Banach Algebras (Modern Analytic and Computational Methods in Science and Mathematics)

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Contents

  1. Mathematics
  2. Graduate Program Course Offerings | Department of Mathematics | University of Pittsburgh
  3. chapter and author info

Topics will include spectral theory of self-adjoint mappings, calculus of matrix valued functions, matrix inequalities, convexity, duality theorem and normed linear spaces. The course is the first term of a two-term graduate algebra sequence. It covers the theory of groups, the theory of fields as well as Galois theory. Highlights of the course will include: Sylow's theorems, the structure of finitely generated abelian groups, the fundamental theorem of Galois theory, and applications such as the solvability of polynomial equations by radicals and geometric constructions with a ruler and a compass.

This is a third course in algebra. It covers the classical results on the structure and representation theory of associative algebras culminating with modern developments such as the theory of quiver algebras and categorification. Highlights of the course include: Wedderburn-Artin theory, the structure of central simple algebras, the structure of finite dimensional algebras, semisimple algebras, the character theory of finite groups, theorems of Maschke, Frobenius, Burnside, quiver algebras and quiver representations, reflection functors, Gabriel's theorem, tensor categories, fusion categories.

This is a fourth course in algebra. It covers introductory topics in algebraic geometry, number theory, and representation theory, selected by the instructor. This course studies the mathematical analysis and practical implementation of discontinuous Galerkin methods for approximating elliptic, parabolic, and hyperbolic partial differential equations. The course will cover several topics in discretizations of partial differential equations and the solution of the resulting algebraic systems.

Topics in discretizations will include mixed finite element methods, finite volume methods, mimetic finite difference methods, and local discontinuous Galerkin methods. Topics in solvers will include domain decomposition methods and multigrid methods. Applications to flow and transport in porous media, as well as coupled fluid and porous media flows will be discussed. The Advanced Scientific Computing sequence covers topics chosen at the leading edge of current computational science and engineering for which there is sufficient interest. The course focuses on the fundamental mathematical aspects of numerical methods for stochastic differential equations, motivated by applications in physics, engineering, biology, economics.

It provides a systematic framework for an understanding of the basic concepts and of the basic tools needed for the development and implementation of numerical methods for SDEs, with focus on time discretization methods for initial value problems of SDEs with Ito diffusions as their solutions. The course material is self-contained. The topics to be covered include background material on probability, stochastic processes and statistics, introduction to stochastic calculus, stochastic differential equations and stochastic Taylor expansions.

The numerical methods for time discretization of ODEs are briefly reviewed, then methods for time discretization for SDEs are introduced and analyzed. A first course in differential geometry. Topics may include the geometry of curves and surfaces eg. Gauss map, fundamental forms, curvature , differentiable manifolds, Lie groups, tangent and tensor bundles, vector fields, and Riemannian structures. This course is a continuation of Differential Geometry 1. Applications may also be included. This course will cover Sobolev spaces, second order elliptic equations, weak solutions, linear evolution equations, semigroup theory, and Hamilton-Jacobi theory, and other topics in nonlinear PDE.

PDE 1 Math is not a pre-requisite but a good background in analysis is necessary. This is the first course in a two-term sequence designed to acquaint students with the fundamental ideas involved in the study of ordinary differential equations. Basic existence and uniqueness of solutions as well as dependence on parameters will be presented. Students will also be introduced to geometric concepts such as stability of fixed points and invariance.

This first term will provide an excellent introduction to ODE theory for students interested in applied mathematics. This course, which follows Math , presents a dynamical systems approach to the study of ordinary differential equations. Topics include geometric theory including proofs of invariant manifold theorems, flows on center manifolds and local bifurcation theory, the method of averaging, Melnikov's method, and an introduction to Smale horseshoes and chaos theory.

The prerequisite for this course is a one undergraduate semester course in complex variables with a grade of B or higher. The grade will be determined from assigned homework problems. This course covers methods that are useful for solving or approximating solutions to problems frequently arising in applied mathematics, including certain theory and techniques relating to the spectral theory of matrices, integral equations, differential operators and distributions, regular perturbation theory, and singular perturbation theory.

This course is for all graduate students not under the direct supervision of a specific faculty member.


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In addition to a student's formal course load, this study is for preparation for the preliminary, comprehensive and overview examinations. This course will introduce students to the subject of calculus of variations and some of its modern applications. Topics to be covered include necessary and sufficient conditions for weak and strong extrema, Hamiltonian vs Lagrangian formulations, principle of least action, conservation laws and direct methods of calculus of variations.

Extensions to the functionals involving higher-order derivatives, variable regions and multiple integrals will be considered. The course will emphasize applications of these ideas to numerical analysis, mechanics and control theory. Prerequisite s : single-variable and multivariable calculus, some exposure to ordinary and partial differential equations.

