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.

Computational complexity theory is a mathematical discipline that explores the boundaries of efficient computation. This course introduces some of the most engaging ideas in complexity theory, showcasing how advanced mathematical methods can address profound philosophical questions. We explore the significance of the P vs NP problem, analyzing approaches like diagonalization and circuit lower bounds, while also examining why progress has been slow. Topics include proof systems such as zero-knowledge proofs, interactive proofs, and probabilistically checkable proofs

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.

The study of the arithmetic of elliptic curves is an extremely rich area of mathematics, incorporating ideas from complex analysis, algebraic geometry, group and Galois theory, and of course number theory. Many fundamental problems in number theory, such as Fermat's Last Theorem and the congruent number problem, lead naturally to the study of elliptic curves. The purpose of this course is to explore some of the basic ideas in this area, including the group law on elliptic curves, the structure of this group over various fields, and the Birch and Swinnerton-Dyer conjeture.

The 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.

Introduction to affine and projective algebraic varieties over fields.

This course will cover topics in Extremal Combinatorics including ones motivated by questions in other areas like Computer Science, Information Theory, Number Theory and Geometry. The subjects that will be covered include Graph powers, the Shannon capacity and the Witsenhausen rate of graphs, Szemeredi's Regularity Lemma and its applications in graph property testing and in the study of sets with no 3 term arithmetic progressions, the Combinatorial Nullstellensatz and its applications, the capset problem, Containers and list coloring, and related topics as time permits.

We study some problems about heights of algebraic cycles motivated by conjectures in Diophantine geometry and special values of L-functions.

We study techniques to determine or bound the average sizes of Selmer sets and Selmer groups in families of elliptic curves and higher-dimensional abelian varieties, with applications to statistical questions about ranks, rational points, and solutions to Diophantine equations.

This course first covers some classical topics about the Euler and Navier-Stokes equations, such as classical existence theories of smooth solutions and weak solutions, global well-posedness in two dimensions and local well-posedness in three dimensions, the Ladyzenskaya-Prodi-Serrin conditional regularity theorem, Caffarelli-Kohn-Nirenberg local regularity theory for Navier-Stokes and various a-priori estimates. We then get to more recent results, such as the Escauriaza-Seregin-Sverak theorem, more recent conditional regularity results, Liouville theorems and results under special symmetry assumptions.

After an introduction to linear and nonlinear harmonic maps we begin discussing the quantitative stratification theory of solutions. The goal is to work our way up to the rectifiable structure and finite measure estimates for singular sets, as well as applications including Schauder type estimates in the nonlinear setting.

This course covers topics of interest concerning nonlinear waves in General Relativity, Hydrodynamics, etc.

The course covers questions of regularity, local existence, uniqueness and long time behavior of solutions of nonlinear nonlocal evolution equations.

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.

The class aims to discuss progress in higher dimensional geometry. The central theme is the classification theory of varieties. It covers recent results in fields including minimal model program, moduli of varieties, K-stability theory of Fano varieties, boundedness of varieties, explicit geometry of special varieties etc. Students are required to have a solid footing on algebraic geometry, e.g. materials in Hartshorne's book; and are recommended to have basic knowledge of the minimal model program, e.g. Kollár-Mori's book.

The course starts with an introduction to the basic theory in second order parabolic equations, including a brief review of the second order elliptic theory, then discusses basic theory about the heat equation, including fundamental solution, Schauder and Lp estimates, maximal principle, Harnack inequality for parabolic equations first on Euclidean space then on manifolds. Later we cover aspects of the Hamilton Ricci Flow on manifolds, including the study of Perelman's W-functional; application to the 'pinching results' along the Ricci Flow for problems in conformal geometry; and recent works of Gursky, Chang-Gursky-S. Zhang, et al.

As convexity of the unit ball characterizes norms among translation-invariant homogeneous metrics on vector spaces, it is natural to study the implications of enhanced convexity assumptions. Those early "isometric" concepts are treated, but the course mainly focuses on deep "isomorphic" theories of enhanced convexity. The term "isomorphic" refers to the fact that the requirements are preserved under large perturbations. A feature of such concepts is the roundabout role that probability plays in their formulation and investigation, as they entail understanding the behavior of vector-valued independent random variables and martingales.

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.

In this course, we study real and complex vector bundles, their characteristic classes and various applications. These include Stiefel-Whitney classes, Chern classes, Pontrjagin classes, the splitting principle, classifying spaces, Grassmanians, Hirzebruch's signature theorem, and the 7-dimensional exotic spheres of Milnor. Additional topics may include Morse Theory, the h-cobordism Theorem, and the Poincare Conjecture in higher dimensions.

This course covers topics in Extremal Combinatorics including ones motivated by questions in other areas like Computer Science, Information Theory, Number Theory and Geometry. The subjects that are covered include Graph powers, the Shannon capacity and the Witsenhausen rate of graphs, Szemeredi's Regularity Lemma and its applications in graph property testing and in the study of sets with no 3 term arithmetic progressions, the Combinatorial Nullstellensatz and its applications, the capset problem, Containers and list coloring, and related topics as time permits.

In this course we focus on results from structural graph theory that have been widely used for designing algorithms. Among the topics covered are path-decompositions, tree-decompositions, induced subgraph detection, the three-in-a-tree theorem, and others (time permitting).

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.