### MATH 5000: Mathematical Pedagogy

**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

Fall 2016 by Fabiana A. Cardetti

Fall 2015 by Fabiana A. Cardetti

Fall 2014 by Fabiana A. Cardetti

Fall 2013 by Fabiana A. Cardetti

Fall 2012 by Patrick Dragon

Fall 2011 by Amit Savkar

Fall 2011 by Kevin Marinelli

Fall 2010 by Kevin Marinelli

Fall 2009 by Kevin Marinelli

Fall 2008 by Sarah Glaz

Fall 2008 by Kevin Marinelli

### MATH 5010 - Section 1: Topics in Analysis I: Analysis on Metric Spaces

**Description:** In recent years, analysis and geometry have seen great advances where first order differential calculus and geometric measure theory have been extended from the classical Euclidean or Riemannian setting to the realm of spaces without a priori smooth structure. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in general settings. In this course we will cover a big part of Juha Heinonen's "Lectures on Analysis on Metric Spaces". Specifically, we will cover: Sobolev spaces, Poincare inequalities, Lipschitz functions and weak gradients, Quasiconformal and quasisymmetric mappings. These notions have naturally appeared in many fields of analysis and geometry such as geometric analysis, geometric function theory and sub-Riemannian geometry. The material will give the attendees a firm groundwork in analysis on metric spaces and will prepare them to study more substantial, related articles.
Instructor: Vyron Vellis

**Prerequisites:** Modern Analysis (or Real Analysis) and familiarity with measure theory.

**Offered:** Spring

**Credits:** 3

Fall 2016 by Behrang Forghani

Spring 2016 by Zhongyang Li

Spring 2014 by Alexander Teplyaev

Spring 2013 by Alexander Teplyaev

Spring 2011 by Alexander Teplyaev

Spring 2009 by Maria Gordina

### MATH 5011: Topics in Analysis II

**Description:** Advanced topics in analysis. With a change of content, this course is repeatable to a maximum of twelve credits.

**Credits:** 3

Fall 2013 by Alexander Teplyaev

### MATH 5016: Topics in Probability

**Description:** Advanced topics in probability theory, theory of random processes, mathematical statistics, and related fields. With a change of content, this course is repeatable to a maximum of twelve credits.

**Credits:** 3

Fall 2017 by Fabrice Baudoin

Fall 2017 by Zhongyang Li

Fall 2016 by Fabrice Baudoin

Fall 2014 by Richard Bass

Fall 2012 by Iddo Ben-Ari

Fall 2010 by Edwin Perkins

Fall 2008 by Richard Bass

### MATH 5020: Topics in Algebra

**Description:** Advanced topics chosen from group theory, ring theory, number theory, Lie theory, combinatorics, commutative algebra, algebraic geometry, homological algebra, and representation theory. With change of content, this course may be repeated to a maximum of twelve credits.

**Prerequisites:** MATH 5211.

**Credits:** 3

Fall 2017 by Alvaro Lozano-Robledo

Spring 2017 by Thomas Roby

Spring 2017 by Kyu-Hwan Lee

Spring 2016 by Thomas Roby

Fall 2015 by Alvaro Lozano-Robledo

Fall 2015 by Jerzy Weyman

Spring 2015 by Thomas Roby

Spring 2015 by Liang Xiao

Fall 2014 by Thomas Roby

Fall 2014 by Sarah Glaz

Spring 2014 by Kyu-Hwan Lee

Spring 2014 by Jerzy Weyman

Fall 2013 by Kyu-Hwan Lee

Fall 2013 by Ralf Schiffler

Spring 2013 by Ralf Schiffler

Spring 2013 by Alvaro Lozano-Robledo

Fall 2012 by Sarah Glaz

Spring 2012 by Keith Conrad

Fall 2011 by Keith Conrad

Spring 2011 by Sarah Glaz

Fall 2010 by Ralf Schiffler

Fall 2009 by Kyu-Hwan Lee

Fall 2009 by Milena Hering

Spring 2009 by Alvaro Lozano-Robledo

Spring 2009 by Thomas Roby

Fall 2008 by Martin Nikolov

### MATH 5026: Topics in Mathematical Logic

**Description:** 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). With a change of content, this course is repeatable to a maximum of twelve credits.

