Courses - Fall 2020

MATH 1011 Academic Support for MATH 1110

Reviews material presented in MATH 1110 lectures, provides problem-solving techniques and tips as well as prelim review. Provides further instruction for students who need reinforcement. Not a substitute for attending MATH 1110 lectures.

Academic Career: UG Instructor: Mark Jauquet (maj29)
Full details for MATH 1011 : Academic Support for MATH 1110
MATH 1012 Academic Support for MATH 1120

Reviews material presented in MATH 1120 lectures, provides problem-solving techniques and tips as well as prelim review. Provides further instruction for students who need reinforcement. Not a substitute for attending MATH 1120 lectures or discussions.

Academic Career: UG Instructor: Brendan Caseria (bjc297)
Full details for MATH 1012 : Academic Support for MATH 1120
MATH 1021 Academic Support for MATH 2210

Reviews material presented in MATH 2210 lectures, provides problem-solving techniques and tips as well as prelim review. Provides further instruction for students who need reinforcement. Not a substitute for attending MATH 2210 lectures or discussions.

Academic Career: UG Instructor: Brendan Caseria (bjc297)
Full details for MATH 1021 : Academic Support for MATH 2210
MATH 1101 Calculus Preparation

Introduces topics in calculus: limits, rates of change, definition of and techniques for finding derivatives, relative and absolute extrema, and applications. The calculus content of the course is similar to 1/3 of the content covered in MATH 1106 and MATH 1110. In addition, the course includes a variety of topics of algebra, with emphasis on the development of linear, power, exponential, logarithmic, and trigonometric functions. Because of the strong emphasis on graphing, students will have a better understanding of asymptotic behavior of these functions.

Academic Career: UG Instructor: Mark Jauquet (maj29)
Full details for MATH 1101 : Calculus Preparation
MATH 1105 Finite Mathematics for the Life and Social Sciences

Introduction to linear algebra, probability, and Markov chains that develops the parts of the theory most relevant for applications. Specific topics include equations of lines, the method of least squares, solutions of linear systems, matrices; basic concepts of probability, permutations, combinations, binomial distribution, mean and variance, and the normal approximation to the binomial distribution. Examples from biology and the social sciences are used.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: James Quigley (jdq27)
Full details for MATH 1105 : Finite Mathematics for the Life and Social Sciences
MATH 1110 Calculus I

Topics include functions and graphs, limits and continuity, differentiation and integration of algebraic, trigonometric, inverse trig, logarithmic, and exponential functions; applications of differentiation, including graphing, max-min problems, tangent line approximation, implicit differentiation, and applications to the sciences; the mean value theorem; and antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, substitution in integration, the area under a curve. Graphing calculators are used, and their pitfalls are discussed, as applicable to the above topics.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Brock Schmutzler (bas386)
Full details for MATH 1110 : Calculus I
MATH 1120 Calculus II

Focuses on integration: applications, including volumes and arc length; techniques of integration, approximate integration with error estimates, improper integrals, differential equations (separation of variables, initial conditions, systems, some applications). Also covers infinite sequences and series: definition and tests for convergence, power series, Taylor series with remainder, and parametric equations.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Robert Connelly (rc46)
Full details for MATH 1120 : Calculus II
MATH 1300 Mathematical Explorations

For students who wish to experience how mathematical ideas naturally evolve. The course emphasizes ideas and imagination rather than techniques and calculations. Homework involves students in actively investigating mathematical ideas. Topics vary depending on the instructor. Some assessment through writing assignments.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Inna Zakharevich (iiz5)
Full details for MATH 1300 : Mathematical Explorations
MATH 1710 Statistical Theory and Application in the Real World

Introductory statistics course discussing techniques for analyzing data occurring in the real world and the mathematical and philosophical justification for these techniques. Topics include population and sample distributions, central limit theorem, statistical theories of point estimation, confidence intervals, testing hypotheses, the linear model, and the least squares estimator. The course concludes with a discussion of tests and estimates for regression and analysis of variance (if time permits). The computer is used to demonstrate some aspects of the theory, such as sampling distributions and the Central Limit Theorem. In the lab portion of the course, students learn and use computer-based methods for implementing the statistical methodology presented in the lectures.

