Extended description: Taught on Mondays and Wednesdays. The Monday classes cover 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. The Wednesday classes cover the IT resources required for someone to become an effective member of our department.
Prerequisites: Open to graduate students in Mathematics, others with consent of instructor. May not be used to satisfy degree requirements in mathematics.
Offered: Fall
Credits: 1

Extended description: With a change of content, this course is repeatable to a maximum of twelve credits. Topics include, but are not restricted to, recursion theory (degree structures, hyperarithmetic hierarchy, applications to computable algebra, reverse mathematics), model theory (quantifier elimination, o-minimality, types, categoricity, indiscernible), set theory (ordinals, cardinals, Martin's axiom, constructible sets, forcing), and proof theory (deductive systems, cut elimination and applications, ordinal analysis). Textbook choices: Recursively Enumerable Sets and Degrees by Soare. Degrees of Unsolvability by Lerman. Higher Recursion Theory by Sacks. Subsystems of Second Order Arithmetic by Simpson. Set Theory by Kunen. Set Theory for the Working Mathematician by Ciesielski. Model Theory: An Introduction by Marker. Proof Theory by Pohlers. Supplementary reading: Countable Structures and the Hyperarithmetic Hierarchy by Ash and Knight. Set Theory by Jech. Model Theory by Chang and Keisler. Model Theory by Hodges. Basic Proof Theory by Troelstra and Schwichtenberg.
Prerequisites: MATH 5260
Credits: 3

Extended description: Complex plane, Riemann sphere, polar coordinates and Euler's formula, complex differentiable functions and Cauchy-Riemann equations, harmonic conjugates, conformal maps. Elementary functions: exp, sin, cos, log, Log, powers. Integration along simple curves, simply connected domains, Cauchy-Goursat theorem, Cauchy integral formula. Power series and the disk of convergence, Taylor and Laurent series, classification of singularities. Cauchy's residue theorem and its use in evaluating real-valued integrals. Textbook choices: Complex Variables and Applications by J.W. Brown and R.V. Churchill. Complex Analysis with Applications by R.A. Silverman. Supplementary reading: Elementary theory of analytic functions of one or several complex variables by H. Cartan.
Prerequisites: Not open to students who have passed MATH 3146. Open for master's credit but not doctoral credit toward degree in Mathematics.
Credits: 3

Extended description: Construction of real numbers, completeness, infima, suprema. Sequences of real numbers, liminf and limsup, limits, big-O and little-o notations. Continuity of functions of one real variable, Intermediate Value theorem, continuous functions on closed bounded sets. Sequences and series of functions of one real variable, uniform convergence. Differentiation of functions of one real variable, Rolle's and Mean Value theorems. Taylor series. Riemann integration of functions of one real variable and the Fundamental Theorem of Calculus. Textbook choices: Elementary Analysis: The Theory of Calculus by K. Ross. Analysis I by T. Tao.
Prerequisites: Not open to students who have passed MATH 3150. Open for master's credit but not doctoral credit toward degree in Mathematics.
Offered: Spring
Credits: 3

Extended description: Real and Complex Number Systems. Basic topology of metric spaces, Bolzano-Weierstrass and Heine-Borel theorems. Sequences and series of functions on compact metric spaces, uniform convergence, Arzela-Ascoli and Stone-Weierstrass theorems. Differentiation and integration of vector valued functions of several real or complex variables. Inverse and implicit function theorems. Contraction mapping and Picard-Lindelof theorems. Differential forms. Textbook choices: Principles of Mathematical Analysis by W. Rudin. Analysis II by T. Tao.
Offered: Spring
Credits: 3

