Faculty of Arts & Sciences

The study of functions and their rates of change. Fundamental ideas of calculus are introduced early and used to provide a framework for the study of mathematical modeling involving algebraic, exponential, and logarithmic functions. Thorough understanding of differential calculus promoted by year long reinforcement. Applications to biology and economics emphasized according to the interests of our students.

Continued investigation of functions and differential calculus through modeling; an introduction to integration with applications; an introduction to differential equations. Solid preparation for Mathematics 1b.

The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to problems from many other disciplines.

Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.

Focus on concepts and techniques of multivariable calculus most useful to those studying the social sciences, particularly economics: functions of several variables; partial derivatives; directional derivatives and the gradient; constrained and unconstrained optimization, including the method of Lagrange multipliers. Covers linear and polynomial approximation and integrals for single variable and multivariable functions; modeling with derivatives. Covers topics from Math 21a most useful to social sciences.

Considers the construction and analysis of mathematical models that arise in the life sciences, ecology and environmental life science. Introduces mathematics that include multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and non-linear dynamical systems. Taught via examples from current literature (both good and bad).

Probability, statistics and linear algebra with applications to life sciences, chemistry, and environmental life sciences. Linear algebra includes matrices, eigenvalues, eigenvectors, determinants, and applications to probability, statistics, dynamical systems. Basic probability and statistics are introduced, as are standard models, techniques, and their uses including the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis.

To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Green's, Stokes's, and Divergence Theorems.

Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as linear transformations and linear spaces, determinants, eigenvalues, and eigenvectors. Applications include dynamical systems, ordinary and partial differential equations, and an introduction to Fourier series.

A rigorous, integrated treatment of linear algebra and multivariable differential calculus, emphasizing topics that are relevant to fields such as physics and economics. Topics: fields, vector spaces and linear transformations, scalar and vector products, elementary topology of Euclidean space, limits, continuity, and differentiation in n dimensions, eigenvectors and eigenvalues, inverse and implicit functions, manifolds, and Lagrange multipliers.

A rigorous, integrated treatment of linear algebra and multivariable calculus. Topics: Riemann and Lebesgue integration, determinants, change of variables, volume of manifolds, differential forms, and exterior derivative. Stokes's theorem is presented both in the language of vector analysis (div, grad, and curl) and in the language of differential forms.

A rigorous treatment of linear algebra. Topics include: Construction of number systems; fields, vector spaces and linear transformations; eigenvalues and eigenvectors, determinants and inner products. Metric spaces, compactness and connectedness.

A rigorous treatment of basic analysis. Topics include: convergence, continuity, differentiation, the Riemann integral, uniform convergence, the Stone-Weierstrass theorem, Fourier series, differentiation in several variables. Additional topics, including the classical results of vector calculus in two and three dimensions, as time allows.

A rigorous treatment of abstract algebra including linear algebra and group theory.

A rigorous treatment of real and complex analysis.

Advanced reading in topics not covered in courses.

Programs of directed study supervised by a person approved by the Department.

Supervised small group tutorial. Topics to be arranged.

An introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, and low-dimensional topology.

Develops the theory of inner product spaces, both finite-dimensional and infinite-dimensional, and applies it to a variety of ordinary and partial differential equations. Topics: existence and uniqueness theorems, Sturm-Liouville systems, orthogonal polynomials, Fourier series, Fourier and Laplace transforms, eigenvalue problems, and solutions of Laplace's equation and the wave equation in the various coordinate systems.

An introduction to mathematical analysis and the theory behind calculus. An emphasis on learning to understand and construct proofs. Covers limits and continuity in metric spaces, uniform convergence and spaces of functions, the Riemann integral.

Analytic functions of one complex variable: power series expansions, contour integrals, Cauchy's theorem, Laurent series and the residue theorem. Some applications to real analysis, including the evaluation of indefinite integrals. An introduction to some special functions.

Lebesgue measure and integration; general topology; introduction to L p spaces, Banach and Hilbert spaces, and duality.

Complex functions; Fourier analysis; Hilbert spaces and operators; Laplace's equations; Bessel and Legendre functions; symmetries; Sturm-Liouville theory.

Develops the theory of convex sets, normed infinite-dimensional vector spaces, and convex functionals and applies it as a unifying principle to a variety of optimization problems such as resource allocation, production planning, and optimal control. Topics include Hilbert space, dual spaces, the Hahn-Banach theorem, the Riesz representation theorem, calculus of variations, and Fenchel duality. Students will be expected to understand and invent proofs of theorems in real and functional analysis.

A self-contained treatment of the theory of probability and random processes with specific application to the theory of option pricing. Topics: axioms for probability, calculation of expectation by means of Lebesgue integration, conditional probability and conditional expectation, martingales, random walks and Wiener processes, and the Black-Scholes formula for option pricing. Students will work in small groups to investigate applications of the theory and to prove key results.

Introduction to dynamical systems theory with a view toward applications. Topics include existence and uniqueness theorems for flows, qualitative study of equilibria and attractors, iterated maps, and bifurcation theory.