Distribution: (MQR-AS, SDS-AS)
Academic Career: UG Instructor: Michael Nussbaum (mn66)
Full details for MATH 1710 : Statistical Theory and Application in the Real World
MATH 1890 FWS: Writing in Mathematics
Academic Career: UG Instructor: Ana Sandu (ass243)
Full details for MATH 1890 : FWS: Writing in Mathematics
MATH 1910 Calculus for Engineers

Essentially a second course in calculus. Topics include techniques of integration, finding areas and volumes by integration, exponential growth, partial fractions, infinite sequences and series, tests of convergence, and power series.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Dan Barbasch (dmb14)
Full details for MATH 1910 : Calculus for Engineers
MATH 1920 Multivariable Calculus for Engineers

Introduction to multivariable calculus. Topics include partial derivatives, double and triple integrals, line and surface integrals, vector fields, Green's theorem, Stokes' theorem, and the divergence theorem.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Reyer Sjamaar (rs73)
Full details for MATH 1920 : Multivariable Calculus for Engineers
MATH 2210 Linear Algebra

Topics include vector algebra, linear transformations, matrices, determinants, orthogonality, eigenvalues, and eigenvectors. Applications are made to linear differential or difference equations. The lectures introduce students to formal proofs. Students are required to produce some proofs in their homework and on exams.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Marie B.Langlois (mb2636)
Full details for MATH 2210 : Linear Algebra
MATH 2220 Multivariable Calculus

Differential and integral calculus of functions in several variables, line and surface integrals as well as the theorems of Green, Stokes, and Gauss.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: James West (jew13)
Full details for MATH 2220 : Multivariable Calculus
MATH 2230 Theoretical Linear Algebra and Calculus

Topics include vectors, matrices, and linear transformations; differential calculus of functions of several variables; inverse and implicit function theorems; quadratic forms, extrema, and manifolds; multiple and iterated integrals.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: John Hubbard (jhh8)
Full details for MATH 2230 : Theoretical Linear Algebra and Calculus
MATH 2310 Linear Algebra with Applications

Introduction to linear algebra for students who wish to focus on the practical applications of the subject. A wide range of applications are discussed and computer software may be used. The main topics are systems of linear equations, matrices, determinants, vector spaces, orthogonality, and eigenvalues. Typical applications are population models, input/output models, least squares, and difference equations.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Andy Borum (adb328)
Full details for MATH 2310 : Linear Algebra with Applications
MATH 2930 Differential Equations for Engineers

Introduction to ordinary and partial differential equations. Topics include: first-order equations (separable, linear, homogeneous, exact); mathematical modeling (e.g., population growth, terminal velocity); qualitative methods (slope fields, phase plots, equilibria, and stability); numerical methods; second-order equations (method of undetermined coefficients, application to oscillations and resonance, boundary-value problems and eigenvalues); and Fourier series. A substantial part of this course involves partial differential equations, such as the heat equation, the wave equation, and Laplace's equation. (This part must be present in any outside course being considered for transfer credit to Cornell as a substitute for MATH 2930.)

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Yuri Berest (yb26)
Full details for MATH 2930 : Differential Equations for Engineers
MATH 2940 Linear Algebra for Engineers

Linear algebra and its applications. Topics include matrices, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and inner product spaces; applications include brief introductions to difference equations, Markov chains, and systems of linear ordinary differential equations. May include computer use in solving problems.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Alex Townsend (ajt253)
Full details for MATH 2940 : Linear Algebra for Engineers
MATH 3040 Prove It!