All other concepts, such as function spaces and the necessary background for the applications, will be introduced in the course. Beginning graduate students and advanced undergraduates are welcome. The course objective is to introduce students to formulating, debugging and solving finite element simulations of practical applications, with a focus on the equations of fluid flow. FEniCS is less tightly integrated, consisting of a collection of functions for specifying the mathematical formulation as well as functions for interfacing with other packages for mesh generation, post processing, and numerical solution.

This course will focus on using Python. Python is a widely-used language with applications far removed from finite element modelling and can be the subject of multiple-semester courses. Although previous experience with Python would be valuable, it is not necessary. The basics of the language plus those features necessary for this course will be presented during the lectures.


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Previous experience with finite element methods will be valuable, but is not required because the theory will be summarized during the lectures. Various boundary conditions and finite elements will be presented, as well as the effect of these choices on solution methods. This course is an introduction to the theoretical and computational aspects of the finite element method for the solution of boundary value problems for partial differential equations.

Emphasis will be on linear elliptic, self-adjoint, second-order problems, and some material will cover time dependent problems as well as nonlinear problems.

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Mathematics

Topics include: Sobolev spaces, variational formulation of boundary value problems, natural and essential boundary conditions, Lax-Milgram lemma, approximation theory, error estimates, element construction, continuous, discontinuous, and mixed finite element methods, and solution methods for the resulting finite element systems. Prerequisite s : Good undergraduate background in linear algebra and advanced calculus.

Familiarity with partial differential equations will be useful. This course provides an introduction to the mathematical subjects required for the mathematical finance program, and assumes that the student has an undergraduate degree with some technical component e. Students are expected to have knowledge of Multivariable Calculus and Linear Algebra, and any sections on these topics will be presented as review. No financial background is required, but many of the examples and llustrations of the mathematics will be drawn from economics and finance.

The course with its pre-sequel MATH present fundamental principles and standard approaches used in mathematical finance. This course will investigate the mathematical modeling, theory and computational methods in modern finance. The main topics will be i basic portfolio theory and optimization, ii the concept of risk versus return and the degree of efficiency of markets, iii discrete models in options. This course describes a number of topics related to mathematical biology.

This year we will cover several areas of interest including pattern formation in reaction-diffusion and advection models with applications to immunology, chemotaxis, etc; evolutionary dynamics such as the evolution of cooperation, some game theory, and replicator dynamics; and some cell physiology modeling such as the cell cycle and simple circadian models.

The prerequisites are some simple differential equations, a bit of Fourier transforms, and some knowledge of software to numerically solve the various equations. The main goal of the course is to understand the structure and classification of complex semisimple Lie algebras as well as their basic representation theory and the relationship with Lie groups. Highlights will include, the theorems of Engel, Cartan and Weyl, root systems, the Harish-Chandra isomorphism and various formulas for characters and weight multiplicities.

This course covers special topics in pure mathematics. The subject matter varies each semester. Topics may include the theory of modular forms, automorphic representations, Galois representations, class field theory, trace formula, special topics in algebraic geometry, the Langland's program, discrete and algorithmic geometry, motivic integration, and the formalization of pure mathematics. Cohomology is an important concept and tool in various areas of pure mathematics, such as topology, differential geometry, algebraic geometry, and representation theory.

This course will start with rational and integral cohomology and then move to survey generalizations such as: topological and algebraic K-theory and elliptic cohomology. We will also describe equivariant and twisted versions. Along the way many techniques and tools will be explained, such as: spectral sequences, mapping spaces, homotopy computations, classification of bundles, topology of Lie groups and of their classifying spaces. As time permits, the associated higher geometric and categorical structures will also be discussed. This experience is to be an integral part of the students individual course of study.

This course is for students normally beyond their first year of graduate study who wish to study in an area not available in a formal course. The work must be under the direct supervision of a faculty member who has approved the proposed work in advance of registration. A brief description of the work should be recorded in the student's file in the department. The theory of water waves embodies the equations of fluid mechanics, the concepts of wave propagation, and the critically important role of boundary dynamics.

It has been a subject of intense research since Euler's derivation of the equations of hydrodynamics. The second part focuses more on the analysis perspective of the water wave equations. Zakharov's Hamiltonian formulation of the irrotational water wave problem will be dis- cussed and applications of this formulation to the issue of wellposedness will be outlined. The course also use some of the asymptotic integrable models as an example to describe the wave-breaking phenomenon.

Graduate Program Course Offerings | Department of Mathematics | University of Pittsburgh

Finally, a particular ow pattern, namely the traveling or steady waves will be addressed. The method of calculus of variation, global bifurcation theory, topological degree theory, Schauder estimates, and Fredholm theory will be introduced to establish the existence of such waves. Mathematically, the above mentioned topics draw on deep ideas from applied mathematics, analysis, and PDEs. The main ingredients and techniques involved include Fourier analysis, harmonic analysis, elliptic theory, and more. Prior exposure to any of these will be helpful but not necessary.

