Mathematics
Departmental Website: Mathematics
07-08
Department Head: Distinguished Professor Michael Neumann
Professors: Abe, Abikoff, R.F. Bass, Blei, Choi, DeFranco, Dey, Dunne, Gine, Glaz, Gui, Haas, Luh, Madych, McKenna, Olshevsky, Ravishanker, Sidney, Spiegel, Tollefson, Turchin, Vadiveloo, Vinsonhaler, Vitale, and Wang
Associate Professors: Bridgeman, Hernandez, Leibowitz, Peters, Roby, Russell, Teplyaev, Terwilland Wang
Assistant Professors: Cardetti, Conrad, Frey, Gordina, Holm, Huber, Kaufmann, Khamsemanan, Lee, Miller, Sellke, Solomon, and Terwilleger
The Department of Mathematics offers work leading to the M.S. and Ph.D. degrees. The master’s program permits a student to emphasize pure and applied mathematics, actuarial science, or numerical methods, with some course work taken in other departments if desired. A professional master’s degree program in Applied Financial Mathematics is also offered. Advanced study at the Ph.D. level is offered in the areas of algebra and number theory, applied mathematics, classical and functional analysis, computational linear algebra, differential geometry, logic, probability, and topology. See the details below.
The Department is one of the few offering graduate study in actuarial science and financial mathematics. Admission requirements differ slightly for this option. For details, write to the Department of Mathematics.
The M.S. Program. A sound undergraduate major in mathematics, including courses in modern algebra and advanced calculus, normally is required for entrance to the master’s program. The Department recommends that students select Plan B. Further details concerning the master’s (and Ph.D.) program may be obtained by writing directly to the Department of Mathematics.
It is recommended that entering graduate students applying for financial aid take the Subject Test in Mathematics of the Graduate Record Examinations.
The Ph.D. Program. Students are admitted to the Ph.D. program only after demonstrating ability and evidence of special aptitude for research in mathematics in their prior work. Although no specified number of course credits is required for the Ph.D., usually at least 24 credits of course work beyond the master’s level is considered necessary. Students must satisfy the doctoral foreign language requirement of the Graduate School. Doctoral students also are expected to possess computer skills necessary for mathematics research. During the first two to three years of the student’s course work, comprehensive examinations covering the major areas of mathematics must be passed. The Ph.D. dissertation contains results of original research in mathematics and makes a substantial contribution to the field. A student normally writes a dissertation in an area in which the Department has faculty actively engaged in research. Such areas are: analysis on fractals, stochastic analysis, symplectic geometry, algebraic geometry, commutative rings theory, hological algebra, combinatorics, Fourier analysis, harmonic analysis, complex analysis, Riemann surfaces, algebraic topology, topological measure theory, probability theory, low-dimensional topology, abelian groups, rings, group rings, discrete groups, number theory, functional analysis, representation theory, logic, computability theory, ordinary and partial differential equations, numerical analysis, approximation theory, differential geometry, numerical linear algebra, matrix theory, inverse problems, tomography, wavelet theory, mathematical physics, mathematical biology, mathematics education, and actuarial science. Further details concerning the Ph.D. (and Master’s) program and faculty research interests may be obtained by writing directly to the Department of Mathematics or by visiting the website: <www.math.uconn.edu>.
Special Facilities. The Homer Babbidge Library has extensive holdings of mathematics books and journals. Subscriptions to numerous mathematical journals are maintained and housed in the Mathematics Department Library.
A weekly colloquium featuring visiting lecturers as well as several area-specific seminars are conducted during the academic year. Colloquia and seminars at neighboring institutions are also held on a regular basis. Because of the easy access to these institutions, there is considerable scholarly interaction.
COURSES OF STUDY
Courses designated by the dagger symbol (†) are approved
for Satisfactory (S) / Unsatisfactory (U) grading.
MATH 300. Investigation of Special Topics
1-6 credits. Lecture.
Students who have well defined mathematical problems worthy of investigation and advanced reading should submit to the department a semester work plan.