In mathematics, the methodology of proof provides a central tool for confirming the validity of mathematical assertions, functioning much as the experimental method does in the physical sciences. In this course, students learn various methods of mathematical proof, starting with basic techniques in propositional and predicate calculus and in set theory and combinatorics, and then moving to applications and illustrations of these via topics in one or more of the three main pillars of mathematics: algebra, analysis, and geometry. Since cogent communication of mathematical ideas is important in the presentation of proofs, the course emphasizes clear, concise exposition.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Ed Swartz (ebs22)
Full details for MATH 3040 : Prove It!
MATH 3110 Introduction to Analysis

Provides a transition from calculus to real analysis. Topics include rigorous treatment of fundamental concepts in calculus: including limits and convergence of sequences and series, compact sets; continuity, uniform continuity and differentiability of functions. Emphasis is placed upon understanding and constructing mathematical proofs.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Alice Nadeau (ann36)
Full details for MATH 3110 : Introduction to Analysis
MATH 3210 Manifolds and Differential Forms

A manifold is a type of subset of Euclidean space that has a well-defined tangent space at every point. Such a set is amenable to the methods of multivariable calculus. After a review of some relevant calculus, this course investigates manifolds and the structures that they are endowed with, such as tangent vectors, boundaries, orientations, and differential forms. The notion of a differential form encompasses such ideas as area forms and volume forms, the work exerted by a force, the flow of a fluid, and the curvature of a surface, space, or hyperspace. The course re-examines the integral theorems of vector calculus (Green, Gauss, and Stokes) in the light of differential forms and applies them to problems in partial differential equations, topology, fluid mechanics, and electromagnetism.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Kathryn Mann (kpm85)
Full details for MATH 3210 : Manifolds and Differential Forms
MATH 3230 Introduction to Differential Equations

A brief one-semester introduction to the theory and techniques of both ordinary and partial differential equations. Topics for ordinary differential equations may include initial-value and two-point boundary value problems, the basic existence and uniqueness theorems, continuous dependence on data, stability of fix-points, numerical methods, special functions. Topics for partial differential equations may include the Poisson, heat and wave equations, boundary and initial-boundary value problems, maximum principles, continuous dependence on data, separation of variables, Fourier series, Green's functions, numerical methods, transform methods.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Jonas Juul (jsj85)
Full details for MATH 3230 : Introduction to Differential Equations
MATH 3320 Introduction to Number Theory

An introductory course on number theory, the branch of algebra that studies the deeper properties of integers and their generalizations. Usually includes most of the following topics: the Euclidean algorithm, continued fractions, Pythagorean triples, Diophantine equations such as Pell's equation, congruences, quadratic reciprocity, binary quadratic forms, Gaussian integers, and factorization in quadratic number fields. May include a brief introduction to Fermat's Last Theorem.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Nicolas Templier (npt27)
Full details for MATH 3320 : Introduction to Number Theory
MATH 3610 Mathematical Modeling

Introduction to the theory and practice of mathematical modeling. This course compares and contrasts different types of mathematical models (discrete vs. continuous, deterministic vs. stochastic), focusing on advantages, disadvantages and limits of applicability for each approach. Case-study format covers a variety of application areas including economics, physics, sociology, traffic engineering, urban planning, robotics, and resource management. Students learn how to implement mathematical models on the computer and how to interpret/describe the results of their computational experiments.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Alexander Vladimirsky (abv8)
Full details for MATH 3610 : Mathematical Modeling
MATH 3840 Introduction to Set Theory

This will be a course on standard set theory (first developed by Ernst Zermelo early in the 20th century): the basic concepts of sethood and membership, operations on sets, functions as sets, the set-theoretic construction of the Natural Numbers, the Integers, the Rational and Real numbers; time permitting, some discussion of cardinality.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Harold Hodes (hth3)
Full details for MATH 3840 : Introduction to Set Theory
MATH 3850 Modal Logic

Modal logic is a general logical framework for systematizing reasoning about qualified and relativized truth. It has been used to study the logic of possibility, time, knowledge, obligation, provability, and much more. This course will explore both the theoretical foundations and the various philosophical applications of modal logic. On the theoretical side, we will cover basic metatheory, including Kripke semantics, soundness and completeness, correspondence theory, and expressive power. On the applied side, we will examine temporal logic, epistemic logic, deontic logic, counterfactuals, two-dimensional logics, and quantified modal logic. 