Extended description: Abstract integration: Lebesgue integration theory, outer measures and Caratheodory's theorem, Fatou's lemma, monotone and dominated convergence theorems. Measure theory: positive Borel measures, Riesz representation theorem for positive linear functionals on C(K), complex measures, Hahn-Jordan and Lebesgue decompositions, Radon-Nykodim theorem and differentiation of measures, Riemann-Stieltjes integral, the Lebesgue measure on R^d. L^p spaces: Cauchy-Bunyakovsky-Schwarz, Hoelder, Minkowski and Jensen inequalities, L² and L^p spaces as Hilbert and Banach spaces, Riesz representation theorem for bounded functionals on L^p. Integration on product spaces, Fubini's and Tonelli's theorems, Fourier transform and Plancherel theorem. See also Real Analysis Prelim Study Guide. Textbook choices: Real and Complex Analysis by W. Rudin. Real Analysis and Probability by R.M. Dudley. Real Analysis: Modern Techniques and Their Applications by G. Folland. Supplementary reading: Real Analysis by H. Royden and P. Fitzpatrick. Lecture notes by Rich Bass. Real Analysis: Measure Theory, Integration, and Hilbert Spaces by E.M. Stein and R. Shakarchi. Measure and Integral by R.L. Wheeden and A. Zygmund.
Prerequisites: MATH 5110
Offered: Spring
Credits: 3

Extended description: Complex plane, Riemann sphere, Euler's formula, complex differentiable functions and Cauchy-Riemann equations, conformal maps, linear fractional transformations. Integration along simple rectifiable curves, Cauchy-Goursat and Morera's theorems, Cauchy integral formula, Cauchy estimates and Schwarz lemma. Power series and the disk of convergence, Taylor and Laurent series, classification of singularities. The argument principle, winding numbers and Rouche's theorem. Cauchy's residue theorem and its use in evaluating real-valued integrals. Maximum modulus, Liouville and Picard theorems, the Fundamental Theorem of Algebra, Schwarz reflection principle. Harmonic functions and harmonic conjugates. Normal families and Montel's theorem. The Riemann mapping theorem. See also Complex Analysis Prelim Study Guide. Textbook choices: Complex analysis by L. V. Ahlfors. Functions of one complex variable by J. B. Conway. Introduction to complex analysis by R. Nevanlinna and V. Paatero. Real and Complex Analysis by W. Rudin. Supplementary reading: Elementary theory of analytic functions of one or several complex variables by H. Cartan. Analytic function theory. Vols. 1 and 2 by E. Hille. Elements of the Theory of Functions, Vols. I and II by K. Knopp. Problem Book in the Theory of Functions, Vols. I and II by K. Knopp. Theory of functions of a complex variable. Vol. I, II, III by A.I. Markushevich. Theory of complex functions by R. Remmert. Complex Analysis by E.M. Stein and R. Shakarchi.
Prerequisites: MATH 5110
Credits: 3

Extended description: Theory of Banach spaces: duality, reflexivity, weak and weak* topologies. Hahn-Banach, Banach-Steinhaus, Banach-Alaoglu theorems. Krein-Milman theorem. Linear operators: compact, integral, trace class, Fredholm, Hilbert-Schmidt, Toepliz, Volterra. Commutative Banach and C* algebras, Gelfand transform and the spectral theorem for bounded normal operators. Compact self-adjoint operators with applications to the classical Sturm-Liouville theory. Other topics in functional analysis at the choice of the instructor (e.g. Banach algebra L¹, Kaplansky density theorem, Gelfand-Naimark-Segal construction, introduction to von Neumann algebras and non-commutative integration, introduction to unbounded self-adjoint operators and the role of the Fourier transform). Textbook choices: Functional Analysis by W. Rudin. Functional Analysis by P.D. Lax. I: Functional Analysis (Methods of Modern Mathematical Physics, Volume 1) by M. Reed and B. Simon. Supplementary reading: Functional Analysis by K. Yosida. A Course in Functional Analysis by J.B. Conway.
Prerequisites: MATH 5111
Credits: 3

