# Mathematical Sciences

Department website: http://www.uvm.edu/~cems/mathstat/

## Overview

The Department of Mathematics and Statistics offers programs towards the Doctor of Philosophy in Mathematical Sciences (the Ph.D.degree under the program heading of MASC). Students are encouraged to take courses in both core mathematics and applied mathematics, thereby gaining an appreciation of the connections between theory and applications.

Opportunities for research arise from the research interests of the Department faculty, which include analysis, algebra, biomathematics, combinatorics, complex systems, computational social science, differential equations, fluid mechanics, graph theory, mathematics education, modeling, network science, and number theory.

The Department also offers Master of Science degrees in Biostatistics, Mathematics, and Statistics.

## Degrees

**Ashikaga, Takamaru; **Professor, Department of Mathematics and Statistics; PHD, University of California Los Angeles

**Bagrow, James; **Assistant Professor, Department of Mathematics & Statistics; PHD, Clarkson University, Potsdam, NY

**Bentil, Daniel E.; **Associate Professor, Department of Mathematics and Statistics; DPHIL, University of Oxford

**Bunn, Janice Yanushka; **Research Associate Professor, Department of Mathematics and Statistics; PHD, Ohio State University

**Buzas, Jeff Sandor; **Professor, Department of Mathematics and Statistics; PHD, North Carolina State University Raleigh

**Callas, Peter W.; **Research Associate Professor, Department of Mathematics and Statistics; PHD, University of Massachusetts Amherst

**Cole, Bernard F.; **Professor, Department of Mathematics and Statistics; PHD, Boston University

**Danforth, Christopher M.; **Associate Professor, Department of Mathematics and Statistics; PHD, University of Maryland College Park

**Dinitz, Jeffrey Howard; **Professor, Department of Mathematics and Statistics; PHD, Ohio State University

**Dodds, Peter S.; **Professor, Department of Mathematics and Statistics; PHD, Massachusetts Institute of Technology

**Foote, Richard Martin; **Professor, Department of Mathematics and Statistics; PHD, University of Cambridge

**Jeffreys, William**; Lecturer I, Department of Mathematics and Statistics; PHD, Yale University

**Lakoba, Taras Igorevich; **Associate Professor, Department of Mathematics and Statistics; PHD, Clarkson University

**Sands, Jonathan Winslow; **Professor, Department of Mathematics and Statistics; PHD, University of California San Diego

**Single, Richard M.; **Associate Professor, Department of Mathematics and Statistics; PHD, SUNY Stony Brook

**Son, Mun Shig; **Professor, Department of Mathematics and Statistics; PHD, Oklahoma State University

**Vincent, Christelle;** Assistant Professor, Department of Mathematics and Statistics; PHD, University of Wisconsin Madison

**Warrington, Gregory S.; **Assistant Professor, Department of Mathematics and Statistics; PHD, Harvard University

**Wilson, James Michael; **Professor, Department of Mathematics and Statistics; PHD, University of California Los Angeles

**Yang, Jianke; **Professor, Department of Mathematics and Statistics; PHD, Massachusetts Institute of Technology

**Yu, Jun; **Professor, Department of Mathematics and Statistics; PHD, University of Washington Seattle

### Courses

**MATH 207. QR: Probability Theory. 3 Credits.**

Distributions of random variables and functions of random variables. Expectations, stochastic independence, sampling and limiting distributions (central limit theorems). Concepts of random number generation. Prerequisites: MATH 121; STAT 151 or STAT 153 recommended. Cross-listed with: STAT 251, BIOS 251.

**MATH 221. QR:Detrministc Modls Oper Rsch. 3 Credits.**

The linear programming problem. Simplex algorithm, dual problem, sensitivity analysis, goal programming. Dynamic programming and network problems. Prerequisites: MATH 122 or MATH 124; MATH 121 desirable. Cross-listed with: CSYS 221.

**MATH 222. QR:Stochastic Models:Oper Rsch. 3 Credits.**

Development and solution of some typical stochastic models. Markov chains, queueing problems, inventory models, and dynamic programming under uncertainty. Prerequisite: MATH 207, STAT 151.

**MATH 230. QR:Ordinary Diffrntl Equation. 3 Credits.**

Solutions of linear ordinary differential equations, the Laplace transformation, and series solutions of differential equations. Prerequisite: MATH 121. Corequisite: MATH 122 or MATH 124. Credit not granted for more than one of the courses MATH 230 or MATH 271.

**MATH 235. QR:Mathematical Models&Anlysis. 3 Credits.**

Techniques of Undergraduate calculus and linear algebra are applied for mathematical analysis of models of natural and human-created phenomena. Students are coached to give presentations. Prerequisites: MATH 121; MATH 122 or MATH 124 or MATH 230 or MATH 271.

**MATH 236. QR: Calculus of Variations. 3 Credits.**

Necessary conditions of Euler, Legendre, Weierstrass, and Jacobi for minimizing integrals. Sufficiency proofs. Variation and eigenvalue problems. Hamilton-Jacobi equations. Prerequisite: MATH 230.

