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# 500-Level Math Courses

### Math 520 - Life Contingencies I

Prerequisites: Math 424 and 425, both with a minimum grade of C-, and permission of instructor 3 credits. Quantifying the financial impact of uncertain events is the central challenge of actuarial mathematics. The goal of this course is to teach the basic actuarial theory of mathematical models for financial uncertainties, mainly the time of death.  In addition to actuarial students, this course is appropriate for anyone interested in mathematical modeling outside of the sciences. The main topics are the development of (1) probability distributions for the future lifetime random variable; (2) probabilistic methods for financial payments on death or survival; and (3) mathematical models of actuarial reserving.

### Math 521 - Life Contingencies II

Prerequisites: Math 520, with a minimum grade of C-, or permission of instructor 3 credits. This course extends the single decrement and single life ideas of Math 520 to multi-decrement and multiple-life applications directly related to life insurance. The sequence Math 520-521 also help students prepare for some of the professional actuarial exams. Topics include multiple life models - joint life, last survivor, contingent insurance; multiple decrement models - disability, withdrawal, retirement, etc.; and reserving models for life insurance. This corresponds to chapters 7-10, 14, and 15 of Bowers et. al.

### Math 522 - Actuarial Theory of Pensions and Social Security

Prerequisites: Math 520, with a minimum grade of C-, or permission of instructor 3 credits This course develops the mathematical models for pre-funded retirement benefit plans. Concepts and calculation are emphasized over proofs. Mathematical models for (1) retirement income, (2) retiree medical benefits, (3) disability benefits, and (4) survivor benefits. There is some coverage of how accounting theory and practice can be explained by these models and of the U.S. laws and regulations that give rise to the models used in practice.

### Math 523 - Loss Models I

Prerequisites: Math 425 with a minimum grade of C- 3 credits. Risk management is of major concern to all financial institutions, especially casualty insurance companies. This course is relevant for students in insurance and provides background for the professional examination in Short-Term Actuarial Modeling offered by the Society of Actuaries (Exam STAM). Students should have a basic knowledge of common probability distributions (Poisson, exponential, gamma, binomial, etc.) and have at least Junior standing. Review of random variables (emphasizing parametric distributions), review of basic distributional quantities, continuous models for insurance claim severity, discrete models for insurance claim frequency, the effect of coverage modification on severity and frequency distributions, aggregate loss models, and simulation.

### Math 524 - Loss Models II

Prerequisites: Math 523, Stats 426 with a minimum grade of C- 3 credits. Risk management is of major concern to all financial institutions, especially casualty insurance companies. This course is relevant for students in insurance and provides background for the professional examination in Short-Term Actuarial Modeling offered by the Society of Actuaries (Exam STAM). Students should have a basic knowledge of common probability distributions (Poisson, exponential, gamma, binomial, etc.) and have at least Junior standing. Frequentist and Bayesian estimation of probability distributions, model selection, credibility, simulation, and other topics in casualty insurance.

### Math 525 (Stats 525) - Probability Theory

Prerequisites: Math 451 (strongly recommended).  Math 425/Stats 425 would be helpful. 3 credits. This course is a thorough and fairly rigorous study of the mathematical theory of probability at an introductory graduate level. The emphasis will be on fundamental concepts and proofs of major results, but the usages of the theorems will be discussed through many examples. This is a core course sequence for the Applied and Interdisciplinary Mathematics graduate pro- gram. This course is the first half of the Math/Stats 525-526 sequence. The following topics will be covered: sample space and events, random variables, concept and definition of probability and expectation, conditional probability and expectation, independence, moment generating functions, Law of large numbers, Central limit theorem, Markov chains, Poisson process and exponential distribution.