**Prerequisites:** MATH 5260.

**Credits:** 3

Spring 2017 by Linda Brown Westrick

Fall 2016 by Damir Dzhafarov

Fall 2015 by Damir Dzhafarov

Spring 2015 by Linda Brown Westrick

Fall 2014 by David R. Solomon

Spring 2014 by Damir Dzhafarov

Fall 2012 by David R. Solomon

Spring 2012 by Henry Towsner

Fall 2011 by David R. Solomon

Spring 2010 by Paul Ellis

Fall 2009 by Paul Ellis

Spring 2009 by Asher Kach

Fall 2008 by David R. Solomon

### MATH 5030: Topics in Geometry and Topology I

**Description:** Advanced topics in geometry and topology. With a change of content this course is repeatable to a maximum of twelve credits.

**Credits:** 3

Fall 2016 by Lan-Hsuan Huang

Fall 2015 by Ovidiu Munteanu

Fall 2013 by Maria Gordina

Fall 2012 by Maria Gordina

Spring 2012 by Arend Bayer

Fall 2010 by Maria Gordina

### MATH 5031: Topics in Geometry and Topology II

**Description:** Advanced topics in geometry and topology. With a change of content this course is repeatable to a maximum of twelve credits.

**Credits:** 3

Spring 2016 by Maria Gordina

Spring 2011 by Maria Gordina

Spring 2010 by Arend Bayer

### MATH 5040: Topics in Applied Analysis I

**Description:** 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. With a change of content, this course is repeatable to a maximum of twelve credits.

**Credits:** 3

Spring 2017 by Jeffrey Connors

Fall 2015 by Vadim Olshevsky

Spring 2013 by Xiaodong Yan

Fall 2011 by Changfeng Gui

### MATH 5041: Topics in Applied Analysis II

**Description:** 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. With a change of content, this course is repeatable to a maximum of twelve credits.

**Prerequisites:** Instructor consent required.

**Credits:** 3

Spring 2012 by Changfeng Gui

### MATH 5046: Introduction to Complex Variables

**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. Not open to students who have passed MATH 3146. Open for master's credit but not doctoral credit toward degree in Mathematics.

**Credits:** 3

Spring 2017 by Angelynn Alvarez

Spring 2017 by Angelynn Alvarez

Fall 2016 by Zhongyang Li

Spring 2016 by Damin Wu

Fall 2015 by Andrew H. Haas

Fall 2014 by Zhongyang Li

Fall 2013 by Benjamin Bailey

Spring 2013 by Wolodymyr R. Madych

Fall 2012 by Alexander Teplyaev

Spring 2012 by John Baber

Fall 2011 by William Abikoff

Spring 2011 by John Baber

Fall 2010 by Wolodymyr R. Madych

Fall 2009 by Wolodymyr R. Madych

Spring 2009 by William Abikoff

Fall 2008 by Maria Gordina

### MATH 5050: Analysis

**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. 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

### MATH 5070: Topics in Scientific Computation

**Credits:** 3

Fall 2014 by Vadim Olshevsky

Spring 2013 by Vadim Olshevsky

Fall 2011 by Vadim Olshevsky

### MATH 5110: Introduction to Modern Analysis

**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.

**Offered:** Fall

**Credits:** 3

Fall 2016 by Luke Rogers

Fall 2015 by Matthew Badger

Fall 2014 by Matthew Badger

Fall 2013 by Yung Choi

Fall 2012 by Patrick J. McKenna

Fall 2011 by Luke Rogers

Fall 2010 by Patrick J. McKenna

Fall 2009 by Erin T. Mullen

Fall 2008 by Michael Neumann

### MATH 5111: Real Analysis

**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,

*L*

^{2}and

*L*spaces as Hilbert and Banach spaces, Riesz representation theorem for bounded functionals on

^{p}*L*. Integration on product spaces, Fubini's and Tonelli's theorems, Fourier transform and Plancherel theorem. For prelim preparation, see the prelim study guide.

^{p}**Prerequisites:** MATH 5110.

**Offered:** Spring

**Credits:** 3

Spring 2017 by Janna Lierl

Spring 2016 by Vasileios Chousionis

Spring 2014 by Iddo Ben-Ari

Spring 2012 by Alexander Teplyaev

Spring 2011 by Evarist Gine-Masdeu

Spring 2010 by Iddo Ben-Ari

Spring 2009 by Iddo Ben-Ari

### MATH 5120: Complex Analysis

**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. A practical purpose of the class is to prepare students to take the qualifying exams.