Skip to main content. University of Pittsburgh. Graduate Program Course Offerings Approximately 35 courses that constitute the department's regular graduate curriculum are offered either annually or biennially. Selected Level Undergraduate Courses Class Course Description Course prerequisites MATH Putnam Seminar The aim of this course is to develop the capacity to solve mathematical problems involving a substantial element of ingenuity and perseverance.

Training will involve the study of problems from previous Putnam competitions, for which this course can be regarded as a useful preparation. An attempt will be made to look for unifying mathematical ideas. General strategies for solving problems will also be discussed. Some applications of number theory will be covered in the course. Special emphasis will be placed on public key cryptosystems, including elliptic curve based systems. Real world applications such as browser security and bitcoin will be discussed. MATH MATH Combinatorial Mathematics Topics covered include the binomial theorem, inclusion exclusion principle, recurrence relations, generating functions, and coloring problems.

MATH or or MATH Classical Numerical Analysis This course, with MATH forms a two term introduction to numerical analysis at the advanced undergraduate level and includes interpolation, numerical differentiation and integration, solution of non-linear equations, numerical solution of systems or ordinary differential equations, and additional topics as time permits. Emphasis is on understanding the algorithms rather than on detailed coding, although some programming will be required. MATH Numerical Linear Algebra This course, with MATH forms a two term introduction to numerical analysis at the advanced undergraduate level and includes interpolation, numerical differentiation and integration, solution of non-linear equations, numerical solution of systems or ordinary differential equations, and additional topics as time permits.

Applications will be emphasized, but some theory will be addressed and proofs will be discussed. As well, students will be taught how to use available software to answer questions. Course topics will include linear programming, integer programming, nonlinear programming, convex and affine sets, convex and concave functions, unconstrained optimization, and combinatorial optimization i.

Network flow problems. Topics covered include physical interpretation of a mathematical model, use of library software, preparation of software, analysis of results, and reporting on findings. It prepares students for the probability exam offered by the Society of Actuaries. Specifically it will present the relevant topics in the theory of interest interest and discount rates, cash flows, annuities, amoritization and sinking funds, bonds and investment stocks, capital asset pricing model, arbitrage pricing theory, portfolios, options.

The material will be presented in the traditional academic format of lectures and help sessions along with optional sessions directed specifically at preparing students for the SOA exam. Specifically it will present the relevant topics in life insurance and life annuities, including multiple decrement models as well as the black and Scholes pricing of derivative securities and risk analysis. Abstract vector spaces, linear transformations, and matrix representations will be studied. Some applications and generalizations will also be investigated. The emphasis is on theory with examples.

Phase plane techniques, perturbation methods, and bifurcation theory are studied. Topics include paths, circuits, trees, planar graphs, coloring problems, digraphs, matching theory, and network flows. MATH or MATH MATH Introduction to Differential Geometry Possible topics are the basic ideas of topology, description of curves in space, definition and local study of smooth surfaces in Euclidean space fundamental forms, geodesics, and curvature , global properties of surfaces, gauss-bonnet formula and applications.

The emphasis is on the model-building process and on developing an understanding of some of the unifying themes of applied mathematics such as equilibria, stability, conservation laws, etc. The material is presented in the form of case studies. Math or Math or Math MATH Topics in Mathematical Modeling This course provides honors recognition to students who wish to digress with the professor to study extra topics not covered in mathematics , mathematical modelling. MATH Modeling in Applied Mathematics 2 This course presents contemporary mathematical theories of neuroscience, including single neurons and neuronal networks.

Attention will be given to the dynamics and the function of neural activity. Models using calculus, ordinary differential equations, partial differential equations, discrete dynamical systems, stochastic dynamics, or a cellular automata framework will be presented and principal methods for their analysis will be described.

Throughout the course, students will have extensive opportunities to practice the development and analysis of mathematical biology models. The objectives of the course are to provide students with the techniques necessary for the formulation and solution of problems involving PDE's and to prepare for further study in PDE's. The three main types of second order linear PDE's - parabolic, elliptic, and hyperbolic are studied. In addition the tools necessary for the solution of PDE's such as Fourier series and Laplace transforms are introduced.

Topics include Fourier transform, maximum principles, and existence, uniqueness and regularity of solutions to PDES. Major topics include random variables, expecation, characteristic functions, conditional probability, and an introduction to Martingales and Markov chains. Math or Math , and one of , or MATH Advanced Calculus 1 This course contains a rigorous development of the calculus of functions of a single variable, including compactness on the real line, continuity, differentiability, integration, and the uniform convergence of sequences and series of functions.