### Math 526 (Stats 526) - Discrete State Stochastic Processes

Prerequisites: Math 525 or EECS 501 3 credits. This is a course on the theory and applications of stochastic processes, mostly on discrete state spaces. It is a second course in probability which should be of interest to students of mathematics and statistics as well as students from other disciplines in which stochastic processes have found significant applications. The material is divided between discrete and continuous time processes. In both, a general theory is developed and detailed study is made of some special classes of processes and their applications. Some specific topics include generating functions; recurrent events and the renewal theorem; random walks; Markov chains; branching processes; limit theorems; Markov chains in continuous time with emphasis on birth and death processes and queueing theory; an introduction to Brownian motion; stationary processes and martingales.

### Math 528 - Topics in Casualty Insurance

Prerequisites: Math 217, 417, 419, or permission of instructor 3 credits. Historically the Actuarial Program has emphasized life, health, and pension topics. This course will provide background in casualty topics for the many students who take employment in this field. Guest lecturers from the industry will provide some of the instruction.  Students are encouraged to take the Casualty Actuarial Society’s Course 3 examination at the completion of the course. The insurance policy is a contract describing the services and protection which the insurance company provides to the insured. This course will develop an understanding of the nature of the coverages provided and the bases of exposure and principles of the underwriting function, how products are designed and modified, and the different marketing systems. It will also look at how claims are settled, since this determines losses which are key components for insurance ratemaking and reserving.  Finally, the course will explore basic ratemaking principles and concepts of loss reserving.

### Math 537 - Introduction to Differentiable Manifolds

Prerequisites: Math 590 and 420 3 credits. This course is an introduction to the theory of smooth manifolds. The prerequisites for this course are a basic knowledge of analysis, algebra, and topology. The following topics will be discussed: smooth manifolds and maps, tangent spaces, submanifolds, vector fields and flows, basic Lie group theory, group actions on manifolds, differential forms, de Rham cohomology, orientation and manifolds with boundary, integration of differential forms, Stokes’ theorem.

### Math 547 - Mathematics of Data

Prerequisites: Flexible.  Math/Stats 425, or Biology 427, or BioChem 451, or basic programming skills desirable; or permission of instructor. 3 credits. This course covers topics in Mathematics of Data This course is open to graduate students and upper-level undergraduate in applied mathemarics, bioinformatics, statistics, and engineering, who are intereteed in learning from data. Students with other backgrounds are also welcome, provided they have maturity in mathematics. Content covered in the course includes linear algebra, multilinear algebra, dynamical systems, and information theory. The course will start with a basic introduction to data representation as vectors, matrices, and tensors. Then the course will move on to geometric methods for dimension reduction, also known as manifold learning, and topological data reduction. Using an application-based approach, the course will cover spectral graph theory, addressing the combinatorial meaning of eigenvalues and eigenvectors of their associated graph matrics and extensions to hypergraphs via tensors. The course will provide an introduction to the application of dynamical systems theory to data including dynamic mode decomposition. Real data examples will be given where possible, along with assistance writing code to implement these algoithms to solve these problems. The methods discussed in this class are shown primarily for biologocal data, but are useful in handling data across many fields. The course will also feature several guest lecturers from the industry and government.

### Math 550 (CMPLXSYS 510) - Introduction to Adaptive Systems

Prerequisites: Math 215, 255, or 285; Math 217; and Math 425 3 credits. This course centers on the construction and use of agent-based adaptive models to study phenomena which are prototypical in the social, biological, and decision sciences. These models are “agent-based” or “bottom-up” in that the structure is placed at the level of the individuals as basic components; they are “adaptive” in that individuals often adapt to their environment through evolution or learning. The goal of these models is to understand how the structure at the individual or micro level leads to emergent behavior at the aggregate or macro level.  Often the individuals are grouped into sub-populations or interesting hierarchies and the researcher may want to understand how the structure or development of these populations affects macroscopic outcomes. The course will start with classical differential equation and game theory approaches. It will then focus on the theory and application of particular models of adaptive systems such as models of neural systems, genetic algorithms, classifier systems, and cellular automata. Time permitting, we will discuss more recent developments such as sugarscape and echo.