MATH 301. Introduction to Modern Analysis
3 credits. Lecture.
Metric spaces, sequences and series, continuity, differentiation, the Riemann-Stielties integral, functions of several variables.
MATH 303. Measure and Integration
3 credits. Lecture. Prerequisite: MATH 301.
Lebesgue measure and integration, differentiation, Lpspaces. Banach spaces, general theory of measure and integration.
MATH 304. Mathematical Modeling
3 credits. Lecture.
Development of mathematical models emphasizing linear algebra, differential equations, graph theory and probability. In-depth study of the model to derive information about phenomena in applied work.
MATH 305. Computerized Modeling in Science
4 credits. Lecture.
Development and computer-assisted analysis of mathematical models in chemistry, physics, and engineering. Topics include chemical equilibrium, reaction rates, particle scattering, vibrating systems, least squares analysis, quantum chemistry and physics.
MATH 307. Introduction to Geometry and Topology I
3 credits. Lecture. Prerequisite: MATH 301, which may be taken concurrently.
Topological spaces, connectedness, compactness, separation axioms, Tychonoff theorem, compact-open topology, fundamental group, covering spaces, simplicial complexes, differentiable manifolds, homology theory and the De Rham theory, intrinsic Riemannian geometry of surfaces.
MATH 308. Introduction to Geometry and Topology II
3 credits. Lecture. Prerequisite: Math 307.
Topological spaces, connectedness, compactness, separation axioms, Tychonoff theorem, compact-open topology, fundamental group, covering spaces, simplicial complexes, differentiable manifolds, homology theory and the De Rham theory, intrinsic Riemannian geometry of surfaces.
MATH 309. Optimization
3 credits. Lecture.
Theory of linear programming: convexity, bases, simplex method, dual and integer programming, assignment, transportation, and flow problems. Theory of nonlinear programming: unconstrained local optimization, Lagrange multipliers, Kuhn-Tucker conditions, computational algorithms. Concrete applications.
MATH 310. Introduction to Applied Mathematics I
3 credits. Lecture.
Banach spaces, linear operator theory and application to differential equations, nonlinear operators, compact sets on Banach spaces, the adjoint operator on Hilbert space, linear compact operators, Fredholm alternative, fixed point theorems and application to differential equations, spectral theory, distributions.
MATH 311. Introduction to Applied Mathematics II
3 credits. Lecture.
Banach spaces, linear operator theory and application to differential equations, nonlinear operators, compact sets on Banach spaces, the adjoint operator on Hilbert space, linear compact operators, Fredholm alternative, fixed point theorems and application to differential equations, spectral theory, distributions.
MATH 313. Numerical Analysis and Approximation Theory I
3 credits. Lecture. Prerequisite: MATH 301, which may be taken concurrently.
The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.
MATH 314. Numerical Analysis and Approximation Theory II
3 credits. Lecture. Prerequisite: MATH 313.
The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.
MATH 315. Abstract Algebra I
3 credits. Lecture.
A study of the fundamental concepts of modern algebra: groups, rings, fields. Also selected topics in linear algebra.
MATH 316. Abstract Algebra II
3 credits. Lecture. Prerequisite: MATH 315.
A study of the fundamental concepts of modern algebra: groups, rings, fields. Also selected topics in linear algebra.
MATH 318. Modern Matrix Theory and Linear Algebra
3 credits. Lecture.
The LU, QR, symmetric, polar, and singular value matrix decompositions. Schur and Jordan normal forms. Symmetric, positive-definite, normal and unitary matrices. Perron-Frobenius theory and graph criteria in the theory of non-negative matrices.
MATH 319. Topics in Scientific Computation
3 credits. Lecture.
MATH 321. Topics in Algebra
3 credits. Lecture. Prerequisite: MATH 316. With a change of content, this course may be repeated to a maximum of 12 credits.
Advanced topics from group theory, abelian groups, rings and homological algebra, Lie algebras, algebraic groups, group rings, combinatorics.
