids in a porous medium, e.g. oil and water in sand. In two dimensions, when the two fluids have the same viscosity but different densities, the problem reduces to a contour dynamics equation. When the interface is the graph of a function, a lot is known; but in the "turnover regime" in which a vertical line may meet the interface in more than one point, hard problems remain. The lectures explain the state of the art as of a few years ago, then present recent work of Jia Shi.

This 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.

We discuss relations local systems and finite groups.

The 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.

This course covers some of the modern techniques and recent developments in knot theory. The focus this semester is on Khovanov homology and knot Floer homology. We discuss applications in knot theory and smooth 4-dimensional topology. We also discuss algebraic and combinatorial methods to compute knot homologies.

This 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.

An 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.

Mathematical methods and terminology which are essential for modern theoretical physics. These include some of the traditional techniques of mathematical analysis, but also more modern tools such as group theory, functional analysis, calculus of variations, non-linear operator theory and differential geometry. Mathematical theories are not treated as ends in themselves; the goal is to show how mathematical tools are developed to solve physical problems.

An introduction to the statistical mechanic of classical and quantum spin systems. Among the topics to be discussed are phase transitions, emergent structures, critical phenomena, and scaling limits. The goal is to present the physics embodied in the subject along with mathematical methods, from probability and analysis, for rigorous results concerning the phenomena displayed by, and within, the subject's essential models. The lectures start with a brisk review of what was covered in the course in Spring 2021, and then continue beyond that with a more extended, though still self contained, discussion of quantum systems.