Other topics may be included, such as the notion of limits and continuity in metric spaces. There will be an emphasis upon problem solving and applications in electromagnetic theory and fluid flow. It is intended for students with a basic knowledge of real analysis including uniform convergence of sequences and series of functions. No knowledge of the Lebasque integral is assumed. One of Math , Math AND one of Math , Math , Math MATH Computer Methods Laboratory This course will introduce students to the use of the computers in our undergraduate computing laboratory and to the numerical methods, symbolic algebra, and computer graphics which are helpful in the study of our upper level subjects in applied mathematics.

Basic notions will be applied to obtain the fundamental existence theorem for first order ordinary differential equations. The course will be run on a theorem proving and problem solving basis. The topic will change each time the course is offered, and will generally reflect current interests of the faculty or a recent trend in mathematics.

MATH Progress in Mathematics This course will deepen the students' understanding of analysis through intensive training in problem solving followed by comprehensive study and dissection of the problems attempted. In particular, enumerative problems are selected by the instructor. MATH Codes and Designs A unified theory of linear codes, combinatorial design and statistical design will be presented.

Hamming, BCH, Golay, quadratic residue and other classes of codes will be constructed. Several decoding schemes, including decoding by way of locator polynomials, will be done. Applications to statistical design and quality control will be described in detail. MATH Numerical Solution of Ordinary Differential Equations 3 Credits This course is an introduction to modern methods for the numerical solution of initial and boundary value problems for systems of ordinary differential equations and differential algebraic equations. MATH Set Theory This is an introductory course intended to prepare the students for the various applications of set theory.

The contents include basic axioms, Boole algebra of classes, algebra of relations, ordinals, transfinite induction, equivalents of the axiom of choice, and of the continuum hypothesis. MATH Logic and Foundations The contents of this course include the propositional calculus, the predicate calculus, model theory and their applications to mathematical systems.

Topics include metric spaces, measures, fractal dimensions, discrete dynamical systems and multifractal analysis. MATH Functions of Several Variables This course covers topics of the calculus of several variables from a more general and theoretical point of view. Topics include convergence, continuity, compactness, inverse and implicit function theorems and differential forms. In particular, the theory of Lebasque integral and some of its many ramifications are studied. This leads to differentiation and integration theories for real valued functions.

Topics include elements of the theory of Banach and Hilbert spaces, Radon Nikodym theorem, duality for the LP spaces, product measures, Fubini's theorem and differentiation. Topics include analytic functions, Cauchy-Riemann equations, elementary conformal mappings, Poisson formula, Taylor and Laurent series, argument principle, residues and the classifications of singularities. The major topics are normal families, the Riemann mapping theorem and harmonic functions. Selected topics, by the instructor, will also be covered. MATH Analytic Number Theory Some of the topics covered in this course include residue classes, unique prime factorization, character mod n, the Riemann zeta function and its analytic continuation, poles and functional equations.

Also the prime number theorem and Dirichlet theorem will be covered. MATH Algebraic Number Theory In this course the theory of numbers will be algebraically, that is, as the study of algebraic numbers. Particular attention will be paid to the development of quadratic and cyclotomic number fields and to factorization theorems. Other topics include ideal theory and the work of Kummer and Fermat's last theorem. Topics include harmonic functions, Poisson integral, maximum principle, classical Dirichlet problem, Harnack's inequality, boundary limit theorem, super harmonic functions and their properties and green's potentials.

Topics include Poisson formula, boundary behavior and Fatou's theorem, factorization theorems and Blaschke products. MATH Analysis 2 3 Credits In this course we continue on from Analysis 1 in Fall Lebesgue measure and integration, and some functional analysis , stirring Fourier analysis, complex analysis, functional analysis and more real analysis into the mix Topics include in that order : 1. Basic theory of Banach and Hilbert spaces. Bounded operators in Banach and Hilbert spaces.

Orthonormal bases in Hilbert spaces, Fourier series and spherical harmonics. Hahn-Banach theorem, separation of convex sets. Reflexive spaces. Weak topology, Tychonov's theorem and Banach-Alaoglu theorem in the general case. Spectral theorem for compact self-adjoint operators. Sobolev spaces and the eigenfunctions of the Laplace operator. Banach algebras. MATH Functional Analysis 1 A first course in the area, the emphasis of the course will be on normed linear spaces and linear operations, their basic properties will be discussed. Also, inner product spaces and their properties will be covered.

Topics to be covered include the spectral theory, compact operators, and distribution theory and Sobolev spaces. The major topic is the Fourier transform and Fourier series. In particular, various kernels and pointwise sum ability of Fourier series will be discussed. MATH Computational Approximation Theory Topics include fundamental theorems, polynomial approximation, splines, surface approximation, domain transformations and applications to science and engineering.