For prelim preparation, see the prelim study guide.

**Prerequisites:** MATH 5110.

**Credits:** 3

Spring 2017 by Iddo Ben-Ari

Spring 2016 by Matthew Badger

Spring 2015 by Damin Wu

Spring 2014 by Ovidiu Munteanu

Spring 2013 by Maria Gordina

Spring 2012 by Luke Rogers

Spring 2011 by William Abikoff

Spring 2010 by Andrew H. Haas

Spring 2009 by Luke Rogers

### MATH 5121: Topics in Complex Function Theory

**Description:** Advanced topics of contemporary interest. These include Riemann surfaces, Kleinian groups, entire functions, conformal mapping, several complex variables, and automorphic functions, among others. With a change of content this course may be repeatable to a maximum of twelve credits.

**Prerequisites:** MATH 5120.

**Credits:** 3

Fall 2017 by Lan-Hsuan Huang

Fall 2016 by Damin Wu

Fall 2015 by Keith Conrad

Fall 2014 by Damin Wu

Fall 2013 by Ovidiu Munteanu

Fall 2008 by William Abikoff

### MATH 5130: Functional Analysis I

**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. unbounded self-adjoint operators, distributions, Banach algebra *L*^{1}, 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).

**Prerequisites:** MATH 5111.

**Credits:** 3

Fall 2015 by Iddo Ben-Ari

Fall 2013 by Richard Bass

Fall 2011 by Richard Bass

Fall 2009 by Richard Bass

### MATH 5131: Functional Analysis II

**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).

**Prerequisites:** MATH 5111.

**Credits:** 3

Spring 2014 by Richard Bass

Spring 2012 by Maria Gordina

Spring 2010 by Richard Bass

### MATH 5140: Fourier Analysis

**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).

**Prerequisites:** MATH 5111.

**Credits:** 3

Fall 2014 by Luke Rogers

Fall 2012 by Richard Bass

Fall 2010 by Luke Rogers

### MATH 5141: Abstract Harmonic Analysis

**Description:** Harmonic analysis on various spaces such as Euclidean spaces, and abelian and non-abelian locally compact groups. Pontryagin duality, the Peter-Weyl theorem, various Fourier transforms and connections to unitary representation theory.With a change of content, this course is repeatable to a maximum of six credits.

**Prerequisites:** MATH 5111.

**Credits:** 3

Spring 2015 by Maria Gordina

Fall 2011 by Maria Gordina

### MATH 5160: Probability Theory and Stochastic Processes I

**Description:** Foundation of probability theory, monotone classes and 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 convergence of probability measures: characteristic functions, Levy-Cramer continuity theorem, tightness and Prohorov's theorem. The Central Limit Theorem: the Lindeberg-Feller theorem, the Levy-Khintchine formula, stable laws. Conditional expectation. Discrete time (sub- and super) martingales: Doob's maximal inequality, Optional Stopping Theorem, uniform integrability, and the a.s. convergence theorem for

*L*

^{1}bounded martingales, convergence in

*L*. 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).

^{p}**Prerequisites:** MATH 5111.

**Credits:** 3

Fall 2016 by Zhongyang Li

Fall 2015 by Joe Chen

Fall 2014 by Richard Bass

Fall 2013 by Evarist Gine-Masdeu

Fall 2011 by Evarist Gine-Masdeu

Fall 2010 by Iddo Ben-Ari

Fall 2009 by Evarist Gine-Masdeu

Fall 2008 by Alexander Teplyaev

### MATH 5161: Probability Theory and Stochastic Processes II

**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). With a change of content, this course is repeatable to a maximum of six credits.

**Prerequisites:** MATH 5160.

**Credits:** 3

Spring 2017 by Fabrice Baudoin

Spring 2016 by Alexander Teplyaev

Spring 2015 by Zhongyang Li

Spring 2014 by Maria Gordina

Spring 2012 by Iddo Ben-Ari

Spring 2011 by Richard Bass

Spring 2009 by Alexander Teplyaev

### MATH 5210: Abstract Algebra I

**Description:** Group theory, ring theory and modules, and universal mapping properties. For prelim preparation, see the prelim study guide.