MATH 322. Probability Theory and Stochastic Processes I
3 credits. Lecture. Prerequisite: MATH 303.
Convergence of random variables and their probability laws, maximal inequalities, series of independent random variables and laws of large numbers, central limit theorems, martingales, Brownian motion. Contemporary theory of stochastic processes, including stopping times, stochastic integration, stochastic differential equations and Markov processes, Gaussian processes, and empirical and related processes with applications in asymptotic statistics.
MATH 323. Probability Theory and Stochastic Processes II
3 credits. Lecture. Prerequisite: MATH 322.
Convergence of random variables and their probability laws, maximal inequalities, series of independent random variables and laws of large numbers, central limit theorems, martingales, Brownian motion. Contemporary theory of stochastic processes, including stopping times, stochastic integration, stochastic differential equations and Markov processes, Gaussian processes, and empirical and related processes with applications in asymptotic statistics.
MATH 324. Advanced Financial Mathematics
3 credits. Lecture.
An introduction to the standard models of modern financial mathematics including martingales, the binomial asset pricing model, Brownian motion, stochastic integrals, stochastic differential equations, continuous time financial models,
completeness of the financial market, the Black-Scholes formula, the fundamental theorem of finance, American options, and term structure models.
MATH 325. Ordinary Differential Equation
3 credits. Lecture. Prerequisite: MATH 303.
Existence and uniqueness of solutions, stability and asymptotic behavior. If time permits: eigenvalue problems, dynamical systems, existence and stability of periodic solutions.
MATH 326. Partial Differential Equations
3 credits. Lecture. Prerequisite: MATH 340.
Cauchy Kowalewsky Theorem, classification of second order equations, systems of hyperbolic equations, the wave equation, the potential equation, the heat equation in Rn.
MATH 327. Topics in Applied Analysis I
3 credits. Lecture.
Advanced topics from the theory of ordinary or partial differential equations. Other possible topics: integral equations, optimization theory, the calculus of variations, advanced approximation theory.
MATH 328. Topics in Applied Analysis II
3 credits. Lecture.
Advanced topics from the theory of ordinary or partial differential equations. Other possible topics: integral equations, optimization theory, the calculus of variations, advanced approximation theory.
MATH 329. Introduction to Representation Theory
3 credits. Lecture. Prerequisite: MATH 315.
Semi-simple rings, Jacobson radical, density theory, Wedderburn’s Theorem, representations and characters of groups, orthogonality relations, Burnside’s theorem.
MATH 330. Algebraic Number Theory
3 credits. Lecture. Prerequisite: MATH 316.
Valuations, p-adic and local fields, ideal theory of Dedekind domains, cyclotomic extensions, units in algebraic number fields.
MATH 332. Topics in Analysis I
3 credits. Lecture.
MATH 333. Topics in Analysis II
3 credits. Lecture. Prerequisite: MATH 332.
MATH 335. Mathematical Logic I
3 credits. Lecture. Prerequisite: MATH 315.
Predicate calculus, completeness, compactness, Lowenheim-Skolem theorems, formal theories with applications to algebra, Godel’s incompleteness theorem. Further topics chosen from: axiomatic set theory, model theory, recursion theory, computational complexity, automata theory and formal languages.
MATH 336. Topics in Mathematical Logic
3 credits. Lecture. Prerequisite: Math 335. May be repeated for credit with a change in content.
Topics include, but are not restricted to, Computability Theory, Model Theory, and Set Theory.
MATH 337. Topics in Geometry and Topology I
3 credits. Lecture.
Advanced topics from uniform spaces, topological groups, Lie groups, fiber spaces, theory of submanifolds, PL topology, differential topology, cohomology operations, complex manifolds, Riemannian manifolds, transformation groups, fixed point theory.
MATH 338. Topics in Geometry and Topology II
3 credits. Lecture. Prerequisite: MATH 337.
Advanced topics from uniform spaces, topological groups, Lie groups, fiber spaces, theory of submanifolds, PL topology, differential topology, cohomology operations, complex manifolds, Riemannian manifolds, transformation groups, fixed point theory.