MATH Matrix Groups The course is an introduction to some of the concepts of lie groups and lie algebra--all done at concrete level of matrix groups. It centers around the isomorphism questions on matrix groups of small dimensions, and it leads to maximal torus, Clifford algebra, and Weyl group. Topology of surfaces. Brief summary of applications to biology, chemistry, and physics. MDL is a discovery-based project course in mathematics.

Students work independently in small groups to explore open-ended mathematical problems and discover original mathematics. Students formulate conjectures and hypotheses; test predictions by computation, simulation, or pure thought; and present their results to classmates. No lecture component; in-class meetings reserved for student presentations, attendance mandatory.

Motivated students with any level of mathematical background are encouraged to apply. Linear algebra for applications in science and engineering: orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares, the condition number of a matrix, algorithms for solving linear systems. Functions of a Complex Variable. Complex numbers, analytic functions, Cauchy-Riemann equations, complex integration, Cauchy integral formula, residues, elementary conformal mappings.

Prerequisite: An introductory course in graph theory establishing fundamental concepts and results in variety of topics. Topics include: basic notions, connectivity, cycles, matchings, planar graphs, graph coloring, matrix-tree theorem, conditions for hamiltonicity, Kuratowski's theorem, Ramsey and Turan-type theorem. Prerequisites: 51 or equivalent and some familiarity with proofs is required. Introduction to Combinatorics and Its Applications.

Topics: graphs, trees Cayley's Theorem, application to phylogony , eigenvalues, basic enumeration permutations, Stirling and Bell numbers , recurrences, generating functions, basic asymptotics. Prerequisites: 51 or equivalent. Applications of the theory of groups. Topics: elements of group theory, groups of symmetries, matrix groups, group actions, and applications to combinatorics and computing. Applications: rotational symmetry groups, the study of the Platonic solids, crystallographic groups and their applications in chemistry and physics.

Honors math majors and students who intend to do graduate work in mathematics should take Applied Number Theory and Field Theory. Number theory and its applications to modern cryptography. Topics: congruences, finite fields, primality testing and factorization, public key cryptography, error correcting codes, and elliptic curves, emphasizing algorithms. Linear Algebra and Matrix Theory. Algebraic properties of matrices and their interpretation in geometric terms. The relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations.

Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; eigenvectors and eigenvalues; diagonalization. Introduction to Scientific Computing. Introduction to Scientific Computing Numerical computation for mathematical, computational, physical sciences and engineering: error analysis, floating-point arithmetic, nonlinear equations, numerical solution of systems of algebraic equations, banded matrices, least squares, unconstrained optimization, polynomial interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations, truncation error, numerical stability for time dependent problems and stiffness.

Functions of a Real Variable. The development of real analysis in Euclidean space: sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Analytic functions, Cauchy integral formula, power series and Laurent series, calculus of residues and applications, conformal mapping, analytic continuation, introduction to Riemann surfaces, Fourier series and integrals. Prerequisites: 52, and or Notions of analysis and algorithms central to modern scientific computing: continuous and discrete Fourier expansions, the fast Fourier transform, orthogonal polynomials, interpolation, quadrature, numerical differentiation, analysis and discretization of initial-value and boundary-value ODE, finite and spectral elements.

Recommended for Mathematics majors and required of honors Mathematics majors. Similar to but altered content and more theoretical orientation. Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Unique factorization domains. Field of fractions, splitting fields, separability, finite fields. Galois groups, Galois correspondence, examples and applications. Modules and Group Representations.

Modules over PID. Tensor products over fields. Group representations and group rings. Maschke's theorem and character theory. Character tables, construction of representations. Also recommended: Partial Differential Equations. An introduction to PDE; particularly suitable for non-Math majors. Topics include physical examples of PDE's, method of characteristics, D'Alembert's formula, maximum principles, heat kernel, Duhamel's principle, separation of variables, Fourier series, Harmonic functions, Bessel functions, spherical harmonics.

Introduction to measure theory, Lp spaces and Hilbert spaces. Random variables, expectation, conditional expectation, conditional distribution. Uniform integrability, almost sure and Lp convergence. Stochastic processes: definition, stationarity, sample path continuity.

Examples: random walk, Markov chains, Gaussian processes, Poisson processes, Martingales. Construction and basic properties of Brownian motion. Mathematical Methods of Classical Mechanics. Newtonian mechanics. Lagrangian formalism. Noether's theorem. Rigid bodies. Introduction to symplectic geometry. Hamiltonian formalism. Legendre transform. Variational principles. Geometric optics. Introduction to the theory of integrable systems.

Mathematically rigorous introduction to the classical N-body problem: the motion of N particles evolving according to Newton's law. Topics include: the Kepler problem and its symmetries; other central force problems; conservation theorems; variational methods; Hamilton-Jacobi theory; the role of equilibrium points and stability; and symplectic methods.