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Alexander Kocurek (awk78)
Full details for MATH 3850 : Modal Logic
MATH 4130 Honors Introduction to Analysis I

Introduction to the rigorous theory underlying calculus, covering the real number system and functions of one variable. Based entirely on proofs. The student is expected to know how to read and, to some extent, construct proofs before taking this course. Topics typically include construction of the real number system, properties of the real number system, continuous functions, differential and integral calculus of functions of one variable, sequences and series of functions.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Camil Muscalu (fm69)
Full details for MATH 4130 : Honors Introduction to Analysis I
MATH 4200 Differential Equations and Dynamical Systems

Covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Richard Rand (rhr2)
Full details for MATH 4200 : Differential Equations and Dynamical Systems
MATH 4220 Applied Complex Analysis

Covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Christian Noack (cjn59)
Full details for MATH 4220 : Applied Complex Analysis
MATH 4250 Numerical Analysis and Differential Equations

Introduction to the fundamentals of numerical analysis: error analysis, approximation, interpolation, numerical integration. In the second half of the course, the above are used to build approximate solvers for ordinary and partial differential equations. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is required to test the theoretical concepts throughout the course.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Alexander Vladimirsky (abv8)
Full details for MATH 4250 : Numerical Analysis and Differential Equations
MATH 4310 Linear Algebra

Introduction to linear algebra, including the study of vector spaces, linear transformations, matrices, and systems of linear equations. Additional topics are quadratic forms and inner product spaces, canonical forms for various classes of matrices and linear transformations.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Michael Stillman (mes15)
Full details for MATH 4310 : Linear Algebra
MATH 4315 Linear Algebra with Supplements

The main focus is on linear algebra, including the study of vector spaces, maps and matrices. Additional topics introduce groups, fields, rings, ideals, and algebraic geometry. The course provides a wide background of the basic concepts in algebra.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Irena Peeva (ivp1)
Full details for MATH 4315 : Linear Algebra with Supplements
MATH 4330 Honors Linear Algebra

Honors version of a course in advanced linear algebra, which treats the subject from an abstract and axiomatic viewpoint. Topics include vector spaces, linear transformations, polynomials, determinants, tensor and wedge products, canonical forms, inner product spaces, and bilinear forms. Emphasis is on understanding the theory of linear algebra; homework and exams include at least as many proofs as computational problems.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Shankar Sen (ss70)
Full details for MATH 4330 : Honors Linear Algebra
MATH 4410 Introduction to Combinatorics I

Combinatorics is the study of discrete structures that arise in a variety of areas, particularly in other areas of mathematics, computer science, and many areas of application. Central concerns are often to count objects having a particular property (e.g., trees) or to prove that certain structures exist (e.g., matchings of all vertices in a graph). The first semester of this sequence covers basic questions in graph theory, including extremal graph theory (how large must a graph be before one is guaranteed to have a certain subgraph) and Ramsey theory (which shows that large objects are forced to have structure). Variations on matching theory are discussed, including theorems of Dilworth, Hall, König, and Birkhoff, and an introduction to network flow theory. Methods of enumeration (inclusion/exclusion, Möbius inversion, and generating functions) are introduced and applied to the problems of counting permutations, partitions, and triangulations.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Karola Meszaros (km626)
Full details for MATH 4410 : Introduction to Combinatorics I
MATH 4530 Introduction to Topology

Topology may be described briefly as qualitative geometry. This course begins with basic point-set topology, including connectedness, compactness, and metric spaces. Later topics may include the classification of surfaces (such as the Klein bottle and Möbius band), elementary knot theory, or the fundamental group and covering spaces.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Martin Kassabov (mdk35)
Full details for MATH 4530 : Introduction to Topology
MATH 4560 Geometry of Discrete Groups

An introduction to the geometric approach to the theory of infinite discrete groups. Topics include group actions, the construction of Cayley graphs, connections to formal language theory, actions on trees, volume growth, and large-scale geometry. Theorems are balanced by specific examples such as Baumslag-Solitar groups, the Lamplighter group, and Thompson's groups.