### Math 551 - Introduction to Real Analysis

Prerequisites: Math 451 & 452, OR MATH 295 &295; AND abstract linear algebra such as Math217 3 Credits The objective of this course is to develop Lebesgue measure theory and to study abstract spaces such as metric spaces and topological spaces. This is an introductory course on real analysis at the level between MATH 451 (Advanced Calusus) and MATH 597 (Real Analysis). If time permits, we may also consider abstract measure theory. Lebesgue measure, measurable functions, Lebegue integral, convergence theorems, metric spaces, topological spaces, Hilbert and Banach spaces

### Math 555 - Introduction to Complex Variables

Prerequisites: Math 450 or 451. Students who had 450 (or equivalent) but not 451 are encouraged to take 451 simultaneously with 555. 3 credits. This course is an introduction to the theory of complex-valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. This is a core course for the AIM graduate program. Differentiation and integration of complex-valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, applications in ideal fluid dynamics. This corresponds to Chapters 1-9 of Churchill & Brown.

### Math 556 - Applied Functional Analysis

Prerequisites: Math 217, 419, or 420; Math 451; and Math 555 3 credits. This is an introduction to methods of applied analysis with emphasis on Fourier analysis and partial differential equations. Students are expected to master both the proofs and applications of major results. The prerequisites include linear algebra, advanced calculus, and complex variables. Topics in functional analysis that are used in the analysis of ordinary and partial differential equations. Metric and normed linear spaces, Banach spaces and the contraction mapping theorem, Hilbert spaces and spectral theory of compact operators, distributions and Fourier transforms, Sobolev spaces and applications to elliptic PDEs.

### Math 557 - Applied Asymptotic Analysis

Prerequisites: Math 217, 419, or 420; Math 451; and Math 555 3 credits. This is an introduction to methods of asymptotic analysis including asymptotic expansions for integrals and solutions of ordinary and partial differential equations. The prerequisites include linear algebra, advanced calculus, and complex variables. Math 556 is not a prerequisite. Topics include stationary phase, steepest descent, characterization of singularities in terms of the Fourier transform, regular and singular perturbation problems, boundary layers, multiple scales, WKB method. Additional topics depend on the instructor but may include non-linear stability theory, bifurcations, applications in fluid dynamics (Rayleigh-Benard convection), combustion (flame speed).

### Math 558 - Applied Nonlinear Dynamics

Prerequisites: Math 451 3 credits. This course is an introduction to Ordinary Differential Equations and Dynamical Systems with emphasis on qualitative analysis. The basic results on qualitative behavior, centered on themes of stability and phase plane analysis will be presented in a context that includes applications to a variety of classic examples. The proofs of the fundamental facts will be presented, along with discussions of examples.

### Math 559 - Topics in Applied Mathematics

Prerequisites: Vary by topic, check with instructor 3 credits. This is an advanced topics course intended for students with strong interests in the intersection of mathematics and the sciences, but not necessarily experience with both applied mathematics and the application field. Mathematical concepts, as well as intuitions arising from the field of application, will be stressed. This course will focus on particular topics in emerging areas of applied mathematics for which the application field has been strongly influenced by mathematical ideas. It is intended for students with interests in mathematical, computational, and/or modeling aspects of interdisciplinary science, and the course will develop the intuitions of the field of application as well as the mathematical concepts. The applications considered will vary with the instructor and may come from physics, biology, economics, electrical engineering, and other fields. Recent examples have been: Nonlinear Waves, Mathematical Ecology, and Computational Neuroscience.

