2019  Spring Semester
GRADUATE COURSE SPRING 2019
This schedule is subject to changes. Please contact the Course Instructor for confirmation.
Course  Class # 
Course Title  Course Day & Time  Rm #  Instructor 
Math 4309  6259  Mathematical Biology  MW, 2:30—4:00PM  SEC 104  R. Azevedo 
12147 
Graph Theory w/Applications  TuTh, 4:00—5:30PM 
SEC 202 
K. Josic  
2895 
Introduction to Real Analysis II  TuTh, 2:30—4:00PM  AH 301  M. Kalantar  
Math 4351  13920  Differential Geometry II  TuTh, 1:00—2:30PM  SEC 203  M. Ru 
Math 4362  13921  Theory of Differential Equations and Nonlinear Dynamics  TuTh, 4:00—5:30PM  AH 302  W. Ott 
Math 4364  9310  Intro. to Numerical Analysis in Scientific Computing  MW, 4:00—5:30PM  D2 LECT2  T.W. Pan 
Math 4364  17945  Intro. to Numerical Analysis in Scientific Computing  Online  Online  J. Morgan 
Math 4365  7697  Numerical Methods for Differential Equations  TuTh, 11:30AM—1PM  CBB 120  J. He 
5199 
Advanced Linear Algebra I  MWF, Noon—1:00PM  SEC 103  D. Wagner  
2896 
Advanced Linear Algebra II  TuTh, 11:30AM—1PM  F 154  A. Mamonov  
Math 4380  2897  A Mathematical Introduction to Options  MW, 1—2:30PM  SEC 105  I. Timofeyev 
Math 4389  2898  Survey of Undergraduate Mathematics  MWF, Noon—1PM  SEC 205  M. Almus 
Math 4397  14519  Statistical & Machine Learning 
MWF, 11am—Noon 
GAR 201 
A. Skripnikov/C. Poliak 
Course  Class #  Course Title  Course Day & Time  Instructor 
Math 5330  4255  Abstract Algebra  Arrange (online course)  K. Kaiser 
Math 5332  2917  Differential Equations  Arrange (online course)  G. Etgen 
Math 5334  13922  Complex Analysis  Arrange (online course)  S. Ji 
Math 5344  18243  Introduction to Scientific Computing  Arrange (online course)  J. Morgan 
Math 5386  5938  Regression and Linear Models  Arrange (online course)  C. Peters 
Course 
Class #  Course Title  Course Day & Time  Rm #  Instructor 
Math 6303  2924  Modern Algebra II  MW, 1—2:30pm  C 105  G. Heier 
Math 6308  4421  Advanced Linear Algebra I  MWF, Noon—1pm  SEC 103  D. Wagner 
Math 6308  8599  Advanced Linear Algebra I (online)  Online  Online  TBA 
Math 6309  4429  Advanced Linear Algebra II  TuTh, 11:30am—1pm  F 154  A. Mamonov 
Math 6313  4420  Introduction to Real Analysis  TuTh, 2:30—4pm  C 106  M. Kalantar 
Math 6321  2941  Theory of Functions of a Real Variable  MWF, 11am—Noon  AH 108  M. Tomforde 
Math 6323  13923  Functional Complex Variable  MWF, 9—10am  AH 301  S. Ji 
Math 6361  4424  Applicable Analysis  MWF, 10—11am  AH 301  B. Bodmann 
Math 6365  12154  Automatic Learning and Data Mining  TuTh, 11:30am—1pm  CBB 124  R. Azencott 
Math 6367  2942  Optimization Theory  MW, 4—5:30pm  SEC 206  R. Hoppe 
Math 6371  2943  Numerical Analysis  MW, 1—2:30pm  AH 303  Y. Kuznetsov 
Math 6383  2944  Probability Statistics  TuTh, 4—5:30pm  AH 7  W. Fu 
Math 6385  12153  ContinuousTime Models in Finance  TuTh, 2:30—4pm  F 162  E. Kao 
Math 6397  13924  Sobolev Calculus & Sobolev Spaces  TuTh, 1—2:30pm  AH 203  G. Auchmuty 
Math 6397  13925  Time Series Analysis  TuTh, 10—11:30am  AH 203  E. Kao 
Math 7321  13926  Functional Analysis  MWF, Noon—1pm  AH 301  D. Blecher 
Math 7350  13927  Geometry of Manifolds  TuTh, 11:30am—1pm  SW 423  A. Török 
Math 7394  13929  Ergodic Theory & Thermodynamic Formalism  MWF, 11am—Noon  AH 2  V. Climenhaga 
Math 7396  13930  Multigrid Methods  MW, 4—5:30pm  AH 301  M. Olshanskiy 
Course Details
SENIOR UNDERGRADUATE COURSES
Math 4309 (6259)  Mathematical Biology 