Distribution: (SMR-AS)
Academic Career: UG Instructor: Jason Manning (jm882)
Full details for MATH 4560 : Geometry of Discrete Groups
MATH 4710 Basic Probability

Introduction to probability theory, which prepares the student to take MATH 4720. The course begins with basics: combinatorial probability, mean and variance, independence, conditional probability, and Bayes formula. Density and distribution functions and their properties are introduced. The law of large numbers and the central limit theorem are stated and their implications for statistics are discussed.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Laurent Saloff-Coste (lps2)
Full details for MATH 4710 : Basic Probability
MATH 4810 Mathematical Logic

First course in mathematical logic providing precise definitions of the language of mathematics and the notion of proof (propositional and predicate logic). The completeness theorem says that we have all the rules of proof we could ever have. The Gödel incompleteness theorem says that they are not enough to decide all statements even about arithmetic. The compactness theorem exploits the finiteness of proofs to show that theories have unintended (nonstandard) models. Possible additional topics: the mathematical definition of an algorithm and the existence of noncomputable functions; the basics of set theory to cardinality and the uncountability of the real numbers.

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Anil Nerode (an17)
Full details for MATH 4810 : Mathematical Logic
MATH 4860 Applied Logic

Propositional and predicate logic, compactness and completeness using tableaux, natural deduction, and/or resolution. Other topics chosen from the following: Equational logic. Herbrand Universes and unification. Rewrite rules and equational logic, Knuth-Bendix method, and the congruence-closure algorithm and lambda-calculus reduction strategies. Modal logics, intuitionistic logic, Prolog, LISP, ML, or Nuprl. Applications to expert systems and program verification. Noncomputability (Turing) and incompleteness (Gödel).

Distribution: (MQR-AS, SMR-AS)
Academic Career: UG Instructor: Bob Constable (rlc7)
Full details for MATH 4860 : Applied Logic
MATH 4900 Supervised Research

An independent research course by arrangement with an individual professor.  The goal is for the student to perform an independent investigation into a specific mathematical question.  The student and professor will set expectations and grading policies at the beginning of the term.

Academic Career: UG Instructor: Marcelo Aguiar (ma18)
Full details for MATH 4900 : Supervised Research
MATH 4901 Supervised Reading

An independent reading course by arrangement with an individual professor. The goal is for the student to master a body of mathematics outside the normal curriculum. The student and professor will set expectations and grading policies at the beginning of the term.

Academic Career: UG Instructor: Marcelo Aguiar (ma18)
Full details for MATH 4901 : Supervised Reading
MATH 5080 Special Study for Teachers

Examines principles underlying the content of the secondary school mathematics curriculum, including connections with the history of mathematics, technology, and mathematics education research. One credit is awarded for attending three of the four Saturday workshops per year. Other credit options are available by permission of instructor for students completing additional work (e.g., independent study projects or presentations).

Academic Career: GR Instructor: Mary Ann Huntley (mh688)
Full details for MATH 5080 : Special Study for Teachers
MATH 6110 Real Analysis

This course covers measure and integration and functional analysis.

Academic Career: GR Instructor: Philippe Sosoe (ps934)
Full details for MATH 6110 : Real Analysis
MATH 6210 Measure Theory and Lebesgue Integration

Covers measure theory, integration, and Lp spaces.

Academic Career: GR Instructor: Terence Harris (tlh236)
Full details for MATH 6210 : Measure Theory and Lebesgue Integration
MATH 6260 Dynamical Systems

Topics include existence and uniqueness theorems for ODEs; Poincaré-Bendixon theorem and global properties of two dimensional flows; limit sets, nonwandering sets, chain recurrence, pseudo-orbits and structural stability; linearization at equilibrium points: stable manifold theorem and the Hartman-Grobman theorem; and generic properties: transversality theorem and the Kupka-Smale theorem. Examples include expanding maps and Anosov diffeomorphisms; hyperbolicity: the horseshoe and the Birkhoff-Smale theorem on transversal homoclinic orbits; rotation numbers; Herman's theorem; and characterization of structurally stable systems.