### Math 561 (TO 518, IOE 510) - Linear Programming I

Prerequisites: Math 214, 217, 417, or 419 3 credits. The allocation of constrained resources such as funds among investment possibilities or personnel among production facilities is a fundamental problem which is very well-suited to mathematical analysis.  Each such problem has as its goal the maximization of some positive objective such as investment return or the minimization of some negative objective such as cost or risk. Such problems are called Optimization Problems.  Linear Programming deals with optimization problems in which both the objective and constraint functions are linear (the word “programming” means “planning” rather than computer programming).  In practice, such problems involve thousands of decision variables and constraints, so a primary focus is the development and implementation of efficient algorithms. However, the subject also has deep connections with higher-dimensional convex geometry. A recent survey showed that most Fortune 500 companies regularly use linear programming in their decision making. This course will present both the classical and modern approaches to the subject and discuss numerous applications of current interest. Formulation of problems from the private and public sectors using the mathematical model of linear programming.  Development of the simplex algorithm; duality theory and economic interpretations.  Postoptimality (sensitivity) analysis; algorithmic complexity; the ellipsoid method; scaling algorithms; applications and interpretations.  Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and to use various computer programs.

### Math 562 (IOE 511) - Continuous Optimization Methods

Prerequisites: Math 214, 217, 417, or 419 3 credits. Optimization is widely used in engineering and science models. The goal of this course is to give a rigorous background to the field. Examples are drawn from engineering and science. Survey of continuous optimization problems.  Unconstrained optimization problems: unidirectional search techniques, gradient, conjugate direction, quasi-Newtonian methods; introduction to constrained optimization using techniques of unconstrained optimization through penalty transformation, augmented Lagrangians, and others; discussion of computer programs for various algorithms.

### Math 563 - Advanced Mathematical Methods for the Biological Sciences

Prerequisites: Math 217, 417, or 419 and Math 450 or 454 3 credits. Natural systems behave in a way that reflects an underlying spatial pattern. This course focuses on discovering the way in which spatial variation influences the motion, dispersion, and persistence of species. The concepts underlying spatially dependent processes and the partial differential equations which model them will be discussed in a general manner with specific applications taken from molecular, cellular, and population biology. This course is centered on modeling in three major areas i) Models of Motion: Diffusion, Convection, Chemotaxis, and Haptotaxis; ii) Biological Pattern Formation; and iii) Delay-differential Equations and Age-structured Models. This course will introduce and explore partial differential equation modeling in biological settings. Students should have some experience with solution techniques for partial differential equations as well as an interest in biomedical applications. There will also be a brief introduction to delay differential equations and age-structured models; however, no previous background in these areas is required. Mathematical topics covered include derivation of relevant PDEs from first principles; reduction of PDEs to ODEs under steady state, quasi-steady state, and traveling wave assumptions; solution techniques for PDEs and concepts of spatial stability and instability. These concepts will be introduced within the setting of classical and current problems in biology and the biomedical sciences such as cell motion, transport of biological substances, and biological pattern formation. Above all, this course aims to enhance the interdisciplinary training of advanced under- graduate and graduate students from mathematics and other disciplines by introducing fundamental properties of partial differential equations in the context of interesting biological phenomena. Grades will be based on the completion of a research project and weekly (or biweekly) homework assignments, computer lab assignments, and in class presentations.

### Math 564 - Topics in Mathematical Biology

Prerequisites: Variable, permission of instructor 3 credits. This is an advanced course on further topics in mathematical biology. Topic will vary according to the instructor. Possible topics include modeling infectious diseases, cancer modeling, mathematical neurosciences or biological oscillators, among others. The sample description below is for a course in biological oscillators from Winter 2006. From sleeping patterns, heartbeats, locomotion, and firefly flashing to the treatment of cancer, diabetes, and neurological disorders, oscillations are of great importance in biology and medicine.  Mathematical modeling and analysis are needed to understand what causes these oscillations to emerge, properties of their period and amplitude, and how they synchronize to signals from other oscillators or from the external world.  The goal of this course will be to teach students how to take real biological data, convert it to a system of equations and simulate and/or analyze these equations. Models will typically use ordinary differential equations. Mathematical techniques introduced in this course include 1) the method of averaging, 2) harmonic balance, 3) Fourier techniques, 4) entrainment and coupling of oscillators, 4) phase plane analysis, and 5) various techniques from the theory of dynamical systems.  Emphasis will be placed on primary sources (papers from the literature) particularly those in the biological sciences.  Consideration will be given in the problem sets and course project to interdisciplinary student backgrounds. Teamwork will be encouraged.