Extended description: Spectral theory of unbounded self-adjoint and normal operators on Hilbert spaces. Quadratic forms. Examples and counterexamples of self-adjoint operators. Spectral theory of differential operators with constant coefficients. Unitary and positivity preserving operator semigroups, resolvents, Trotter product formula, Hille-Yosida theorem. Other topics in functional analysis at the choice of the instructor (e.g. applications to probability and quantum mechanics, introduction to von Neumann algebras and non-commutative integration). Textbook choices: Functional Analysis by W. Rudin. Functional Analysis by P.D. Lax. I: Functional Analysis, Volume 1 (Methods of Modern Mathematical Physics) by M. Reed and B. Simon. Supplementary reading: Functional Analysis by K. Yosida. A Course in Functional Analysis by J.B. Conway. Functional Analysis, Sobolev Spaces and Partial Differential Equations by H. Brezis. Lecture notes by A.J. Wassermann and by V.F.R. Jones. Lectures on Quantum Mechanics for Mathematics Students by L.D. Faddeev and O.A. Yakubovskii.
Prerequisites: MATH 5111
Credits: 3

Extended description: Basic properties of Fourier series, convergence of Fourier series, applications of Fourier series. Fourier transform and distributions. Fourier transform in L^p spaces. Hardy-Littlewood maximal inequality. Marcinkiewicz and Riesz-Thorin interpolation theorem. Hilbert and Riesz transforms, singular integrals, Calderon-Zygmund operators. Other topics in harmonic and Fourier analysis at the choice of the instructor (e.g. Littlewood-Paley theory, Marcinkiewicz multiplier theorem, fast Fourier transform, wavelets). Textbook choices: Fourier Analysis: An Introduction by E.M. Stein and R. Shakarchi. Singular Integrals and Differentiability Properties of Functions by E.M. Stein. Real and Complex Analysis by W. Rudin. Introduction to Fourier Analysis and Wavelets by M.A. Pinsky. Classical and Modern Fourier Analysis by L. Grafakos.
Prerequisites: MATH 5111
Credits: 3

Extended description: With a change of content, this course is repeatable to a maximum of six credits. Harmonic analysis on Abelian and non-Abelian locally compact groups, Pontryagin duality, the Peter-Weyl theorem, various Fourier transforms and connections to unitary representation theory. Textbook choices: A Course in Abstract Harmonic Analysis by G.B. Folland.
Prerequisites: MATH 5111
Credits: 3

Extended description: Foundation of probability theory, pi-lambda theorem, Kolmogorov extension theorem and infinite product spaces, Kolmogorov zero-one law, a.s. convergence, convergence in probability and in L^p of random variables, Borell-Cantelli lemma. Convergence of series of independent random variables: the theorems of Kolmogorov and Levy. Weak and Strong Laws of Large Numbers. Weak convergence of probability measures, characteristic functions, tightness and Prohorov's theorem, Levy's continuity theorem, the Central Limit Theorem, Lindberg-Feller theorem, Levy-Khintchine formula, stable laws. Conditional expectation and related results, sub- and super- discrete time martingales, uniform integrability and the martingale a.s. convergence theorem, Doob's maximal inequality and convergence in L^p, Optional Stopping Theorem. Definition, existence and basic properties of the Brownian Motion. Other topics in probability theory at the choice of the instructor (e.g. Markov chains, Birkhoff-Khinchine and Kigman ergodic theorems, Levy's arcsine law, Law of Iterated Logarithm, convergence to stable laws). See also Probability Prelim Study Guide. Textbook choices: Real Analysis and Probability by R.M. Dudley. Probability, Theory and Examples, 4th edition by R. Durrett. Probability Theory by S.R.S. Varadhan. Foundations of Modern Probability by O. Kallenberg. Supplementary reading: Lecture Notes by R. Bass. A Course in Probability by K.L. Chung. Probability with Martingales by D. Williams.
Prerequisites: MATH 5111
Credits: 3