Prerequisites: 53, and or Geometry of curves and surfaces in three-space and higher dimensional manifolds. Parallel transport, curvature, and geodesics. Surfaces with constant curvature. Minimal surfaces. An introduction to the methods and concepts of algebraic geometry. The point of view and content will vary over time, but include: affine varieties, Hilbert basis theorem and Nullstellensatz, projective varieties, algebraic curves. Required: Strongly recommended: additional mathematical maturity via further basic background with fields, point-set topology, or manifolds.

Differentiable manifolds, tangent space, submanifolds, implicit function theorem, differential forms, vector and tensor fields. Frobenius' theorem, DeRham theory. Prerequisite: 62CM or 52 and familiarity with linear algebra and analysis arguments at the level of and respectively. Smooth manifolds, transversality, Sards' theorem, embeddings, degree of a map, Borsuk-Ulam theorem, Hopf degree theorem, Jordan curve theorem. Prerequisite: or Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots.

Introduction to Probability Theory. Counting; axioms of probability; conditioning and independence; expectation and variance; discrete and continuous random variables and distributions; joint distributions and dependence; central limit theorem and laws of large numbers. Prerequisite: 52 or consent of instructor. Elementary Theory of Numbers. Euclid's algorithm, fundamental theorems on divisibility; prime numbers; congruence of numbers; theorems of Fermat, Euler, Wilson; congruences of first and higher degrees; quadratic residues; introduction to the theory of binary quadratic forms; quadratic reciprocity; partitions.

Properties of number fields and Dedekind domains, quadratic and cyclotomic fields, applications to some classical Diophantine equations. Prerequisites: and , especially modules over principal ideal domains and Galois theory of finite fields. Introduction to Dirichlet series and Dirichlet characters, Poisson summation, Gauss sums, analytic continuation for Dirichlet L-functions, applications to prime numbers e.

Calculus of random variables and their distributions with applications. Review of limit theorems of probability and their application to statistical estimation and basic Monte Carlo methods. Introduction to Markov chains, random walks, Brownian motion and basic stochastic differential equations with emphasis on applications from economics, physics and engineering, such as filtering and control.

Prerequisites: exposure to basic probability. Discrete Probabilistic Methods. Modern discrete probabilistic methods suitable for analyzing discrete structures of the type arising in number theory, graph theory, combinatorics, computer science, information theory and molecular sequence analysis. Typically in alternating years. Informal and axiomatic set theory: sets, relations, functions, and set-theoretical operations. The Zermelo-Fraenkel axiom system and the special role of the axiom of choice and its various equivalents.

Well-orderings and ordinal numbers; transfinite induction and transfinite recursion. Equinumerosity and cardinal numbers; Cantor's Alephs and cardinal arithmetic. Open problems in set theory. Prerequisite: students should be comfortable doing proofs. Fundamental Concepts of Analysis. Properties of Riemann integrals, continuous functions and convergence in metric spaces; compact metric spaces, basic point set topology. Prerequisite: 61CM or 61DM or or consent of the instructor. Lebesgue Integration and Fourier Analysis. Similar to A, but for undergraduate Math majors and graduate students in other disciplines.

Prerequisite: or consent of instructor. Theory of Partial Differential Equations. A rigorous introduction to PDE accessible to advanced undergraduates. Elliptic, parabolic, and hyperbolic equations in many space dimensions including basic properties of solutions such as maximum principles, causality, and conservation laws. Methods include the Fourier transform as well as more classical methods.

The Lebesgue integral will be used throughout, but a summary of its properties will be provided to make the course accessible to students who have not had or A. Prerequisite: or equivalent. Elementary Functional Analysis. Linear operators on Hilbert space. Spectral theory of compact operators; applications to integral equations. Elements of Banach space theory. Hamiltonian systems and their geometry. Structural stability and hyperbolic dynamical systems. Completely integrable systems.

Perturbation theory. Polya Problem Solving Seminar. Topics in mathematics and problem solving strategies with an eye towards the Putnam Competition. Topics may include parity, the pigeonhole principle, number theory, recurrence, generating functions, and probability. Students present solutions to the class.

Open to anyone with an interest in mathematics. Honors math major working on senior honors thesis under an approved advisor carries out research and reading. Satisfactory written account of progress achieved during term must be submitted to advisor before term ends. May be repeated 3 times for a max of 9 units. Contact department student services specialist to enroll. Only for undergraduate students majoring in mathematics.

Students obtain employment in a relevant industrial or research activity to enhance their professional experience. Students submit a concise report detailing work activities, problems worked on, and key results. May be repeated for credit up to 3 units. Prerequisite: qualified offer of employment and consent of department. Prior approval by Math Department is required; you must contact the Math Department's Student Services staff for instructions before being granted permission to enroll.

