*Note: Unless otherwise noted, all prerequisite
courses must be passed with a grade of C- or higher.
Under normal circumstances it is assumed that a student has taken basic courses on the 8000 level before entering any of the 9000-level courses.
*

**5000. Special Topics in Math (3 s.h.) **

**5001. Linear Algebra (3 s.h.) **

Vector spaces and subspaces over the real and complex numbers; linear independence and bases; linear mappings; dual and quotient spaces; fields and general vector spaces; polynomials, ideals and factorization of polynomials; determinant; Jordan canonical form. Fundamentals of multilinear algebra.

**5041. Concepts of Analysis I
(3 s.h.) **

This two-semester course covers advanced calculus in one and several real variables. Topics include topology of metric spaces, continuity, sequences and series of numbers and functions, convergence, including uniform convergence. Ascoli and Stone-Weierstrass theorems. Integration and Fourier series. Inverse and implicit function theorems, differential forms, Stokes theorem.

**5043. Introduction to
Numerical Analysis (3 s.h.) **

Roots of nonlinear equations, errors, their source and propagation, linear systems, approximation and interpolation of functions, numerical integration.

**5045. Ordinary Differential
Equations (3 s.h.) **

Existence and uniqueness theorems, continuous and smooth dependence on parameters, linear differential equations, asymptotic behavior of solutions, isolated singularities, nonlinear equations, Sturm-Liouville problems, numerical solution of ODEs.

**8001. Candidates Seminar (3 s.h.) **

Challenging problems from many different areas of mathematics are posed and discussed.

**8002. Candidates Seminar (3 s.h.) **

Challenging problems from many different areas of mathematics are posed and discussed.

**8003. Number Theory I
(3 s.h.) **

This two-semester course gives an introduction to the ideas and techniques of number theory, elementary, analytic, and algebraic. The object of the course is to demonstrate how real and complex analysis and modern algebra can be applied to classical problems in number theory. References: H. Rademacher, "Lectures on elementary number theory"; H. Davenport, "Multiplicative number theory"; Rosen and Ireland, "A classical introduction to algebraic number theory".

**8007. Introduction to Methods in Applied Mathematics I
(3 s.h.) **

This two-semester course gives a general overview of mathematical concepts and tools for applied mathematics. Topics covered in the first semester (Math 8007) include modeling and derivation of equations of continuum mechanics; solution methods for linear PDE in special domains, such as Fourier and Laplace transforms as well as Green's functions; calculus of variations and control theory. Material covered in the second semester (Math 8008) of the course is largely independent of the first semester; it includes the following topics: dynamical systems and bifurcation theory; asymptotic analysis and perturbation theory; systems of hyperbolic conservation laws.

**8011. Abstract Algebra I
(3 s.h.) **

This two-semester course sequence gives a rigorous introduction to the terminology and methods of modern abstract algebra. The main topics covered are: groups, rings, fields, Galois theory, modules, and (multi-)linear algebra. The course is a prerequisite for many of the higher-level courses below, and it provides the background needed for the PhD qualifying exam in Algebra.

**8013. Numerical Linear
Algebra I (3 s.h.) **

The syllabus of this year-long course includes iterative methods, classical methods, nonnegative matrices. Semi-iterative methods. Multigrid methods. Conjugate gradient methods. Preconditioning. Domain decomposition. Direct Methods. Sparse Matrix techniques. Graph theory. Eigenvalue Problems.

**8023. Numerical Differential Equations I
(3 s.h.) **

Analysis and numerical solution of ordinary and partial differential equations. Elliptic, parabolic and hyperbolic systems. Constant and variable coefficients. Finite difference methods. Finite element methods. Convergence analysis. Practical applications. This course is usually offered alternating with the Math 8013/8014 course sequence.

**8031. Probability Theory (3
s.h.) **

With a rigorous approach the course covers the axioms, random variables, expectation and variance. Limit theorems are developed through characteristic functions.

**8032. Stochastic Processes (3
s.h.) **

Random sequences and functions; linear theory; limit theorems; Markov processes; branching processes; queuing processes.

**8041. Real Analysis I (3 s.h.) **

*Prerequisite: Math 5041 or equivalent. *

**8042. Real Analysis II (3 s.h.) **

*Prerequisite: Math 8041 or equivalent. *

The syllabus coincides with the syllabus for the PhD
qualifying exam in Real Analysis.