**Offered:** Fall

**Credits:** 3

Fall 2016 by Alvaro Lozano-Robledo

Fall 2015 by Ralf Schiffler

Fall 2014 by Jerzy Weyman

Fall 2013 by Keith Conrad

Fall 2012 by Kyu-Hwan Lee

Fall 2011 by Ralf Schiffler

Fall 2010 by Milena Hering

Fall 2009 by Ralf Schiffler

Fall 2008 by Kyu-Hwan Lee

### MATH 5211: Abstract Algebra II

**Description:** Linear and multilinear algebra, Galois theory, category theory, and commutative algebra.

**Prerequisites:** MATH 5210.

**Offered:** Spring

**Credits:** 3

Spring 2017 by Charles Paquette

Spring 2016 by Ralf Schiffler

Spring 2015 by Jerzy Weyman

Spring 2014 by Keith Conrad

Spring 2013 by Keith Conrad

Spring 2012 by Ralf Schiffler

Spring 2011 by Ralf Schiffler

Spring 2010 by Ralf Schiffler

Spring 2009 by Kyu-Hwan Lee

### MATH 5220: Introduction to Representation Theory

**Description:** Introduction to the representation theory of finite groups and Lie algebras. Characters, induced representations, representations of the symmetric and general linear groups, symmetric functions, Schur-Weyl duality, representations of complex semi-simple Lie algebras, and the Weyl character formulae.

**Prerequisites:** MATH 5210.

**Credits:** 3

Fall 2012 by Ralf Schiffler

Spring 2010 by Ryan Kinser

### MATH 5230: Algebraic Number Theory

**Description:** Algebraic integers, ideal class group, ramification, Frobenius elements in Galois groups, Dirichlet's unit theorem, localization, and completion. Further topics (zeta-functions, function fields, non-maximal orders) as time permits.

**Prerequisites:** MATH 5211.

**Credits:** 3

Fall 2014 by Liang Xiao

Fall 2012 by Keith Conrad

Fall 2010 by Alvaro Lozano-Robledo

Fall 2008 by Keith Conrad

### MATH 5250: Modern Matrix Theory and Linear Algebra

**Description:** 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.

**Offered:** Fall

**Credits:** 3

Spring 2015 by Vadim Olshevsky

Fall 2013 by Vadim Olshevsky

Fall 2012 by Vadim Olshevsky

Fall 2010 by Michael Neumann

### MATH 5260: Mathematical Logic I

**Description:** Predicate calculus, completeness theorem, compactness and applications, Lowenheim-Skolem theorems, formal theories with applications to algebra, Goedel'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).

**Prerequisites:** MATH 5210.

**Credits:** 3

Spring 2016 by David R. Solomon

Fall 2013 by David R. Solomon

Spring 2011 by David R. Solomon

### MATH 5310: Introduction to Geometry and Topology I

**Description:** Topological spaces, maps, induced topologies, separation axioms, compactness, connectedness, classification of surfaces, the fundamental group and its applications, covering spaces. For prelim preparation, see the prelim study guide.

**Prerequisites:** MATH 5110, which may be taken concurrently.

**Offered:** Fall

**Credits:** 3

Fall 2016 by Ovidiu Munteanu

Fall 2015 by Lan-Hsuan Huang

Fall 2014 by Lan-Hsuan Huang

Fall 2013 by Iddo Ben-Ari

Fall 2012 by Ovidiu Munteanu

Fall 2011 by Milena Hering

Fall 2010 by Arend Bayer

Fall 2009 by Jeffrey L. Tollefson

Fall 2008 by Jeffrey L. Tollefson

### MATH 5311: Introduction to Geometry and Topology II

**Description:** 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. With a change of content, this course is repeatable to a maximum of six credits.

**Prerequisites:** MATH 5310.

**Credits:** 3

Spring 2017 by Fabrice Baudoin

Spring 2015 by Ovidiu Munteanu

Spring 2014 by Lan-Hsuan Huang

Spring 2012 by James Bridgeman

Spring 2011 by Alexander Teplyaev

### MATH 5320: Algebraic Geometry I

**Description:** This course is an introduction to algebraic varieties: affine and projective varieties, dimension of varieties and subvarieties, algebraic curves, singular points, divisors and line bundles, differentials, intersections.