MATH 340. Complex Function Theory I
3 credits. Lecture. Prerequisite: MATH 301.
An introduction to the theory of analytic functions, with emphasis on modern points of view.
MATH 341. Topics in Complex Function Theory
3 credits. Lecture. Prerequisite: MATH 340. May be repeated for credit to a maximum of 12 credits with a change in content and consent of the instructor.
Advanced topics of contemporary interest. These include Riemann surfaces, Kleinian groups, entire functions, conformal mapping, several complex variables, and automorphic functions, among others.
MATH 342. Finite Element Solution Methods I
3 credits. Lecture.
Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.
MATH 343. Finite Element Solution Methods II
3 credits. Lecture. Prerequisite: MATH 342
Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.
MATH 347. Tensor Calculus I
3 credits. Lecture.
An introduction to tensor algebra and tensor calculus with applications chosen from the fields of the physical sciences and mathematics.
MATH 348. Tensor Calculus II
3 credits. Lecture. Prerequisite: MATH 347.
An introduction to tensor algebra and tensor calculus with applications chosen from the fields of the physical sciences and mathematics.
MATH 352. Introduction to Complex Variables
3 credits. Lecture. Not open to students who have passed MATH 252. Not open for graduate credit toward degrees in Mathematics.
Functions of a complex variable, integration in the complex plane, conformal mapping.
MATH 354. Functional Analysis I
3 credits. Lecture. Prerequisites: MATH 303 and MATH 316.
Normed linear spaces and algebras, the theory of linear operators, spectral analysis.
MATH 355. Functional Analysis II
3 credits. Lecture. Prerequisite: MATH 354.
Normed linear spaces and algebras, the theory of linear operators, spectral analysis.
MATH 357. Differential Geometry
3 credits. Lecture.
An introduction to the study of differentiable manifolds on which various differential and integral calculi are developed. A special emphasis is placed on the global aspects of modern differential geometry.
MATH 360. Mathematical Pedagogy
1 credit. Seminar. Open to graduate students in Mathematics, others with consent of instructor. May not be used to satisfy degree requirements in mathematics.
The theory and practice of teaching mathematics at the college level. Basic skills, grading methods, cooperative learning, active learning, use of technology, classroom problems, history of learning theory, reflective practice.
MATH 365. Financial Mathematics I
3 credits. Lecture. Not open to students who have passed MATH 285Q
The mathematics of measurement of interest, accumulation and discount, present value, annuities, loans, bonds, and other securities.
MATH 366. Introduction to Operations Research
3 credits. Lecture. Not open to students who have passed MATH 286, STAT 286, or STAT 356.
Introduction to the use of mathematical and statistical techniques to solve a wide variety of organizational problems. Topics include linear programming, project scheduling, queuing theory, decision analysis, dynamic and integer programming and computer simulation.
MATH 369. Financial Mathematics II
3 credits. Lecture. Not open to students who have passed MATH 289.
The continuation of MATH 365. Measurement of financial risk, the mathematics of capital budgeting, mathematical analysis of financial decisions and capital structure, and option pricing theory.
MATH 373. Algebraic Topology I
3 credits. Lecture. Prerequisite: MATH 316 and MATH 307, which may be taken concurrently.
Complexes, homology and cohomology groups, homotopy theory.
MATH 374. Algebraic Topology II
3 credits. Lecture. Prerequisite: MATH 373.
Complexes, homology and cohomology groups, homotopy theory.
MATH 375. Analysis
3 credits. Lecture. Not open to students who have passed MATH 273. Not open for graduate credit toward degrees in Mathematics.
Introduction to the theory of functions of a real variable.
MATH 377. Applied Analysis
3 credits. Lecture. Not open to students who have passed MATH 277. Not open for graduate credit toward degrees in Mathematics.
Convergence of Fourier Series, Legendre and Hermite polynomials, existence and uniqueness theorems, two point boundary value problems and Green’s functions.