**8051. Functions of a Complex Variable I (3 s.h.) **

*Prerequisite: Math 4051 or equivalent. *

**8052. Functions of a Complex Variable II (3 s.h.) **

*Prerequisite: Math 8051 or equivalent. *

Analytic functions. Conformal mapping. Analytic continuation. Topics
in univalent functions, elliptic functions, Riemann surfaces, analytic
number theory. Nevanlinna theory, several complex variables.
This course
provides the background needed for the PhD qualifying exam in Complex
Analysis.

** 8061. Differential Geometry and Topology I
(3 s.h.) **

This course is a year-long primer on differentiable manifolds and their topology; it provides the background needed for the PhD qualifying exam in Differential Geometry & Topology. The material covered in the first semester (Math 8061) includes the elementary theory of smooth manifolds; singular cohomology and DeRham's theorem; fundamental group and covering spaces; and Hodge theory. The second semester will cover the fundamental tools in algebraic topology needed to distinguish manifolds from one another. These tools include the fundamental group, van Kampen's theorem, covering spaces, homology and cohomology. Time permitting, the course will also cover one or both of Poincaré duality and the rudiments of Hodge theory.

**8107. Mathematical Modeling for Science, Engineering, and Industry
(3 s.h.) **

In this course, students work in groups on projects that arise in industry, engineering, or in other disciplines of science. In addition to being advised by the course instructors, in all projects an external partner is present. The problems are formulated in non-mathematical language, and the final results need to be formulated in a language accessible to the external partner. This means in particular that the mathematical and computational methods must be selected or created by the students themselves. Students disseminate their progress and achievements in weekly presentations, a mid-term and a final project report, and a final presentation. Group work with and without the instructor's involvement is a crucial component in this course.

**8141. Partial Differential
Equations I (3 s.h.) **

**8142. Partial Differential Equations II
(3 s.h.) **

This two-semester course covers the material needed for the PhD qualifying exam in PDEs: the classical theory of partial differential equations; elliptic, parabolic, and hyperbolic operations.

**8161. Topology (3
s.h.) **

Point set topology through the Urysohn Metrization Theorem; fundamental group and covering spaces. Differential forms; the DeRham groups.

**8200. Topics in Applied
Mathematics (3 s.h.) **

*Prerequisite: Permission of instructor. *

Variable topics, such as control theory and transform theory,
will be treated.

**8210. Topics in Applied Mathematics (3 s.h.) **

*Prerequisite: Permission of instructor. *

Variable topics, such as control theory and transform theory, will be
treated.

**8700. Topics in
Computer Programming (3 s.h.) **

**8710. Topics in
Computer Programming (3 s.h.) **

**8985. Teaching in Higher Education (1-3 s.h)**

This course is required for any student seeking Temple's Teaching in Higher Education Certificate. The course focuses on the research on learning theory and the best teaching practices, with the aim of preparing students for effctive higher education teaching. All educational topics will be considered through the lens of teaching mathematics and quantitative thinking. This is a repeatable course, offered for variable credit.

**9000. Topics in Number Theory (3-6 s.h.) **

Analytic and algebraic number theory. Classical results and methods and special topics such as partition theory, asymptotic, Zeta functions, transcendence, modular functions.

**9003. Modular Functions I
(3 s.h.) **

This two-semester course focuses upon the modular group and its subgroups, the corresponding fundamental region and their invariant functions. Included will be a discussion of the basic properties of modular forms and their construction by means of Eisenstein and Poincaré series and theta series. Other topics: the Hecke correspondence between modular forms and Dirichlet series with functional equations, the Peterson inner product, the Hecke's operators. Emphasis will be placed upon applications to number theory. References: M. Knopp, "Modular functions in analytic number theory"; J. Lehner, "A short course in automorphic forms"; B. Schoeneberg, "Elliptic modular forms".

**9005****. Combinatorial Mathematics (3-6 s.h.) **

Topics include: Enumeration, Trees, Graphs, Codes, Matchings, Designs, Chromatic Polynomials, Coloring, Networks.

**9010. Topics in Number Theory (3-6 s.h.) **

Analytic and algebraic number theory. Classical results and methods and special topics such as partition theory, asymptotic, Zeta functions, transcendence, modular functions.

**9011. Homological
Algebra (3 s.h.) **

In this course students will learn about some fundamental notions of homological algebra such as chain complexes, Abelian categories, derived functors, and spectral sequences. A portion of this course is also devoted to the rudiments of category theory.