**Prerequisites:** MATH 5211 and 5310, which may be taken concurrently.

**Credits:** 3

Spring 2016 by Liang Xiao

Fall 2013 by Damin Wu

Fall 2011 by Arend Bayer

Fall 2008 by Kinetsu Abe

### MATH 5321: Algebraic Geometry II

**Description:** This course introduces further concepts and methods of modern algebraic geometry, including schemes and cohomology.

**Prerequisites:** MATH 5320.

**Credits:** 3

Spring 2012 by Milena Hering

### MATH 5360: Differential Geometry

**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.

**Prerequisites:** MATH 5310.

**Credits:** 3

Spring 2016 by Lan-Hsuan Huang

Spring 2013 by Lan-Hsuan Huang

Spring 2009 by Kinetsu Abe

### MATH 5410: Introduction to Applied Mathematics I

**Description:** 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.

**Credits:** 3

Spring 2017 by Xiaodong Yan

Spring 2016 by Yung Choi

Spring 2015 by Xiaodong Yan

Spring 2014 by Yung Choi

Spring 2013 by Patrick J. McKenna

Spring 2012 by Xiaodong Yan

Spring 2011 by Yung Choi

Spring 2010 by Changfeng Gui

Spring 2009 by Yung Choi

### MATH 5411: Introduction to Applied Mathematics II

**Description:** 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.

**Credits:** 3

Fall 2016 by Patrick J. McKenna

Fall 2015 by Yung Choi

Fall 2014 by Patrick J. McKenna

Fall 2013 by Patrick J. McKenna

Fall 2012 by Xiaodong Yan

Fall 2011 by Yung Choi

Fall 2010 by Changfeng Gui

Fall 2009 by Changfeng Gui

Fall 2008 by Greg Huber

### MATH 5420: Ordinary Differential Equations

**Description:** Existence and uniqueness of solutions, stability and asymptotic behavior. If time permits: eigenvalue problems, dynamical systems, existence and stability of periodic solutions.

**Prerequisites:** MATH 5111.

**Credits:** 3

### MATH 5430: Applied Analysis

**Description:** Convergence of Fourier series, Legendre and Hermite polynomials, existence and uniqueness theorems, two-point boundary value problems and Green's functions.

**Prerequisites:** MATH 5111 and 5140 are helpful but not required.

**Credits:** 3

### MATH 5435: Introduction to Partial Differential Equations

**Description:** Solution of first and second-order partial differential equations with applications to engineering and science.

**Prerequisites:** Not open to students who have passed MATH 3435. Not open for graduate credit toward degrees in mathematics.

**Credits:** 3

### MATH 5440: Partial Differential Equations

**Description:** Cauchy-Kovalevskaya Theorem, classification of second-order equations, systems of hyperbolic equations, the wave equation, the potential equation, the heat equation in **R**^{n}.

**Prerequisites:** MATH 5120.

**Credits:** 3

Fall 2016 by Xiaodong Yan

Fall 2015 by Xiaodong Yan

Fall 2012 by Yung Choi

Fall 2011 by Xiaodong Yan

Fall 2010 by Changfeng Gui

Fall 2008 by Changfeng Gui

### MATH 5510: Numerical Analysis and Approximation Theory I

**Description:** The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.

**Prerequisites:** MATH 5110, which may be taken concurrently.

**Offered:** Fall

**Credits:** 3

Fall 2016 by Dmitriy Leykekhman

Fall 2015 by Jeffrey Connors

Fall 2014 by Vadim Olshevsky

Fall 2013 by Dmitriy Leykekhman

Fall 2012 by Vadim Olshevsky

Fall 2011 by Vadim Olshevsky

Fall 2010 by Vadim Olshevsky

Fall 2009 by Michael Neumann

Fall 2008 by Vadim Olshevsky

### MATH 5511: Numerical Analysis and Approximation Theory II

**Description:** The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.

**Prerequisites:** MATH 5510.

**Offered:** Spring

**Credits:** 3

Spring 2012 by Vadim Olshevsky

### MATH 5520: Finite Element Solution Methods I

**Description:** Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.

**Credits:** 3

Spring 2016 by Jeffrey Connors

Spring 2014 by Jeffrey Connors

Spring 2011 by Dmitriy Leykekhman

Spring 2009 by Dmitriy Leykekhman

### MATH 5521: Finite Element Solution Methods II

**Description:** Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.

**Prerequisites:** MATH 5520.

**Credits:** 3

### MATH 5530: Mathematical Modeling

**Description:** 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.

**Credits:** 3

### MATH 5540: Computerized Modeling in Science

**Description:** 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.

**Credits:** 4

### MATH 5580: Optimization

**Description:** 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.

**Credits:** 3

### MATH 5620: Financial Mathematics I

**Description:** The mathematics of measurement of interest, accumulation and discount, present value, annuities, loans, bonds, and other securities. Not open to students who have passed MATH 2620.