For Math majors only. Undergraduates pursue a reading program under the direction of a Math faculty member; topics limited to those not in regular department course offerings. Credit can fulfill the elective requirement for Math majors. Departmental approval required; please contact the Student Services Specialist for the enrollment proposal form at least 2 weeks before the final study list deadline.

May be repeated for credit. Enrollment beyond a third section requires additional approval. Basic measure theory and the theory of Lebesgue integration.

Point set topology, basic functional analysis, Fourier series, and Fourier transform. Prerequisites: and A or equivalent. Basic commutative ring and module theory, tensor algebra, homological constructions, linear and multilinear algebra, canonical forms and Jordan decomposition. Prerequisite: and or equivalent. Continuation of A. Topics in field theory, commutative algebra, algebraic geometry, and finite group representations. Prerequisites: A, and or equivalent. Topics: fundamental group and covering spaces, basics of homotopy theory, homology and cohomology simplicial, singular, cellular , products, introduction to topological manifolds, orientations, Poincare duality.

Prerequisites: , , and Prerequisite: A. This course will be an introduction to Riemannian Geometry. Topics will include the Levi-Civita connection, Riemann curvature tensor, Ricci and scalar curvature, geodesics, parallel transport, completeness, geodesics and Jacobi fields, and comparison techniques. Prerequisites or B.

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Introduction to Algebraic Geometry. Algebraic varieties, and introduction to schemes, morphisms, sheaves, and the functorial viewpoint. Prerequisites: AB or equivalent. Complex Differential Geometry. Complex structures, almost complex manifolds and integrability, Hermitian and Kahler metrics, connections on complex vector bundles, Chern classes and Chern-Weil theory, Hodge and Dolbeault theory, vanishing theorems, Calabi-Yau manifolds, deformation theory.

Partial Differential Equations of Applied Mathematics. First-order partial differential equations; method of characteristics; weak solutions; elliptic, parabolic, and hyperbolic equations; Fourier transform; Fourier series; and eigenvalue problems. Mathematical Methods of Imaging. Image denoising and deblurring with optimization and partial differential equations methods. Imaging functionals based on total variation and l-1 minimization. Fast algorithms and their implementation. Array imaging using Kirchhoff migration and beamforming, resolution theory for broad and narrow band array imaging in homogeneous media, topics in high-frequency, variable background imaging with velocity estimation, interferometric imaging methods, the role of noise and inhomogeneities, and variational problems that arise in optimizing the performance of array imaging algorithms.

Numerical Solution of Partial Differential Equations. Hyperbolic partial differential equations: stability, convergence and qualitative properties; nonlinear hyperbolic equations and systems; combined solution methods from elliptic, parabolic, and hyperbolic problems. Examples include: Burger's equation, Euler equations for compressible flow, Navier-Stokes equations for incompressible flow.

Partial Differential Equations and Diffusion Processes. Parabolic and elliptic partial differential equations and their relation to diffusion processes. First order equations and optimal control. Emphasis is on applications to mathematical finance. Stochastic Methods in Engineering.

The basic limit theorems of probability theory and their application to maximum likelihood estimation. Basic Monte Carlo methods and importance sampling. Markov chains and processes, random walks, basic ergodic theory and its application to parameter estimation. Discrete time stochastic control and Bayesian filtering.

Diffusion approximations, Brownian motion and an introduction to stochastic differential equations. Examples and problems from various applied areas. Prerequisites: exposure to probability and background in analysis. Probability, Stochastic Analysis and Applications. Diffusion approximations, Brownian motion and basic stochastic differential equations. Mathematical tools: sigma algebras, measure theory, connections between coin tossing and Lebesgue measure, basic convergence theorems.

Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Large deviations. Weak convergence; central limit theorems; Poisson convergence; Stein's method. Conditional expectations, discrete time martingales, stopping times, uniform integrability, applications to laws, Radon-Nikodym Theorem, ruin problems, etc. Other topics as time allows selected from i local limit theorems, ii renewal theory, iii discrete time Markov chains, iv random walk theory,n v ergodic theory.

Theory of Probability III. Continuous time stochastic processes: martingales, Brownian motion, stationary independent increments, Markov jump processes and Gaussian processes. Markov and strong Markov property. Infinitely divisible laws. Some ergodic theory. Mathematics and Statistics of Gambling. Probability and statistics are founded on the study of games of chance. Nowadays, gambling in casinos, sports and the Internet is a huge business.

This course addresses practical and theoretical aspects. Topics covered: mathematics of basic random phenomena physics of coin tossing and roulette, analysis of various methods of shuffling cards , odds in popular games, card counting, optimal tournament play, practical problems of random number generation. Prerequisites: Statistics and Topics in Probability: Percolation Theory.