**9012. Representation Theory I
(3 s.h.) **

*Prerequisites: Math 8011 and 8012 or permission of instructor. *

**9013. Representation Theory II
(3 s.h.) **

*Prerequisites: Math 9012 or permission of instructor. *

This is a two-semester course on the principal methods
and results of algebraic representation theory. The first semester (Math 9012)
of the course will start with an
introduction to the fundamental
notions, tools and general results of representation theory in the setting of
associative algebras. This will be followed by a thorough coverage of the
classical representation theory of finite groups over an algebraically closed
field
of characteristic zero. If time permits, the semester will conclude
with a brief introductory discussion of the representation
theory of the general linear group.
The main focus in the second semester will be on
representations of finite-dimensional Lie algebras, with
particular emphasis on
the case of semisimple Lie algebras. Time permitting, the course will conclude
with an introduction to the representation theory of Hopf algebras.

**9014. Commutative Algebra and Algebraic Geometry I
(3 s.h.) **

This is a two-semester course on the fundamental concepts of commutative algebra and classical as well as modern algebraic geometry. Topics for the first semester include: ideals of commutative rings, modules, Noetherian and Artinian rings, Noether normalization, Hilbert's Nullstellensatz, rings of fractions, primary decomposition, discrete valuation rings and the rudiments of dimension theory. Topics for the second semester include: affine and projective varieties, morphisms of algebraic varieties, birational equivalence, and basic intersection theory. In the second semester, students will also learn about schemes, morphisms of schemes, coherent sheaves, and divisors.

**9021. Riemannian Geometry
(3 s.h.) **

*Prerequisites: Math 8061 and knowledge of some topics in
algebraic topology such as the fundamental group and covering spaces.
Taking Math 8062 concurrently will be sufficient for the
background in topology. *

The main goal of this one-semester course is to provide a
solid introduction to the two central
concepts of Riemannian Geometry, namely, geodesics and
curvature and their relationship.
After taking this course, students will have an intimate
acquaintance with the tools and
concepts that are needed for
pursuing research in Riemannian Geometry or applying its
ideas to other fields of mathematics
such as analysis, topology, and algebraic geometry. The topics covered include
Riemannian metrics, Riemannian connections, geodesics, curvature
(sectional, Ricci, and
scalar curvatures), the Jacobi equation, the second fundamental
form, and global results
such as the Gauss-Bonnet Theorem, the theorems of Hopf-Rinow
and Hadamard, variations
of energy, the theorems of Bonnet-Myers and of
Synge-Weinstein, and the Rauch comparison theorem.

**9023. Knot Theory and Low-Dimensional Topology I
(3 s.h.) **

*Prerequisites: Math 8061-62 or permission of the instructor. *

**9024. Knot Theory and Low-Dimensional Topology II
(3 s.h.) **

*Prerequisites: Math 9023 or permission of instructor. *

This is a two-semester course surveying the modern theory of knots and
providing
an introduction to some fundamental results and techniques of
low-dimensional topology. The course
will start at the very beginning of knot theory; it will then
proceed to several classical knot invariants
(Alexander, Jones, HOMFLY polynomials). The first semester
will also touch on braid groups and
mapping class groups, and use these groups to show that
every (closed, orientable)
3-manifold can be constructed via knots.
in particular exploring the connection between knots and braid
groups. It will also use Dehn surgery techniques to extend
construct quantum invariants of closed 3-dimensional manifolds.
Finally, the course will survey several results in 4-dimensional topology
and their connection to knot theory.

**9031. Advanced Probability Theory (3 s.h.) **

This course is a continuation of Math 8031 and is based on measure theory. It covers advanced topics in probability theory: martingales, Brownian motion, Markov chains, continuous time Markov processes, ergodic theory and their applications.

**9041. Functional Analysis I
(3 s.h.) **

This is a year-long sequence. Topics covered include: Banach and Hilbert spaces, Banach-Steinhaus theorem, Hahn-Banach theorem, Stone-Weierstrass theorem, Operator theory, self-adjointness, compactness. Also covered are Sobolev spaces, embedding theorems, Schwartz distributions, Paley-Wiener theory. If time permits, Banach and C algebras will be covered.