**Credits:** 3

Fall 2017 by Terryn Boucher

Fall 2017 by Vladimir Pozdnyakov

Spring 2017 by Terryn Boucher

Fall 2016 by Terryn Boucher

Fall 2016 by Vladimir Pozdnyakov

Spring 2016 by Terryn Boucher

Fall 2015 by Terryn Boucher

Spring 2015 by Bruce Campbell

Fall 2014 by Terryn Boucher

Spring 2014 by Bruce Campbell

Fall 2013 by John Dinius

Fall 2013 by Terryn Boucher

Spring 2013 by Bruce Campbell

Fall 2012 by John Dinius

Spring 2012 by Bruce Campbell

Fall 2011 by John Dinius

Spring 2011 by Bruce Campbell

Fall 2010 by Gregory Smith

Spring 2010 by Gregory Smith

Fall 2009 by Gregory Smith

Fall 2009 by Bruce Campbell

Spring 2009 by Gregory Smith

Fall 2008 by Gregory Smith

### MATH 5621: Financial Mathematics II

**Description:** The continuation of Math 5620, focusing on the mathematics of finance: measurement of financial risk and the opportunity cost of capital, the mathematics of capital budgeting and securities valuation, mathematical analysis of financial decisions and capital structure, and option pricing theory. Provides VEE credit in the Corporate Finance subject area for Society of Actuaries and Casualty Actuarial Society requirements. Not open to students who have passed MATH 3650.

**Credits:** 4

Spring 2017 by Edward Perry

Fall 2016 by Christopher MacKlem

Spring 2016 by James Bridgeman

Fall 2015 by James Bridgeman

Spring 2015 by James Bridgeman

Fall 2014 by James Bridgeman

Spring 2014 by James Bridgeman

Fall 2013 by James Bridgeman

Spring 2013 by James Bridgeman

Fall 2012 by James Bridgeman

Spring 2012 by James Bridgeman

Fall 2011 by James Bridgeman

Spring 2011 by James Bridgeman

Fall 2010 by James Bridgeman

Spring 2010 by James Bridgeman

Fall 2009 by James Bridgeman

Spring 2009 by James Bridgeman

Fall 2008 by James Bridgeman

Fall 2008 by James Bridgeman

### MATH 5630: Actuarial Mathematics I

**Description:** 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. Not open to students who have passed MATH 3630.

**Prerequisites:** MATH 2620 or 5620, which may be taken concurrently.

**Offered:** Fall

**Credits:** 4

Fall 2016 by Jeyaraj Vadiveloo

Fall 2015 by Jeyaraj Vadiveloo

Fall 2014 by Jeyaraj Vadiveloo

Fall 2013 by Jeyaraj Vadiveloo

Fall 2012 by Jeyaraj Vadiveloo

Fall 2011 by Jeyaraj Vadiveloo

Fall 2010 by Jeyaraj Vadiveloo

Fall 2009 by Jeyaraj Vadiveloo

Fall 2008 by Jeyaraj Vadiveloo

### MATH 5631: Actuarial Mathematics II

**Description:** Lecture. 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. Not open to students who have passed MATH 3631.

**Prerequisites:** MATH 5630.

**Offered:** Spring

**Credits:** 4

Spring 2017 by Jeyaraj Vadiveloo

Spring 2016 by Jeyaraj Vadiveloo

Spring 2015 by Jeyaraj Vadiveloo

Spring 2014 by Jeyaraj Vadiveloo

Spring 2013 by Jeyaraj Vadiveloo

Spring 2012 by Jeyaraj Vadiveloo

Spring 2011 by Jeyaraj Vadiveloo

Spring 2010 by Jeyaraj Vadiveloo

Spring 2009 by Jeyaraj Vadiveloo

### MATH 5633: Survival Models

**Description:** Analysis, estimation, and validation of lifetime tables.

**Prerequisites:** MATH 5630 or STAT 3445.

**Credits:** 3

### MATH 5635: Introduction to Operations Research

**Description:** 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. Not open to students who have passed MATH 4535, STAT 4535, or STAT 5535.