Prerequisites:  
Text(s):  A Biologist's Guide to Mathematical Modeling in Ecology and Evolution by Sarah P. Otto and Troy Day; ISBN13:9780691123448 
Description: 
Topics in mathematical biology, epidemiology, population models, models of genetics and evolution, network theory, pattern formation, and neuroscience. Students may not receive credit for both MATH 4309 and BIOL 4309. 
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Math 4315 (12147)  Graph Theory w/Applications


Prerequisites:  Either MATH 3330 or MATH 3336 and three additional hours of 30004000 level Mathematics 
Text(s):  Networks, Crowds, and Markets: Reasoning About a Highly Connected World. By David Easley and Jon Kleinberg. This text is availabe at this link: https://www.cs.cornell.edu/home/kleinber/networksbook/ 
Description: 
Introduction to basic concepts, results, methods, and applications of graph theory. Additional Description: How does information propagate between friends and acquaintances on social media? How do diseases spread, and when do epidemics start? How should we design power grids to avoid failures, and systems of roads to optimize traffic flow? These questions can be addressed using network theory . Students in the course will develop a sound knowledge of the basics of graph theory, as well as some of the computational tools used to address the questions above. Course topics include basic structural features of networks, generative models of networks, centrality, random graphs, clustering, and dynamical processes on graphs. 
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Math 4332 (2895)  Introduction to Real Analysis II


Prerequisites:  MATH 4331 or consent of instructor 
Text(s):  Real Analysis with Real Applications  Edition: 1; Allan P. Donsig, Allan P. Donsig; ISBN: 9780130416476 
Description: 
Further development and applications of concepts from MATH 4331. Topics may vary depending on the instructor's choice. Possibilities include: Fourier series, pointset topology, measure theory, function spaces, and/or dynamical systems. 
Math 4351 (13920)  Differential Geometry II


Prerequisites:  MATH 4350. 
Text(s):  Instructor's notes will be provided. 
Description: 
Continuation of the study of Differential Geometry from MATH 4350. Holonomy and the GaussBonnet theorem, introduction to hyperbolic geometry, surface theory with differential forms, calculus of variations and surfaces of constant mean curvature, abstract surfaces (2D Riemannian manifolds). 
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Math 4362 (13921)  Theory of Differential Equations an Nonlinear Dynamics


Prerequisites:  MATH 3331, or equivalent, and three additional hours of 30004000 level Mathematics. 
Text(s):  Nonlinear Dynamics and Chaos (2nd Ed.) by Strogatz. ISBN: 9780813349107 
Description: 
ODEs as models for systems in biology, physics, and elsewhere; existence and uniqueness of solutions; linear theory; stability of solutions; bifurcations in parameter space; applications to oscillators and classical mechanics. 
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Math 4364 (9310) Numerical Analysis in Scientific Computing


Prerequisites: 
MATH 3331 and COSC 1410 or equivalent or consent of instructor. Instructor's Prerequisite Notes: 1. MATH 2331, In depth knowledge of Math 3331 (Differential Equations) or Math 3321 (Engineering Mathematics) 2. Ability to do computer assignments in FORTRAN, C, Matlab, Pascal, Mathematica or Maple. 
Text(s): 
Numerical Analysis (9th edition), by R.L. Burden and J.D. Faires, BrooksCole Publishers, ISBN:9780538733519 
Description:  This is an one semester course which introduces core areas of numerical analysis and scientific computing along with basic themes such as solving nonlinear equations, interpolation and splines fitting, curve fitting, numerical differentiation and integration, initial value problems of ordinary differential equations, direct methods for solving linear systems of equations, and finitedifference approximation to a twopoints boundary value problem. This is an introductory course and will be a mix of mathematics and computing. 
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Math 4364 (17945) Numerical Analysis in Scientific Computing