MATH 378. Introduction to Partial Differential Equations
3 credits. Lecture. Not open to students who have passed MATH 278. Not open for graduate credit toward degrees in Mathematics.
Solution of first and second order partial differential equations with applications to engineering and science.
MATH 381. Fourier Analysis
3 credits. Lecture. Prerequisites: MATH 303 and MATH341.
Foundations of harmonic analysis developed through the study of Fourier series and Fourier transforms.
MATH 382. Fourier Analysis on Groups
3 credits. Lecture. Prerequisites: MATH 303 and MATH341.
MATH 385. Vector Field Theory I
3 credits. Lecture.
Vector algebra and vector calculus with particular emphasis on invariance. Classification of vector fields. Solution of the partial differential equations of field theory.
MATH 386. Vector Field Theory II
3 credits. Lecture. Prerequisite: MATH385.
Vector algebra and vector calculus with particular emphasis on invariance. Classification of vector fields. Solution of the partial differential equations of field theory.
MATH 387. Actuarial Mathematics I
3 credits. Lecture. Prerequisite: MATH 285 or MATH 365, which may be taken concurrently. Not open to students who have passed MATH 287.
Survival distributions, claim frequency and severity distributions, life tables, life insurance, life annuities, net premiums, net premium reserves, multiple life functions, and multiple decrement models.
MATH 388. Actuarial Mathematics II
3 credits. Lecture. Prerequisite: MATH 387. Not open to students who have passed MATH 288.
Survival distributions, claim frequency and severity distributions, life tables, life insurance, life annuities, net premiums, net premium reserves, multiple life functions, and multiple decrement models.
MATH 390. Graduate Field Study Internship
1-3 credits. Practicum
Participation in internship and paper describing experiences.
MATH 392. Advanced Topics in Actuarial Mathematics I
3 credits. Lecture.
Survival models, mathematical graduation, or demography.
MATH 393. Advanced Topics in Actuarial Mathematics II
3 credits. Lecture.
Credibility theory or advanced theory of interest.
MATH 394. Survival Models
3 credits. Lecture. Prerequisite: MATH 387.
Analysis, estimation, and validation of lifetime tables.
MATH 395. Risk Theory
3 credits. Lecture.
Individual risk theory, distribution theory, ruin theory, stoploss, reinsurance and Monte Carlo methods. Emphasis is on problems in insurance.
†GRAD 395. Master’s Thesis Research
1 - 9 credits.
†GRAD 396. Full-Time Master’s Research
3 credits.
†GRAD 397. Full-Time Directed Studies (Master’s Level)
3 credits.
GRAD 398. Special Readings (Master’s)
Non-credit.
GRAD 399. Thesis Preparation
Non-credit.
MATH 401. Seminar in Current Mathematical Literature
1-6 credits. Seminar.
Participation and presentation of mathematical papers in joint student faculty seminars. Variable topics.
†MATH 410. Seminar in Algebra
1-6 credits. Seminar. Prerequisite: MATH 316.
†MATH 430 Seminar in Geometry
1-6 credits. Seminar. Prerequisite: MATH 357.
†MATH 435. Seminar in Mathematical Logic
1-6 credits. Seminar. Prerequisite: MATH 335.
†MATH 450. Seminar in Analysis
1-6 credits. Seminar.
MATH 460. Computers in Mathematical Research
1 credit. Lecture.
†MATH 470. Seminar in Topology
1-6 credits. Seminar. Prerequisite: MATH 374.
†MATH 471. Seminar in Set Theory
1-6 credits. Seminar. Prerequisite: MATH 307.
†MATH 480. Seminar in Applied Mathematics
1-6 credits. Seminar.
†GRAD 495. Doctoral Dissertation Research
1 - 9 credits.
†GRAD 496. Full-Time Doctoral Research
3 credits.
†GRAD 497. Full-Time Directed Studies (Doctoral Level)
3 credits.
GRAD 498. Special Readings (Doctoral)
Non-credit.
GRAD 499. Dissertation Preparation
Non-credit.