Real and complex vector spaces, linear transformations, determinants, inner products, dual spaces, and eigenvalue problems. Applications to geometry, systems of linear differential equations, electric circuits, optimization, and Markov processes. Emphasizes learning to understand and write proofs. Students will work in small groups to solve problems and develop proofs.

Groups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups.

Rings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules.

Factorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; Pell's equation; selected Diophantine equations; theory of integral quadratic forms.

Algebraic number theory: number fields, unique factorization of ideals, finiteness of class group, structure of unit group, Frobenius elements, local fields, ramification, weak approximation, adeles, and ideles.

Presents several classical geometries, these being the affine, projective, Euclidean, spherical and hyperbolic geometries. They are viewed from many different perspectives, some historical and some very topical. Emphasis on reading and writing proofs.

Abstract topological spaces; compactness, connectedness, continuity. Homeomorphism and homotopy, fundamental groups, covering spaces. Introduction to combinatorial topology.

Differential manifolds, smooth maps and transversality. Winding numbers, vector fields, index and degree. Differential forms, Stokes' theorem, introduction to cohomology.

The exterior differential calculus and its applications to curves and surfaces in 3-space and to various notions of curvature. Introduction to Riemannian geometry in higher dimensions.

Affine and projective spaces, plane curves, Bezout's theorem, singularities and genus of a plane curve, Riemann-Roch theorem.

An introduction to mathematical logic with applications to computer science and algebra. Formal languages. Completeness and compactness of first order logic. Definability and interpolation. Decidability. Unsolvable problems. Computable functions and Turing machines. Recursively enumerable sets. Transfinite induction.

An introduction to model theory with applications to fields and groups. First order languages, structures, and definable sets. Compactness, completeness, and back-and-forth constructions. Quantifier elimination for algebraically closed, differentially closed, and real closed fields. Omitting types, prime extensions, existence and uniqueness of the differential closure, saturation, and homogeneity. Forking, independence, and rank.

An introduction to set theory covering the fundamentals of ZFC (cardinal arithmetic, combinatorics, descriptive set theory) and the independence techniques (the constructible universe, forcing, the Solovay model). We will demonstrate the independence of CH (the Continuum Hypothesis), SH (Suslin's Hypothesis), and some of the central statements of classical descriptive set theory.

An introduction to large cardinals and their inner models, with special emphasis on Woodin's recent advances toward finding an ultimate version of Godel's L. Topics include: Weak extender models, the HOD Dichotomy Theorem, and the HOD Conjecture.

An introduction to sets, logic, finite groups, finite fields, finite geometry, combinatorics, discrete probability, and graph theory. A unifying theme of the course is the symmetry group of the regular icosahedron, whose elements can be realized as permutations, as linear transformations of vector spaces over finite fields, as collineations of a finite plane, or as vertices of a graph. Taught in a seminar format: students will gain experience in presenting proofs at the blackboard.

Introduces basic concepts of mathematical biology and evolutionary dynamics: evolution of genomes, quasi-species, finite and infinite population dynamics, chaos, game dynamics, evolution of cooperation and language, spatial models, evolutionary graph theory, infection dynamics, somatic evolution of cancer.

An introduction to probability theory. Discrete and continuous random variables; distribution and density functions for one and two random variables; conditional probability. Generating functions, weak and strong laws of large numbers, and the central limit theorem. Geometrical probability, random walks, and Markov processes.

An introduction to counting techniques and other methods in finite mathematics. Possible topics include: the inclusion-exclusion principle and Mobius inversion, graph theory, generating functions, Ramsey's theorem and its variants, probabilistic methods.

Presents the probability theory and statistical principles which underly the tools that are built into the open-source programming language R. Each class presents the theory behind a statistical tool, then shows how the implementation of that tool in R can be used to analyze real-world data. The emphasis is on modern bootstrapping and resampling techniques, which rely on computational power. Topics include discrete and continuous probability distributions, permutation tests, the central limit theorem, chi-square and Student t tests, linear regression, and Bayesian methods.

An interactive introduction to problem solving with an emphasis on subjects with comprehensive applications. Each class will be focused around a group of questions with a common topic. Possible topics: logic, information, number theory, probability, and algorithms.

An introduction to categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads, and other topics as time permits with the aim of revisiting a broad range of mathematical examples from the categorical perspective.

Banach spaces, Hilbert spaces and functional analysis. Distributions, spectral theory and the Fourier transform.

This class will be an introduction to harmonic analysis and singular integral. The textbook is Classical and Multilinear Harmonic Analysis, Volume 1, by Muscalu and Schlag. The topics covered in the course include maximum functions, interpolation of operators, Calderon-Zygmund theory and Littlewood-Paley theory. Some elementary probability theory will also be included. Good references of this course are Stein's book on singular integrals and Fourier analysis.

A second course in complex analysis: series, product and partial fraction expansions of holomorphic functions; Hadamard's theorem; conformal mapping and the Riemann mapping theorem; elliptic functions; Picard's theorem and Nevanlinna Theory.