**Credits:** 3

### MATH 5637: Risk Theory

**Description:** Individual and collective risk theory, distribution theory, ruin theory, stoploss, reinsurance and Monte Carlo methods. Emphasis is on problems in insurance. For prelim preparation, see the prelim study guide.

**Offered:** Fall

**Credits:** 3

Fall 2016 by Gao Niu

Fall 2016 by Gao Niu

Fall 2015 by James Bridgeman

Fall 2014 by James Bridgeman

Fall 2013 by James Bridgeman

Fall 2012 by James Bridgeman

Fall 2011 by James Bridgeman

Fall 2010 by James Bridgeman

Fall 2009 by James Bridgeman

Fall 2008 by James Bridgeman

### MATH 5640: Advanced Topics in Actuarial Mathematics I

**Description:** Survival models, mathematical graduation, or demography.

**Credits:** 3

Fall 2016 by Abdul Sharif

Fall 2015 by Abdul Sharif

Fall 2014 by Abdul Sharif

Fall 2013 by Gregory Smith

Fall 2012 by Gregory Smith

Fall 2011 by Gregory Smith

Fall 2010 by Gregory Smith

Fall 2009 by Gregory Smith

Fall 2008 by Gregory Smith

### MATH 5641: Advanced Topics in Actuarial Mathematics II

**Description:** Credibility theory or advanced theory of interest.

**Credits:** 3

Spring 2016 by Abdul Sharif

Spring 2015 by Abdul Sharif

Spring 2012 by Gregory Smith

Spring 2011 by Gregory Smith

Spring 2010 by Gregory Smith

Spring 2009 by Gregory Smith

### MATH 5650: Financial Mathematics II

**Description:** The continuation of Math 5620, focusing on the mathematics of finance: measurement of financial risk and the opportunity cost of capital, the mathematics of capital budgeting and securities valuation, mathematical analysis of financial decisions and capital structure, and option pricing theory. Provides VEE credit in the Corporate Finance subject area for Society of Actuaries and Casualty Actuarial Society requirements. Not open to students who have passed MATH 3650.

**Credits:** 4

### MATH 5660: Advanced Financial Mathematics

**Description:** 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.

**Offered:** Spring

**Credits:** 3

Fall 2017 by Edward Perry

Spring 2017 by Oleksii Mostovyi

Fall 2016 by Edward Perry

Spring 2016 by Oleksii Mostovyi

Fall 2015 by Edward Perry

Spring 2015 by James Bridgeman

Fall 2014 by Edward Perry

Spring 2014 by James Bridgeman

Fall 2013 by Edward Perry

Spring 2013 by James Bridgeman

Fall 2012 by Edward Perry

Spring 2012 by Edward Perry

Spring 2011 by James Bridgeman

Spring 2010 by James Bridgeman

Spring 2009 by James Bridgeman

### MATH 5661: Yield Curve Models

**Credits:** 3

### MATH 5670: Financial Programming and Modeling

**Description:** Online

**Credits:** 3

### MATH 5671: Financial Data Mining with Big Data

**Description:** Online

**Credits:** 3

### MATH 5710: Tensor Calculus I

**Description:** An introduction to tensor algebra and tensor calculus with applications chosen from the fields of the physical sciences and mathematics.

**Credits:** 3

### MATH 5711: Tensor Calculus II

**Description:** An introduction to tensor algebra and tensor calculus with applications chosen from the fields of the physical sciences and mathematics.

**Prerequisites:** MATH 5710.

**Credits:** 3

### MATH 5720: Vector Field Theory I

**Description:** Vector algebra and vector calculus with particular emphasis on invariance. Classification of vector fields. Solution of the partial differential equations of field theory.

**Credits:** 3

### MATH 5721: Vector Field Theory II

**Description:** Vector algebra and vector calculus with particular emphasis on invariance. Classification of vector fields. Solution of the partial differential equations of field theory.