An introduction to first passage percolation and related general tools and models. Topics include early results on shape theorems and fluctuations, more modern development using hyper-contractivity, recent breakthrough regarding scaling exponents, and providing exposure to some fundamental long-standing open problems. Course prerequisite: graduate-level probability. A topics course in combinatorics and related areas.

The topic will be announced by the instructor. Combinatorial estimates and the method of types. Large deviation probabilities for partial sums and for empirical distributions, Cramer's and Sanov's theorems and their Markov extensions. Applications in statistics, information theory, and statistical mechanics. Offered every years. This advanced course in extremal combinatorics covers several major themes in the area. These include extremal combinatorics and Ramsey theory, the graph regularity method, and algebraic methods.

This is a graduate-level course on the use and analysis of Markov chains. Emphasis is placed on explicit rates of convergence for chains used in applications to physics, biology, and statistics. A variety of card shuffling processes will be studies. Central Limit and concentration. Classical functional inequalities Nash, Faber-Krahn, log-Sobolev inequalities , comparison of Dirichlet forms. Random walks and isoperimetry of amenable groups with a focus on solvable groups. Entropy, harmonic functions, and Poisson boundary following Kaimanovich-Vershik theory.

Introduction to Stochastic Differential Equations. Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Random walk approximation of diffusions. Prerequisite: or equivalent and differential equations.

Introduction to mathematical models of complex static and dynamic stochastic systems that undergo sudden regime change in response to small changes in parameters. Examples from materials science phase transitions , power grid models, financial and banking systems. Special emphasis on mean field models and their large deviations, including computational issues.

Dynamic network models of financial systems and their stability. Topics in Financial Math: Market microstructure and trading algorithms. Introduction to market microstructure theory, including optimal limit order and market trading models. Random matrix theory covariance models and their application to portfolio theory.

Statistical arbitrage algorithms. Stochastic models of financial markets. Forward and futures contracts. European options and equivalent martingale measures. Hedging strategies and management of risk. Term structure models and interest rate derivatives. Optimal stopping and American options. Computation and Simulation in Finance. Monte Carlo, finite difference, tree, and transform methods for the numerical solution of partial differential equations in finance. Emphasis is on derivative security pricing. Functions of Several Complex Variables. Holomorphic functions in several variables, Hartogs phenomenon, d-bar complex, Cousin problem.

Domains of holomorphy. Plurisubharmonic functions and pseudo-convexity. Stein manifolds. Levi problem and its solution. Riemann surfaces and holomorphic maps, algebraic curves, maps to projective spaces. Calculus on Riemann surfaces. Elliptic functions and integrals. Riemann-Hurwitz formula. Riemann-Roch theorem, Abel-Jacobi map. Uniformization theorem. Hyperbolic surfaces. Suitable for advanced undergraduates. Topics in Algebraic Geometry. Topics of contemporary interest in algebraic geometry.

Topics in number theory: L-functions. The Riemann Zeta function and Dirichlet L-functions, zero-free regions and vertical distribution of the zeros, primes in arithmetic progressions, the class number problem, Hecke L-functions and Tate's thesis, Artin L-functions and the Chebotarev density theorem, Modular forms and Maass forms.

Introduction to Ergodic Theory.


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  4. Topics may include 1 subadditive and multiplicative ergodic theorems, 2 notions of mixing, weak mixing, spectral theory, 3 metric and topological entropy of dynamical systems, 4 measures of maximal entropy. Topics of contemporary interest in number theory. The theory of linear and nonlinear partial differential equations, beginning with linear theory involving use of Fourier transform and Sobolev spaces. Topics: Schauder and L2 estimates for elliptic and parabolic equations; De Giorgi-Nash-Moser theory for elliptic equations; nonlinear equations such as the minimal surface equation, geometric flow problems, and nonlinear hyperbolic equations.

    Symplectic Geometry and Topology. Linear symplectic geometry and linear Hamiltonian systems. Symplectic manifolds and their Lagrangian submanifolds, local properties. Symplectic geometry and mechanics. Contact geometry and contact manifolds. Relations between symplectic and contact manifolds. Hamiltonian systems with symmetries.

    Momentum map and its properties. Topics in Geometric Analysis. Introduction to the mathematics of the Fourier transform and how it arises in a number of imaging problems. Mathematical topics include the Fourier transform, the Plancherel theorem, Fourier series, the Shannon sampling theorem, the discrete Fourier transform, and the spectral representation of stationary stochastic processes. Applications include Fourier imaging the theory of diffraction, computed tomography, and magnetic resonance imaging and the theory of compressive sensing.

    Algebraic Combinatorics and Symmetric Functions. Symmetric function theory unifies large parts of combinatorics. Theorems about permutations, partitions, and graphs now follow in a unified way. Topics: The usual bases monomial, elementary, complete, and power sums. Schur functions. Representation theory of the symmetric group.