Fundamentals of Riemann surfaces. Topics may include sheaves and cohomology, potential theory, uniformization, and moduli.

A first course in commutative algebra: Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, discrete valuation rings, filtrations, completions and dimension theory.

Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.

A graduate introduction to algebraic number theory. Topics: the structure of ideal class groups, groups of units, a study of zeta functions and L-functions, local fields, Galois cohomology, local class field theory, and local duality.

Continuation of Mathematics 223a. Topics: adeles, global class field theory, duality, cyclotomic fields. Other topics may include: Tate's thesis or Euler systems.

Structure theory of reductive Lie groups, unitary representations, Harish Chandra modules, characters, the discrete series, Plancherel theorem.

Fundamental methods, results, and problems of analytic number theory. Riemann zeta function and the Prime Number Theorem; Dirichlet's theorem on primes in arithmetic progressions; lower bounds on discriminants from functional equations; sieve methods, analytic estimates on exponential sums, and their applications.

Connections on the tangent bundle, Levi-Civita's theorem, Gauss's lemma, curvature, distance and volume, general relativity, connections on principle bundles.

A continuation of Mathematics 230a. Topics in differential geometry: Analysis on manifolds. Laplacians. Hodge theory. Spin structures. Clifford algebras. Dirac operators. Index theorems. Applications.

Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Kunneth formulas. Hurewicz theorem. Manifolds and Poincare duality.

Continuation of Mathematics 231a. Vector bundles and characteristic classes. Bott periodicity. K-theory, cobordism and stable cohomotopy as examples of cohomology theories.

Introduction to complex algebraic curves, surfaces, and varieties.

The course will cover the classification of complex algebraic surfaces.

An introduction to the theory and language of schemes. Textbooks: Algebraic Geometry by Robin Hartshorne and Geometry of Schemes by David Eisenbud and Joe Harris. Weekly homework will constitute an important part of the course.

A continuation of Mathematics 233a. Will cover the theory of schemes, sheaves, and sheaf cohomology.

Advanced topics of evolutionary dynamics. Seminars and research projects.

Topics in classical and high-dimensional knot theory, with a focus on invariants related to the Alexander module. Possible topics: Seifert surfaces and pairings, Tristram-Levine signatures, the Blanchfield pairing, classification of simple n-knots, singularities of algebraic hypersurfaces, connections to arithmetic invariant theory.

The goal of this course is to give a detailed account of the recent advances concerning the local statistics of eigenvalue distributions of random matrices. Basic knowledge of probability theory and measure theory are required.

A study of (positive existential) definability problems in number theory. The main topics to be considered will be definability of multiplication, interpretations and undecidability.

An investigation of geometric aspects of mirror symmetry in the SYZ approach using Langrangian intersection theory.

A discussion of the complex Monge-Ampere equation, and its applications in the geometry of Kahler manifolds. Topics: Yau's solution of the Calabi Conjecture, and the geometry of Gromov-Hausdorff limits of Ricci flat metrics. Further topics may include the degenerate Monge-Ampere equation and singular Calabi-Yau metrics, as well as Ricci flat metrics on non-compact manifolds, particularly conical Calabi-Yau metrics and their connection to the geometry of Fano varieties.

Basic analysis of Riemannian manifolds and their applications in geometry and theoretical physics including general relativity and string theory.

An introduction to topological K-theory followed by recent applications. Specific topics may include: twisted K-theory and representations of loop groups, differential K-theory and the index theorem, Ramond-Ramond fields in superstring theory, topological insulators.

An introduction to the theory of reductive groups, beginning with their structure theory over algebraically closed fields, discussing rationality questions, and a study of special phenomena that occur when the field of definition is a local or global field.

A second course in algebraic geometry, centered around intersection theory but intended in addition to introduce the student to basic tools of algebraic geometry, such as deformation theory, characteristic classes, Hilbert schemes and specialization methods.

An introduction to PDE's for statistical physics out of thermodynamic equilibrium. 1. Mathematical tools for the study of kinetic transport equations. 2. Mean field approximation: the case of the Vlasov-Poisson system. 3. Collisional kinetic theory: an introduction to the Boltzmann equation.

Topics: Cohen Lenstra heuristics, prehomogeneous vector spaces, applications to statistics of number fields and class groups, Poonen-Rains heuristics, and ranks of elliptic curves. Tools will include Davenport and Bhargava's geometry-of-numbers' methods.

A survey of fundamental results and current research. Topics may include: ergodic theory, hyperbolic manifolds, Mostow rigidity, Kazhdan's property T, Ratner's theorem, and dynamics over moduli space.

An introduction to the algebraic K-theory of rings and ring spectra, emphasizing connections with simple homotopy theory and the topology of manifolds.

Become an effective instructor. This course focuses on observation, practice, feedback, and reflection providing insight into teaching and learning. Involves iterated videotaped micro-teaching sessions, accompanied by individual consultations. Required of all mathematics graduate students.