**Prerequisites:** MATH 5720.

**Credits:** 3

### MATH 5800 - Section 31:

**Credits:** 0

Fall 2017 by Edward Perry

Fall 2017 by Fabiana A. Cardetti

Fall 2017 by Oleksii Mostovyi

Fall 2017 by Cuong Do

Fall 2017 by Cuong Do

Spring 2017 by Edward Perry

Spring 2017 by Cuong Do

Spring 2017 by Cuong Do

Spring 2017 by Gao Niu

Spring 2017 by Gao Niu

Fall 2016 by Fabiana A. Cardetti

Fall 2016 by Oleksii Mostovyi

Fall 2016 by Cuong Do

Fall 2016 by Cuong Do

Spring 2016 by Damin Wu

Spring 2016 by Lan-Hsuan Huang

Spring 2016 by Thomas Roby

Spring 2016 by Ralf Schiffler

Spring 2016 by Edward Perry

Spring 2016 by Alexander Teplyaev

Spring 2016 by Jeyaraj Vadiveloo

Spring 2016 by Maria Gordina

Spring 2016 by David R. Solomon

Spring 2016 by Yung Choi

Spring 2016 by Cuong Do

Spring 2016 by Cuong Do

Spring 2016 by Patrick J. McKenna

Spring 2016 by Xiaodong Yan

Spring 2016 by Kyu-Hwan Lee

Spring 2016 by Luke Rogers

Spring 2016 by Alan Parry

Fall 2015 by Fabiana A. Cardetti

Fall 2015 by Cuong Do

Fall 2015 by Cuong Do

Fall 2015 by Lan-Hsuan Huang

Spring 2015 by Edward Perry

Spring 2015 by Fabiana A. Cardetti

Spring 2015 by Cuong Do

Spring 2015 by Cuong Do

Fall 2014 by Cuong Do

Fall 2014 by Fabiana A. Cardetti

Fall 2014 by James Bridgeman

Fall 2014 by Cuong Do

Spring 2014 by Yung Choi

Spring 2014 by Lan-Hsuan Huang

Spring 2014 by Thomas Roby

Spring 2014 by Ralf Schiffler

Spring 2014 by Edward Perry

Spring 2014 by Alexander Teplyaev

Spring 2014 by Yung Choi

Spring 2014 by Jeyaraj Vadiveloo

Spring 2014 by Maria Gordina

Spring 2014 by Cuong Do

Spring 2014 by Patrick J. McKenna

Spring 2014 by Alan Parry

Spring 2014 by Patrick Dragon

Fall 2013 by James Bridgeman

Fall 2013 by Cuong Do

Spring 2013 by Edward Perry

Spring 2013 by Cuong Do

Fall 2012 by James Bridgeman

Spring 2012 by James Bridgeman

Fall 2011 by James Bridgeman

Fall 2009 by James Bridgeman

Fall 2009 by Fabiana A. Cardetti

Spring 2009 by James Bridgeman

Spring 2009 by Emiliano Valdez

Spring 2009 by Jeyaraj Vadiveloo

Spring 2009 by James Bridgeman

Spring 2009 by Louis J. Lombardi

### MATH 5850: Graduate Field Study Internship

**Description:** Participation in internship and paper describing experiences.

**Credits:** 1 to 3

Summer 2017 by Jeyaraj Vadiveloo

Summer 2017 by Vladimir Pozdnyakov

Summer 2017 by Emiliano Valdez

Fall 2017 by Vladimir Pozdnyakov

Fall 2017 by Jeyaraj Vadiveloo

Spring 2017 by Jeyaraj Vadiveloo

Spring 2017 by Vladimir Pozdnyakov

Spring 2017 by Emiliano Valdez

Summer 2016 by Edward Perry

Fall 2016 by Vladimir Pozdnyakov

Fall 2016 by Jeyaraj Vadiveloo

Spring 2016 by James Bridgeman

Summer 2015 by James Bridgeman

Summer 2015 by James Bridgeman

Summer 2015 by James Bridgeman

Summer 2014 by James Bridgeman

Summer 2014 by James Bridgeman

Summer 2014 by James Bridgeman

Fall 2014 by James Bridgeman

Spring 2014 by James Bridgeman

Fall 2013 by James Bridgeman

Spring 2012 by James Bridgeman

Fall 2008 by James Bridgeman