**MATH 237. QR:Intro to Numerical Analysis. 3 Credits.**

Error analysis, root-finding, interpolation, least squares, quadrature, linear equations, numerical solution of ordinary differential equations. Prerequisites: MATH 121; MATH 122 or MATH 124 or MATH 271; CS 020 or CS 021. Cross-listed with: CS 237.

**MATH 238. QR:Appld Computational Methods. 3 Credits.**

Direct and iterative methods for solving linear systems; numerical solution of ordinary and partial differential equations. Focus will be on application of numerical methods. Prerequisites: MATH 121; MATH 122 or MATH 124 or MATH 271.

**MATH 240. QR:Fourier Series&Intgrl Trans. 3 Credits.**

Fourier series, orthogonal functions, integral transforms and boundary value problems. Prerequisite: MATH 230 or MATH 271.

**MATH 241. QR:Anyl in Several Real Vars I. 3 Credits.**

Properties of the real numbers, basic topology of metric spaces, infinite sequences and series, continuity. Prerequisites: MATH 052 or Math 141 or MATH 151; MATH 121; MATH 122 or MATH 124.

**MATH 242. QR:Anyl Several Real Vrbes II. 3 Credits.**

Differentiation and integration in n-space, uniform convergence of functions, fundamental theorem of calculus, inverse and implicit function theorems. Prerequisite: MATH 241.

**MATH 251. QR: Abstract Algebra I. 3 Credits.**

Basic theory of groups, rings, fields, homomorphisms, and isomorphisms. Prerequisites: MATH 052 or MATH 141 or MATH 151; MATH 122 or MATH 124.

**MATH 252. QR: Abstract Algebra II. 3 Credits.**

Modules, vector spaces, linear transformations, rational and Jordan canonical forms. Finite fields, field extensions, and Galois theory leading to the insolvability of quintic equations. Prerequisite: MATH 251.

**MATH 255. QR:Elementary Number Theory. 3 Credits.**

Divisibility, prime numbers, Diophantine equations, congruence of numbers, and methods of solving congruences. Prerequisite: MATH 052 or MATH 054.

**MATH 257. QR:Topics in Group Theory. 3 Credits.**

Topics may include abstract group theory, representation theory, classical groups, Lie groups. Prerequisite: MATH 251.

**MATH 259. QR: Cryptography. 3 Credits.**

A survey of classical and modern cryptography. The strengths and weaknesses of various cryptosystems are discussed. Topics include specific public-key and private-key cryptosystems such as RSA, ElGamal, and elliptic curve cryptosystems, as well as digital signatures and key exchange. Prerequisite: MATH 052 or CS 064; any 100-level MATH course.

**MATH 260. QR: Foundations of Geometry. 3 Credits.**

Geometry as an axiomatic science; various non-Euclidean geometries; relationships existing between Euclidean plane geometry and other geometries; invariant properties. Prerequisite: MATH 022 and either MATH 052 or MATH 054.

**MATH 264. QR: Vector Analysis. 3 Credits.**

Gradient, curl and divergence, Green, Gauss, and Stokes Theorems, applications to physics, tensor analysis. PrerequisiteS: MATH 121; MATH 122 or MATH 124 or MATH 271.

**MATH 266. QR:Chaos,Fractals&Dynmcal Syst. 3 Credits.**

Discrete and continuous dynamical systems, Julia sets, the Mandelbrot set, period doubling, renormalization, Henon map, phase plane analysis and Lorenz equations. Co-requisite: MATH 271 or MATH 230. Cross-listed with: CSYS 266.

**MATH 268. QR:Mathematical Biology&Ecol. 3 Credits.**

Mathematical modeling in the life sciences. Topics include population modeling, dynamics of infectious diseases, reaction kinetics, wave phenomena in biology, and biological pattern formation. Prerequisite: MATH 122 or MATH 124; MATH 230; or Instructor permission. Cross-listed with: CSYS 268.

**MATH 271. QR:Adv Engineering Mathematics. 3 Credits.**

Differential equations, Laplace transforms, matrix theory, and systems of differential equations; brief introduction to Fourier series. Examples from engineering and physical sciences. Credit not granted for both MATH 230 and MATH 271. No credit for Mathematics majors. Prerequisite: MATH 121.

**MATH 272. QR: Applied Analysis. 3 Credits.**

Basics of Fourier series, partial differential equations of mathematical physics, functions of a complex variable, Cauchy's theorem, integral formula. Prerequisites: MATH 230 or MATH 271.

**MATH 273. QR:Combinatorial Graph Theory. 3 Credits.**

Paths and trees, connectivity, Eulerian and Hamiltonian cycles, matchings, edge and vertex colorings, planar graphs, Euler's formula and the Four Color Theorem, networks. Prerequisite: MATH 052 or MATH 054.

**MATH 274. QR:Numerical Linear Algebra. 3 Credits.**

Direct and iterative methods for solving linear equations, least square factorization methods, eigenvalue computations, ill-conditioning and stability. Prerequisite: MATH 237.