Prerequisites: 
MATH 3331 and COSC 1410 or equivalent or consent of instructor. Instructor's Prerequisite Notes: 1. MATH 2331, In depth knowledge of Math 3331 (Differential Equations) or Math 3321 (Engineering Mathematics) 2. Ability to do computer assignments in FORTRAN, C, Matlab, Pascal, Mathematica or Maple. 
Text(s): 
Numerical Analysis (9th edition), by R.L. Burden and J.D. Faires, BrooksCole Publishers, ISBN:9780538733519 
Description:  This is an one semester course which introduces core areas of numerical analysis and scientific computing along with basic themes such as solving nonlinear equations, interpolation and splines fitting, curve fitting, numerical differentiation and integration, initial value problems of ordinary differential equations, direct methods for solving linear systems of equations, and finitedifference approximation to a twopoints boundary value problem. This is an introductory course and will be a mix of mathematics and computing. 
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Math 4365 (7397)  Numerical Methods for Differential Equations


Prerequisites:  MATH 3331, or equivalent, and three additional hours of 3000–4000 level Mathematics. 
Text(s):  TITLE:TBA, AUTHOR:TBA, ISBN:TBA 
Description:  Numerical differentiation and integration, multistep and RungeKutta methods for ODEs, finite difference and finite element methods for PDEs, iterative methods for linear algebraic systems and eigenvalue computation. 
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Math 4377 (5199)  Advanced Linear Algebra I


Prerequisites:  MATH 2331 or equivalent, and three additional hours of 3000–4000 level Mathematics. 
Text(s):  Linear Algebra  Edition: 4; Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence; ISBN: 9780130084514 
Description: 
Linear systems of equations, matrices, determinants, vector spaces and linear transformations, eigenvalues and eigenvectors. Additional Notes: This is a proofbased course. It will cover Chapters 14 and the first two sections of Chapter 5. Topics include systems of linear equations, vector spaces and linear transformations (developed axiomatically), matrices, determinants, eigenvectors and diagonalization. 
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Math 4378 (2896)  Advanced Linear Algebra II


Prerequisites:  MATH 4377 
Text(s):  Linear Algebra, Fourth Edition, by S.H. Friedberg, A.J Insel, L.E. Spence,Prentice Hall, ISBN 0130084514; 9780130084514 
Description: 
Similarity of matrices, diagonalization, Hermitian and positive definite matrices, normal matrices, and canonical forms, with applications. Instructor's Additional notes: This is the second semester of Advanced Linear Algebra. I plan to cover Chapters 5, 6, and 7 of textbook. These chapters cover Eigenvalues, Eigenvectors, Diagonalization, CayleyHamilton Theorem, Inner Product spaces, GramSchmidt, Normal Operators (in finite dimensions), Unitary and Orthogonal operators, the Singular Value Decomposition, Bilinear and Quadratic forms, Special Relativity (optional), Jordan Canonical form. 
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Math 4380 (2897)  A Mathematical Introduction to Options  
Prerequisites:  MATH 2433 and MATH 3338. 
Text(s):  An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation  Edition: 1; Desmond Higham; 9780521547574 
Description:  Arbitragefree pricing, stock price dynamics, callput parity, BlackScholes formula, hedging, pricing of European and American options. 
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Math 4389 (2898)  Survey of Undergraduate Mathematics  
Prerequisites:  MATH 3330, MATH 3331, MATH 3333, and three hours of 4000level Mathematics. 
Text(s):  Instructor will use his own notes 
Description:  A review of some of the most important topics in the undergraduate mathematics curriculum. 
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ONLINE GRADUATE COURSES
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MATH 5330 (4255)  Abstract Algebra


Prerequisites:  Graduate standing. 
Text(s): 
Abstract Algebra , A First Course by Dan Saracino. Waveland Press, Inc. ISBN 0881336653 
Description: 
Groups, rings and fields; algebra of polynomials, Euclidean rings and principal ideal domains. Does not apply toward the Master of Science in Mathematics or Applied Mathematics. Other Notes: This course is meant for students who wish to pursue a Master of Arts in Mathematics (MAM). Please contact me in order to find out whether this course is suitable for you and/or your degree plan. Notice that this course cannot be used for MATH 3330, Abstract Algebra. 
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MATH 5332 (2917)  Differential Equations


Prerequisites:  Graduate standing. MATH 5331. 
Text(s):  TBA 
Description: 
Linear and nonlinear systems of ordinary differential equations; existence, uniqueness and stability of solutions; initial value problems; higher dimensional systems; Laplace transforms. Theory and applications illustrated by computer assignments and projects. Applies toward the Master of Arts in Mathematics degree; does not apply toward the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees. 
MATH 5334 (13922)  Complex Analysis