Academic Career: GR Instructor: William Clark (wac76)
Full details for MATH 6260 : Dynamical Systems
MATH 6310 Algebra

MATH 6310-MATH 6320 are the core algebra courses in the mathematics graduate program. MATH 6310 covers group theory, especially finite groups; rings and modules; ideal theory in commutative rings; arithmetic and factorization in principal ideal domains and unique factorization domains; introduction to field theory; tensor products and multilinear algebra. (Optional topic: introduction to affine algebraic geometry.)

Academic Career: GR Instructor: David Zywina (djz44)
Full details for MATH 6310 : Algebra
MATH 6330 Noncommutative Algebra

An introduction to the theory of noncommutative rings and modules. Topics vary by semester and include semisimple modules and rings, the Jacobson radical and Artinian rings, group representations and group algebras, characters of finite groups, representations of the symmetric group, central simple algebras and the Brauer group, representation theory of finite-dimensional algebras, Morita theory.

Academic Career: GR Instructor: Marcelo Aguiar (ma18)
Full details for MATH 6330 : Noncommutative Algebra
MATH 6340 Commutative Algebra with Applications in Algebraic Geometry

Covers Dedekind domains, primary decomposition, Hilbert basis theorem, and local rings.

Academic Career: GR Instructor: Irena Peeva (ivp1)
Full details for MATH 6340 : Commutative Algebra with Applications in Algebraic Geometry
MATH 6520 Differentiable Manifolds

MATH 6510-MATH 6520 are the core topology courses in the mathematics graduate program. MATH 6520 is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle, and a section of the tangent bundle is a vector field. Alternatively, vector fields can be viewed as first-order differential operators. Students study flows of vector fields and prove the Frobenius integrability theorem. In the presence of a Riemannian metric, the notions of parallel transport, curvature, and geodesics are development. Students examine the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits, de Rham cohomology, Morse theory, or other optional topics are introduced.

Academic Career: GR Instructor: James West (jew13)
Full details for MATH 6520 : Differentiable Manifolds
MATH 6530 K-Theory and Characteristic Classes

An introduction to topological K-theory and characteristic classes. Topological K-theory is a generalized cohomology theory which is surprisingly simple and useful for computation while still containing enough structure for proving interesting results. The class will begin with the definition of K-theory, Chern classes, and the Chern character. Additional topics may include the Hopf invariant 1 problem, the J-homomorphism, Stiefel-Whitney classes and Pontrjagin classes, cobordism groups and the construction of exotic spheres, and the Atiyah-Singer Index Theorem.

Academic Career: GR Instructor: Inna Zakharevich (iiz5)
Full details for MATH 6530 : K-Theory and Characteristic Classes
MATH 6630 Symplectic Geometry

Symplectic geometry is a branch of differential geometry which studies manifolds endowed with a nondegenerate closed 2-form. The field originated as the mathematics of classical (Hamiltonian) mechanics and it has connections to (at least!) complex geometry, algebraic geometry, representation theory, and mathematical physics. In this introduction to symplectic geometry, the class will begin with linear symplectic geometry, discuss canonical local forms (Darboux-type theorems), and examine related geometric structures including almost complex structures and Kähler metrics. Further topics may include symplectic and Hamiltonian group actions, the orbit method, the topology and geometry of momentum maps, toric symplectic manifolds, Hamiltonian dynamics, symplectomorphism groups, and symplectic embedding problems.

Academic Career: GR Instructor: Tara Holm (tsh33)
Full details for MATH 6630 : Symplectic Geometry
MATH 6710 Probability Theory I

A mathematically rigorous course in probability theory which uses measure theory but begins with the basic definitions of independence and expected value in that context. Law of large numbers, Poisson and central limit theorems, and random walks.

Academic Career: GR Instructor: Lionel Levine (ll432)
Full details for MATH 6710 : Probability Theory I
MATH 6740 Mathematical Statistics II

Focuses on the modern theory of statistical inference, with an emphasis on nonparametric and asymptotic methods. Topics include empirical Bayes and shrinkage estimators,  unbiased estimation of risk, adaptive estimation, as well as oracle inequalities, a powerful concept which has  applications in classification and machine learning. An optional topic is the use of Markov random fields for image restoration. The course includes a discussion of the general asymptotic theory for statistical models, as a tool for finding optimal decisions, based on the concepts of contiguity and local asymptotic normality.