### Math 565 - Combinatorics and Graph Theory

Prerequisites: Math 465 or equivalent experience with abstract mathematics. 3 credits. Math 565 and 566 introduce the basic notions and techniques of combinatorics and graph theory at the beginning graduate level. These two courses can be taken in any order. Math 565 emphasizes the aspects of combinatorics connected with computer science, geometry, and topology. Graph theory: connectivity, planarity, graph coloring, extremal graph theory, and Ramsey theory. Other topics: combinatorial designs, partially ordered sets, lattices, matroids, simplicial complexes, and convex polytopes.

### Math 566 - Combinatorial Theory

Prerequisites: Math 465 or equivalent experience with abstract mathematics. 3 credits. Math 565 and 566 introduce the basic notions and techniques of combinatorics and graph theory at the beginning graduate level. These two courses can be taken in any order. Math 566 emphasizes the enumerative and algebraic aspects of the subject. Algebraic graph theory: graph eigenvalues, enumeration of walks and tilings, electric networks, the matrix-tree theorem. Partitions and Young tableaux: generating functions, q-analogues, hooklength formula, the Schensted correspondence.

### Math 567 - Introduction to Coding Theory

Prerequisites: Math 412 or equivalent experience with abstract mathematics. 3 credits. Mathematical foundations of modern coding and information theory. When data is being transmitted over a noisy channel, the goal is to find efficient ways to encode/decode it to deal with data loss and errors. Topics include: entropy, Huffman codes, channel capacity, Shannon’s theorem, error correcting block codes, various constructions of linear codes over finite fields (Hamming codes, Golay codes, Reed-Muller codes, cyclic codes, etc.), bounds for codes, and more.

### Math 568 (BIOINF 568) - Mathematical & Computational Neuroscience

Prerequisites: Math 463 or 462 (for undergrad students) or Graduate standing. Advisory: Math 214, 217, 417 or 419; AND Math 216, 286 or 316 3 credits. In this course, students will derive, interpret and solve mathematical models of neural systems. The neural models consist of ordinary and partial differential equations and students are required to analytically and numerically solve the equations. Additional mathematical analysis techniques of phase plane analysis, linear stability of equilibria and bifurcation analysis will also be covered. Computational neuroscience provides a set of quantitative approaches to investigate the biophysical mechanisms and computational principles underlying the function of the nervous system. This course introduces students to mathematical modeling and quantitative techniques used to investigate neural systems at many different scales, from single neuron activity to the dynamics of large neuronal networks.

### Math 571 - Numerical Linear Algebra

Prerequisites: Math 214, 217, 417, 419, or 420 and one of Math 450, 451, or 454; or permission of instructor 3 credits. This course is an introduction to numerical linear algebra, which is at the foundation of much of scientific computing. Numerical linear algebra deals with (1) the solution of linear systems of equations, (2) computation of eigenvalues and eigenvectors, and (3) least squares problems. We will study accurate, efficient, and stable algorithms for matrices that could be dense, or large and sparse, or even highly ill-conditioned. The course will emphasize both theory and practical implementation. This course is an introduction to numerical linear algebra, a core subject in scientific computing. Three types of problems are considered: (1) linear systems, (2) eigenvalues and eigenvectors, and (3) least squares problems. Topics include: Gram-Schmidt orthogonalization, QR factorization, singular value decomposition (SVD), normal equations, vector and matrix norms, condition number, backward error analysis, LU factorization, Cholesky factorization, reduction to Hessenberg and tridiagonal form, power method, inverse iteration, Rayleigh quotient iteration, QR algorithm, Krylov subspace methods, Arnoldi iteration, GMRES, steepest descent, conjugate gradient method, preconditioning, applications to image compression, finite-difference schemes for two-point boundary value problems, Dirichlet problem for the Laplace equation, least squares data fitting.