Prerequisites:  Graduate standing. and two semesters of calculus. 
Text(s):  The course will be based on my notes. 
Description: 
This course is an introduction to complex analysis. It will cover the theory of holomorphic functions, Cauchy theorem and Cauchy integral formula, residue theorem, harmonic and subharmonic functions, and other topics. In each week, some lecture notes will be posted in Blackboard Learn, including homework assignment. 
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MATH 5344 (13922)  Introduction to Scientific Computing w/Excel


Prerequisites: 
Graduate standing and three semesters of Calculus. MATH 2331, In depth knowledge of Math 3331 (Differential Equations) or Math 3321 (Engineering Mathematics) (see the description for more prerequisite details) 
Text(s): 
Numerical Analysis (9th edition), by R.L. Burden and J.D. Faires, BrooksCole Publishers, 9780538733519 
Description: 
The students in this online section will be introduced to topics in scientific computing, including numerical solutions to nonlinear equations, numerical differentiation and integration, numerical solutions of systems of linear equations, least squares solutions and multiple regression, numerical solutions of nonlinear systems of equations, numerical optimization, numerical solutions to discrete dynamical systems, and numerical solutions to initial value problems and boundary value problems. Computations in this course will primarily be illustrated directly in an Excel spreadsheet, or via VBA programming, but students who prefer to do their computations using Matlab, Julia, Python or some other programming language are also welcome. For students who want to do their computing in Excel, there will be tutorials associated with the use of Excel, and programming in VBA. Students who decide to use Excel are expected to have access and basic familiarity with Excel, but they are not expected to know advanced spreadsheet functionality or have programming experience with VBA. Students will not be tested over Excel or VBA, and students using Matlab, Julia or Python will also receive some help materials. 
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MATH 5386 (5938)  Regression and Linear Models (VEE approved course)


Prerequisites:  Graduate standing. Two semesters of calculus, one semester of linear algebra, and MATH 5385, or consent of instructor. 
Text(s):  Introduction to Linear Regression Analysis  Edition:5; Montgomery, Peck, Vining; ISBN: 9780470542811; Wiley 
Description: 
Simple and multiple linear regression, linear models, inferences from the normal error model, regression diagnostics and robust regression, computing assignments with appropriate software. Applies toward Master of Arts in Mathematics degree; does not apply toward the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees. Note: This course is VEE approved for the regression component only. Approval Code: 445811008. For more information on VEE approved courses, click here. 
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GRADUATE COURSES
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MATH 6303 (2924)  Modern Algebra II


Prerequisites: 
Graduate standing. MATH 4333 or MATH 4378 Additional Prerequisites: students should be comfortable with basic measure theory, groups rings and fields, and pointset topology 
Text(s): 
No textbook is required. 
Description: 
Topics from the theory of groups, rings, fields, and modules. Additional Description: This is primarily a course about analysis on topological groups. The aim is to explain how many of the techniques from classical and harmonic analysis can be extended to the setting of locally compact groups (i.e. groups possessing a locally compact topology which is compatible with their algebraic structure). In the first part of the course we will review basic point set topology and introduce the concept of a topological group. The examples of padic numbers and the Adeles will be presented in detail, and we will also spend some time discussing SL_2(R). Next we will talk about characters on topological groups, Pontryagin duality, Haar measure, the Fourier transform, and the inversion formula. We will focus on developing details in specific groups (including those mentioned above), and applications to ergodic theory and to number theory will be discussed. 
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MATH 6308 (4421)  Advanced Linear Algebra I


Prerequisites:  Graduate standing. MATH 2331 and a minimum of 3 semester hours transformations, eigenvalues and eigenvectors. 
Text(s):  Linear Algebra  Edition: 4; Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence; ISBN: 9780130084514 
Description: 
Transformations, eigenvalues and eigenvectors. Additional Notes: This is a proofbased course. It will cover Chapters 14 and the first two sections of Chapter 5. Topics include systems of linear equations, vector spaces and linear transformations (developed axiomatically), matrices, determinants, eigenvectors and diagonalization. 
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MATH 6308 (8599)  Advanced Linear Algebra I (online)


Prerequisites:  Graduate standing. MATH 2331 and a minimum of 3 semester hours transformations, eigenvalues and eigenvectors. 
Text(s):  Linear Algebra  Edition: 4; Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence; ISBN: 9780130084514 
Description: 
Transformations, eigenvalues and eigenvectors. An expository paper or talk on a subject related to the course content is required 
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MATH 6309 (4429)  Advanced Linear Algebra II