Academic Career: GR Instructor: Michael Nussbaum (mn66)
Full details for MATH 6740 : Mathematical Statistics II
MATH 6870 Set Theory

First course in axiomatic set theory at the level of the book by Kunen.

Academic Career: GR Instructor: Slawomir Solecki (ss3777)
Full details for MATH 6870 : Set Theory
MATH 7150 Fourier Analysis

An introduction to (mostly Euclidean) harmonic analysis. Topics usually include convergence of Fourier series, harmonic functions and their conjugates, Hilbert transform, Calderon-Zygmund theory, Littlewood-Paley theory, pseudo-differential operators, restriction theory of the Fourier transform, connections to PDE. Applications to number theory and/or probability theory may also be discussed, as well as Fourier analysis on groups.

Academic Career: GR Instructor: Camil Muscalu (fm69)
Full details for MATH 7150 : Fourier Analysis
MATH 7290 Seminar on Scientific Computing and Numerics

Talks on various methods in scientific computing, the analysis of their convergence properties and computational efficiency, and their adaptation to specific applications.

Academic Career: GR Instructor: Anil Damle (asd239)
Alex Townsend (ajt253)
Full details for MATH 7290 : Seminar on Scientific Computing and Numerics
MATH 7510 Berstein Seminar in Topology

A seminar on an advanced topic in topology or a related subject. Content varies. The format is usually that the participants take turns to present.

Academic Career: GR Instructor: Jason Manning (jm882)
Full details for MATH 7510 : Berstein Seminar in Topology
MATH 7610 Topics in Geometry

Selection of advanced topics from modern geometry. Content varies.

Academic Career: GR Instructor: John Hubbard (jhh8)
Full details for MATH 7610 : Topics in Geometry
MATH 7670 Topics in Algebraic Geometry

Selection of topics from algebraic geometry. Content varies.  

Academic Career: GR Instructor: Michael Stillman (mes15)
Full details for MATH 7670 : Topics in Algebraic Geometry
MATH 7710 Topics in Probability Theory

Selection of advanced topics from probability theory. Content varies.

Academic Career: GR Instructor: Lionel Levine (ll432)
Full details for MATH 7710 : Topics in Probability Theory
MATH 7740 Statistical Learning Theory: Classification, Pattern Recognition, Machine Learning

The course aims to present the developing interface between machine learning theory and statistics. Topics include an introduction to classification and pattern recognition; the connection to nonparametric regression is emphasized throughout. Some classical statistical methodology is reviewed, like discriminant analysis and logistic regression, as well as the notion of perception which played a key role in the development of machine learning theory. The empirical risk minimization principle is introduced, as well as its justification by Vapnik-Chervonenkis bounds. In addition, convex majoring loss functions and margin conditions that ensure fast rates and computable algorithms are discussed. Today's active high-dimensional statistical research topics such as oracle inequalities in the context of model selection and aggregation, lasso-type estimators, low rank regression and other types of estimation problems of sparse objects in high-dimensional spaces are presented.

Academic Career: GR Instructor: Marten Wegkamp (mhw73)
Full details for MATH 7740 : Statistical Learning Theory: Classification, Pattern Recognition, Machine Learning
MATH 7810 Seminar in Logic

A twice weekly seminar in logic. Typically, a topic is selected for each semester, and at least half of the meetings of the course are devoted to this topic with presentations primarily by students. Opportunities are also provided for students and others to present their own work and other topics of interest.

Academic Career: GR Instructor: Justin Moore (jtm237)
Full details for MATH 7810 : Seminar in Logic
MATH 7900 Supervised Reading and Research

Supervised research for the doctoral dissertation.

Academic Career: GR Instructor: Karola Meszaros (km626)
Full details for MATH 7900 : Supervised Reading and Research