**9043. Calculus of Variations (3 s.h.) **

**9044. Harmonic Analysis (3 s.h.) **

A year long course to explore the real-variable techniques developed in Harmonic Analysis to study smoothness properties of functions and the behavior of certain spaces under the action of some operators. These techniques are also essential in many applications to PDE's and several complex variables. Offered every two years.

**9051. Several Complex Variables (3 s.h.) **

Holomorphic functions of several complex variables, domains of holomorphy, pseudoconvexity, analytic varieties, CR manifolds.

**9052. Several Complex Variables (3 s.h.) **

Holomorphic functions of several complex variables, domains of holomorphy, pseudoconvexity, analytic varieties, CR manifolds.

**9053. Harmonic Analysis (3 s.h.) **

A year long course to explore the real-variable techniques developed in Harmonic Analysis to study smoothness properties of functions and the behavior of certain spaces under the action of some operators. These techniques are also essential in many applications to PDE's and several complex variables. Offered every two years.

**9061. Lie Groups (3-6 s.h.) **

This course develops Lie theory from the ground up. Starting with basic definitions of Lie group-manifolds and Lie algebras, the course develops structure theory, analytic and algebraic aspects, and representation theory. Interactions with other fields, e.g., differential equations and geometry are also discussed.

**9062. Lie Groups (3-6 s.h.) **

This course develops Lie theory from the ground up. Starting with basic definitions of Lie group-manifolds and Lie algebras, the course develops structure theory, analytic and algebraic aspects, and representation theory. Interactions with other fields, e.g., differential equations and geometry are also discussed.

**9063. Riemann
Surfaces (3 s.h.) **

**9064. Riemann Surfaces (3
s.h.)**

**9071.
Differential Topology (3 s.h.)**

**9072.
Differential Topology (3 s.h.)**

**9082. Independent Study (1-3 s.h.) **

**9083. Independent Study (1-3 s.h.) **

**9100. Topics in Algebra (3-6 s.h.) **

Variable topics in theory of commutative and non-commutative rings, groups, algebraic number theory, algebraic geometry.

**9110. Topics in Algebra (3-6 s.h.) **

Variable topics in theory of commutative and non-commutative rings, groups, algebraic number theory, algebraic geometry.

**9200. Topics in Numerical Analysis (3-6 s.h.) **

These courses cover some basic, as well as advanced topics in numerical analysis. The topics can be changed from time to time. The usual topics include: scientific computing, numerical methods for differential equations, computational fluid dynamics, Monte Carlo simulation, Optimization, Spare matrices, Fast Fourier transform and applications, etc.

**9210. Topics in Numerical Analysis (3-6 s.h.) **

These courses cover some basic, as well as advanced topics in numerical analysis. The topics can be changed from time to time. The usual topics include: scientific computing, numerical methods for differential equations, computational fluid dynamics, Monte Carlo simulation, Optimization, Spare matrices, Fast Fourier transform and applications, etc.

**9300. Seminar in Probability (3-6 s.h.) **

Research topics related to probability theory are presented in the seminar. Topics vary depending on the interests of the students and the instructor. Current topics include stochastic calculus with applications in mathematical finance, statistical mechanics, interacting particle systems, percolation, and probability models in mathematical physics.

**9310. Seminar in Probability (3-6 s.h.) **

Research topics related to probability theory are presented in the seminar. Topics vary depending on the interests of the students and the instructor. Current topics include stochastic calculus with applications in mathematical finance, statistical mechanics, interacting particle systems, percolation, and probability models in mathematical physics.

**9400. Topics in Analysis (3 s.h.) **

Variable content course.

**9410. Topics in Functional Analysis (3 s.h.) **

This is a year long sequence with 9041-9042 or its equivalent as a prerequisite. The content varies from time to time depending on the interests of the students. Typical topics include some of the following: pseudodifferential operators, Fourier integral operators, singular integral operators, applications to partial differential equations.

**9420. Topics in Differential
Equations (3-6 s.h.) **

This is a year long sequence with 8141-8142 or its equivalent as a prerequisite. Topics covered may include the theory of elliptic partial differential equations in divergence form and non-divergence form, and nonlinear PDEs. These courses may also focus on pseudodifferential operators and Fourier integral operators.

**9994. Preliminary Examination
Preparation (1-6 s.h.) **

**9996. Master's Thesis Project
(3 s.h.) **

**9998. Pre-Dissertation
Research (1-6 s.h.) **

**9999.
Dissertation Research (1-6 s.h.)**