### Math 572 - Numerical Methods for Differential Equations

Prerequisites: Math 214, 217, 417, 419, or 420 and one of Math 450, 451, or 454; or permission of instructor 3 credits. This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. Graduate students from engineering and science departments and strong undergraduates are also welcome. The course is an introduction to numerical methods for solving ordinary differential equations and hyperbolic and parabolic partial differential equations. Fundamental concepts and methods of analysis are emphasized.  Students should have a strong background in linear algebra and analysis, and some experience with computer programming. Content varies somewhat with the instructor. Numerical methods for ordinary differential equations; Lax’s equivalence theorem; finite difference and spectral methods for linear time dependent PDEs: diffusion equations, scalar first order hyperbolic equations, symmetric hyperbolic systems.

### Math 573 - Financial Mathematics I

Prerequisites: Math 526 3 credits. This is a core course for the quantitative finance and risk management Masters program and introduces students to the main concepts of Financial Mathematics. This course emphasizes the application of mathematical methods to the relevant problems of financial industry and focuses mainly on developing skills of model building. This is an introductory course in Financial Mathematics. This course starts with the basic version of Mathematical Theory of Asset Pricing and Hedging (Fundamental Theorem of Asset Pricing in discrete time and discrete space). This theory is applied to problems of Pricing and Hedging of simple Financial Derivatives. Finally, the continuous time version of the proposed methods is presented, culminating with the BlackScholes model. A part of the course is devoted to the problems of Optimal Investment in discrete time (including Markowitz Theory and CAPM) and Risk Management (VaR and its extensions). This course shows how one can formulate and solve relevant problems of financial industry via mathematical (in particular, probabilistic) methods.

### Math 574 - Financial Mathematics II

Prerequisites: Math 526 and Math 573 3 credits. This is a core course for the quantitative finance and risk management Masters program and is a sequel to Math 573. This course emphasizes the application of mathematical methods to the relevant problems of financial industry and focuses mainly on developing skills of model building. This is a continuation of Math 573. This course discusses Mathematical Theory of Continuous-time Finance. The course starts with the general Theory of Asset Pricing and Hedging in continuous time and then proceeds to specific problems of Mathematical Modeling in Continuous- time Finance. These problems include pricing and hedging of (basic and exotic) Derivatives in Equity, Foreign Exchange, Fixed Income and Credit Risk markets. In addition, this course discusses Optimal Investment in Continuous time (Merton’s problem), Highfrequency Trading (Optimal Execution), and Risk Management (e.g. Credit Value Adjustment).

### Math 575 - Introduction to Theory of Numbers

Prerequisites: Math 451 and one of Math 420 or 494, or permission of instructor 3 Credits.  1 credit after Math 475 Many of the results of algebra and analysis were invented to solve problems in number theory. This field has long been admired for its beauty and elegance and recently has turned out to be extremely applicable to coding problems. This course is a survey of the basic techniques and results of elementary number theory. Students should have significant experience in writing proofs at the level of Math 451 and should have a basic understanding of groups, rings, and fields, at least at the level of Math 412 and prefer- ably Math 493. Proofs are emphasized, but they are often pleasantly short. This is a first course in number theory. Topics covered include divisibility and prime numbers, congruences, quadratic reciprocity, quadratic forms, arithmetic functions, and Diophantine equations. Other topics may be covered as time permits or by request.