Extended description: The course material changes with each occurrence of the course and may be taken for credit repeatedly with the instructor's permission. Continuous time random processes, Kolmogorov's continuity theorem. Brownian Motion: the Donsker invariance principle, Holder continuity, quadratic variation. Continuous-time martingales and square integrable martingales. Markov processes and the strong Markov property. Properties of Brownian Motion: strong Markov property, Blumenthal zero-one law, Law of Iterated Logarithm. Stochastic integration with respect to continuous local martingales, Ito's formula, Levy's Characterization of Brownian motion, Girsanov transformation, Stochastic Differential Equations with Lipschitz Coefficients. Other topics in probability theory and stochastic processes at the choice of the instructor (e.g. connection to PDEs, local time, Skorokhod's embedding theorem, zeros of the BM, empirical processes, concentration inequalities and applications in non-parametric statistics, infinite dimensional analysis, processes with stationary independent increments and infinitely divisible processes, jump measures and Levy measures, random orthogonal measures, symmetric Markov processes, stochastic analysis for jump process). Textbook choices: Foundations of Modern Probability by O. Kallenberg. Brownian Motion and Stochastic Calculus, 2nd Ed. by I. Karatzas and S. Shreve. Continuous Martingales and Brownian Motion, 3rd Ed. by D. Revuz and M. Yor. Supplementary reading: Lecture Notes by R. Bass. Stochastic Processes by S.R.S. Varadhan. Introduction to the Theory of Random Processes by N.V. Krylov.
Prerequisites: MATH 5160
Credits: 3

Extended description: Predicate calculus, completeness theorem, compactness and applications, Lowenheim-Skolem theorems, formal theories with applications to algebra, Godel's incompleteness theorems. Further topics chosen from: axiomatic set theory (ordinals, cardinals, infinite combinatorics, independence), model theory (quantifier elimination, types), recursion theory (reducibilities, degree structures, arithmetic hierarchy, Post's problem) or proof theory (deductive systems, cut elimination). Textbook choices: Mathematical Logic by Ebbinghaus, Flum and Thomas. A Mathematical Introduction to Logic by Enderton. Mathematical Logic by Shoenfield. Supplementary reading: Set Theory by Kunen. Model Theory: An Introduction by Marker. Recursively Enumerable Sets and Degrees by Soare. Degrees of Unsolvability by Lerman. Basic Proof Theory by Troelstra and Schwichtenberg.
Prerequisites: MATH 5210
Credits: 3

Extended description: Topological spaces, maps, induced topologies, separation axioms, compactness, connectedness, classification of surfaces, the fundamental group and its applications, covering spaces. Syllabus: see the Geometry/Topology prelim study guide. Textbook choices: Algebraic Topology by A. Hatcher, also available online. Introduction to Topological Manifolds by J. M. Lee. Topology, 2nd ed. by J. R. Munkres.
Prerequisites: MATH 5110, which may be taken concurrently.
Offered: Fall
Credits: 3

Extended description: With a change of content, this course is repeatable to a maximum of six credits. This course is offered either as an introduction to algebraic topology or differential topology. Differential topology: review of topology; smooth manifolds; vector fields, tangent and cotangent bundles; submersions, immersions and embeddings, submanifolds; Lie groups; differential forms, orientations, integration on manifolds; De Rham cohomolgy, singular cohomology and de Rham theorem; other topics in geometry at the choice of the instructor (e.g. intrinsic Riemannian geometry of surfaces). Algebraic topology: simplical homology, singular homology; Eilenberg-Steenrod axioms; Mayer-Vietoris sequences; CW-homology; cohomology; cup products; Hom and Tensor products; Ext and Tor; the Universal Coefficient Theorems; Cech cohomology and Steenrod homology; Poincare, Alexander, Lefschetz, Alexander-Pontryagin dualities. Textbook choices: Introduction to Smooth Manifolds by J.M. Lee. Topology and Geometry by G.E. Bredon. Algebraic Topology by A. Hatcher, also available online. Elements of algebraic topology by J.R. Munkres.
Prerequisites: Math 5310
Credits: 3

Extended description: This course is an introduction to the study of differentiable manifolds on which various differential and integral calculi are developed. The topics include covariant derivatives and connections, geodesics and exponential map, Riemannian metrics, curvature tensor, Ricci and scalar curvature. Textbook choices: Riemannian manifolds. An introduction to curvature by J.M. Lee. Riemannian geometry by M. do Carmo. Riemannian geometry by P. Petersen.
Prerequisites: MATH 5310
Credits: 3