**MATH 295. Special Topics. 1-18 Credits.**

For advanced students in the indicated fields. Lectures, reports, and directed readings on advanced topics. Credit as arranged. Offered as occasion warrants.

**MATH 300. Principles of Complex Systems. 3 Credits.**

Introduction to fundamental concepts of complex systems. Topics include: emergence, scaling phenomena, and mechanisms, multi-scale systems, failure, robustness, collective social phenomena, complex networks. Students from all disciplines welcomed. Pre/co-requisites: Calculus and statistics required; Linear Algebra, Differential Equations, and Computer programming recommended but not required. Cross-listed with: CSYS 300.

**MATH 303. Complex Networks. 3 Credits.**

Detailed exploration of distribution, transportation, small-world, scale-free, social, biological, organizational networks; generative mechanisms; measurement and statistics of network properties; network dynamics; contagion processes. Students from all disciplines welcomed. Pre/co-requisites: MATH 300/CSYS 300, Calculus, and Statistics required. Cross-listed with: CSYS 303.

**MATH 330. Adv Ordinary Diff Equations. 3 Credits.**

Linear and nonlinear systems, approximate solutions, existence, uniqueness, dependence on initial conditions, stability, asymptotic behavior, singularities, self-adjoint problems. Prerequisite: MATH 230.

**MATH 331. Theory of Func of Complex Var. 4 Credits.**

Differentiation, integration, Cauchy-Riemann equations, infinite series, properties of analytic continuation, Laurent series, calculus of residues, contour integration, meromorphic functions, conformal mappings, Riemann surfaces. Prerequisite: MATH 242.

**MATH 332. Approximation Theory. 3 Credits.**

Interpolation and approximation by interpolation, uniform approximation in normed linear spaces, spline functions, orthogonal polynomials. Least square, and Chebychev approximations, rational functions. Prerequisites: MATH 122 or MATH 124; MATH 237.

**MATH 333. Thry Functions Real Variables. 4 Credits.**

The theory of Lebesgue integration, Lebesgue measure, sequences of functions, absolute continuity, properties of LP-spaces. Prerequisite: MATH 242.

**MATH 335. Advanced Real Analysis. 3 Credits.**

L2-spaces, LP-spaces; Hilbert, Banach spaces; linear functionals, linear operators; completely continuous operators (including symmetric); Fredholm alternative; Hilbert-Schmidt theory; unitary operators; Bochner's Theorem; Fourier-Plancherel, Watson transforms. Prerequisites: MATH 333.

**MATH 336. Advanced Real Analysis. 3 Credits.**

L2-spaces, LP-spaces; Hilbert, Banach spaces; linear functionals, linear operators; completely continuous operators (including symmetric); Fredholm alternative; Hilbert-Schmidt theory; unitary operators; Bochner's Theorem; Fourier-Plancherel, Watson transforms. Prerequisite: MATH 333 and MATH 335.

**MATH 337. Numerical Diff Equations. 3 Credits.**

Numerical solution and analysis of differential equations: initial-value and boundary-value problems; finite difference and finite element methods. Prerequisites: MATH 121; MATH 122 or MATH 124; MATH 230 or MATH 271 or MATH 237 recommended.

**MATH 339. Partial Differential Equations. 3 Credits.**

Classification of equations, linear equations, first order equations, second order elliptic, parabolic, and hyperbolic equations, uniqueness and existence of solutions. Prerequisite: MATH 230; MATH 242.

**MATH 351. Topics in Algebra. 3 Credits.**

Topics will vary each semester and may include algebraic number theory, algebraic geometry, and the arithmetic of elliptic curves. Repeatable for credit with Instructor permission. Prerequisite: MATH 252.

**MATH 353. Point-Set Topology. 3 Credits.**

Topological spaces, closed and open sets, closure operators, separation axioms, continuity, connectedness, compactness, metrization, uniform spaces. Prerequisite: MATH 241.

**MATH 354. Algebraic Topology. 3 Credits.**

Homotopy, Seifert-van Kampen Theorem; simplicial, singular, and Cech homology. Prerequisite: MATH 241 or MATH 353.

**MATH 373. Topics in Combinatorics. 3 Credits.**

Topics will vary each semester and may include combinatorial designs, coding theory, topological graph theory, cryptography. Prerequisite: MATH 251 or MATH 273.

**MATH 382. Seminar. 1 Credit.**

Topical discussions with assigned reading. Required of M.S. degree candidates.

**MATH 392. Independent Study. 1-18 Credits.**

A course which is tailored to fit the interests of a specific student, which occurs outside the traditional classroom/laboratory setting under the supervision of a faculty member, for which credit is awarded. Offered at department discretion.

**MATH 395. Advanced Special Topics. 1-18 Credits.**

Subject will vary from year to year. May be repeated for credit.

**MATH 492. Independent Study. 1-18 Credits.**

A course which is tailored to fit the interests of a specific student, which occurs outside the traditional classroom/laboratory setting under the supervision of a faculty member, for which credit is awarded.

**MATH 496. Advanced Special Topics. 1-18 Credits.**

See Schedule of Courses for specific titles.