### Math 582 - Introduction to Set Theory

Prerequisites: Math 412 or 451 or equivalent experience with abstract mathematics 3 credits. One of the great discoveries of modern mathematics was that essentially every mathematical concept may be defined in terms of sets and membership. Thus Set Theory plays a special role as a foundation for the whole of mathematics.  One of the goals of this course is to develop some understanding of how Set Theory plays this role. The analysis of common mathematical concepts (e.g., function, ordering, infinity) in set-theoretic terms leads to a deeper understanding of these concepts. At the same time, the student will be introduced to many new concepts (e.g., transfinite ordinal and cardinal numbers, the Axiom of Choice) which play a major role in many branches of mathematics. The development of Set Theory will be largely axiomatic with the emphasis on proving the main results from the axioms.  Students should have substantial experience with theorem-proof mathematics; the listed pre- requisites are minimal and stronger preparation is recommended. No course in mathematical logic is presupposed. The main topics covered are set algebra (union, intersection), relations and functions, orderings (partial, linear, well), the natural numbers, finite and denumerable sets, the Axiom of Choice, and ordinal and cardinal numbers.

### Math 590 - An Introduction to Topology

Prerequisites: Math 451 3 credits. The purpose of this course is to introduce basic concepts of topology. Most of the course will be devoted to the fundamentals of general (point set) topology. Topics include metric spaces, topological spaces, continuous functions and homeomorphisms, separation axioms, quotient and product topology, compactness, and connectedness. We will also cover a bit of algebraic topology (e.g., fundamental groups) as time permits.

### Math 591 - General and Differential Topology

Prerequisites: Math 451, 452, 590 3 credits. This is one of the basic courses for students beginning the PhD program in mathematics. The approach is rigorous and emphasizes abstract concepts and proofs. The first 2-3 weeks of the course will be devoted to general topology, and the remainder of the course will be devoted to differential topology. Topics include: Product and quotient topology, CW-complexes, group actions, topological groups, topological manifolds, smooth manifolds, manifolds with boundary, smooth maps, partitions of unity, tangent vectors and differentials, the tangent bundle, submersions, immersions and embeddings, smooth submanifolds, Sard's Theorem, the Whitney embedding theorem, transversality, Lie groups, vector fields, Lie brackets, Lie algebra, multilinear algebra, vector bundles, differential forms, exterior derivatives, orientation, De Rham cohomology groups, homotopy invariance, degree theory.

### Math 592 - An Introduction to Algebraic Topology

Prerequisites: Math 591 3 credits. This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, Eilenberg-Steenrod axioms, Brouwer’s and Lefschetz’ fixed-point theorems, and other topics.

### Math 593 - Algebra I

Prerequisites: Math 494; or Math 412, 420, and 451 3 credits. This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. This course, together with Math 594, offer excellent preparation for the PhD Qualifying exam in algebra. Students should have had a previous course equivalent to Math 493 (Honors Algebra I). Topics include basics about rings and modules, including Euclidean rings, PIDs, UFDs. The structure theory of modules over a PID will be an important topic, with applications to the classification of finite abelian groups and to Jordan and rational canonical forms of matrices. The course will also cover tensor, symmetric, and exterior algebras, and the classification of bilinear forms with some emphasis on the field case.

### Math 594 - Algebra II

Prerequisites: Math 593 3 credits This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Topics include group theory, permutation representations, simplicity of alternating groups for n>4, Sylow theorems, series in groups, solvable and nilpo- tent groups, Jordan-Hölder Theorem for groups with operators, free groups and presentations, fields and field extensions, norm and trace, algebraic closure, Galois theory, and transcendence degree.

### Math 596 - Analysis I (Complex)

Prerequisites: Math 451 3 credits.  2 credits after Math 555 This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Review of analysis in R2 including metric spaces, differentiable maps, Jacobians; analytic functions, Cauchy-Riemann equations, conformal mappings, linear fractional transformations; Cauchy’s theorem, Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families, Riemann mapping theorem.

### Math 597 - Analysis II (Real)

Prerequisites: Math 451 and 420 3 credits. This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Topics include:  Lebesgue measure on the real line; measurable functions and integration on R; differentiation theory, fundamental theorem of calculus; function spaces, Lp(R), C(K), Hölder and Minkowski inequalities, duality; general measure spaces, product measures, Fubini’s Theorem; Radon-Nikodym Theorem, conditional expectation, signed measures, introduction to Fourier transforms.