Prerequisites:  Graduate standing and MATH 6308 
Text(s):  Linear Algebra, Fourth Edition, by S.H. Friedberg, A.J Insel, L.E. Spence,Prentice Hall, ISBN 0130084514; 9780130084514 
Description:  Similarity of matrices, diagonalization, hermitian and positive definite matrices, canonical forms, normal matrices, applications. An expository paper or talk on a subject related to the course content is required. 
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MATH 6313 (4420)  Introduction to Real Analysis II


Prerequisites:  Graduate standing and MATH 6312. 
Text(s):  TBA 
Description:  Properties of continuous functions, partial differentiation, line integrals, improper integrals, infinite series, and Stieltjes integrals. An expository paper or talk on a subject related to the course content is required. 
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MATH 6321 (2941)  Theory of Functions of a Real Variable II


Prerequisites: 
Graduate standing. MATH 4332 or consent of instructor. Instructor's Prerequisite Notes: MATH 6320 
Text(s): 
Primary (Required): Real Analysis for Graduate Students, Richard F. Bass Supplementary (Recommended): Real Analysis: Modern Techniques and Their Applications, Gerald Folland (2nd edition); ISBN: 9780471317166. 
Description: 
Lebesque measure and integration, differentiation of real functions, functions of bounded variation, absolute continuity, the classical Lp spaces, general measure theory, and elementary topics in functional analysis. Instructor's Additional Notes: Math 6321 is the second course in a twosemester sequence intended to introduce the theory and techniques of modern analysis. The core of the course covers elements of functional analysis, Radon measures, elements of harmonic analysis, the Fourier transform, distribution theory, and Sobolev spaces. Additonal topics will be drawn from potential theory, ergodic theory, and the calculus of variations. 
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MATH 6323 (13923)  Functional Complex Variable


Prerequisites: 
Graduate standing. Math 6322 or consent of instructor. 
Text(s): 
No textbook required. Lecture notes provided. 
Description: 
Classical examples, Schwartz lemma, Riemann mapping theorem, complex hyperbolic geometry, Little and Picard theorems, Riemann surface theory and others. 
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MATH 6361 (4424)  Applicable Analysis


Prerequisites:  Graduate standing. MATH 4332 or consent of instructor. 
Text(s): 
The instructor will provide lecture notes on the material. A reference text is L.D. Berkowitz, Convexity and Optimization in Rn, WileyInterscience 2002.

Description: 
This course provides an introduction to the mathematical analysis of finite dimensional optimization problems. Topics to be studied include the existence of, and the extremality conditions that hold at, solutions of constrained and unconstrained optimaization problems. Elementary theory of convex sets, functions and constructions from convex analysis will be introduced and used. Concepts include subgradients, conjugate functions and some duality theory. Specific problems to be studied include energy and least squares methods for solving linear equations, important inequalities, eigenproblems and some nonlinear programming problems from applications.

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MATH 6365 (12154)  Automatic Learning and Data Mining


Prerequisites:  Graduate standing. MATH 3338 and MATH 3339. 
Text(s): 
The instructor will provide lecture notes on the material. A reference text is L.D. Berkowitz, Convexity and Optimization in Rn, WileyInterscience 2002.

Description: 
Automatic learning and data mining cluster highdimension inputs to predict their impact on decision outputs. Kernel based Clustering and Learning enable dictionary generation, pattern classification, non linear regression. Applications: shape recognition, genes expression analysis, etc.

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MATH 6365 (13334 )  Automatic Learning and Data Mining


Prerequisites:  Graduate standing. MATH 3338 and MATH 3339. 
Text(s): 
The instructor will provide lecture notes on the material. A reference text is L.D. Berkowitz, Convexity and Optimization in Rn, WileyInterscience 2002.

Description: 
Automatic learning and data mining cluster highdimension inputs to predict their impact on decision outputs. Kernel based Clustering and Learning enable dictionary generation, pattern classification, non linear regression. Applications: shape recognition, genes expression analysis, etc.

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MATH 6367 (2942)  Optimization Theory


Prerequisites:  Graduate standing. MATH 4331 and MATH 4377. 
Text(s): 
 D.P. Bertsekas; Dynamic Programming and Optimal Con trol, Vol. I, 4th Edition. Athena Scientific, 2017, ISBN10: 1886529434  J.R. Birge and F.V. Louveaux; Introduction to Stochastic Programming. Springer, New York, 1997, ISBN: 038798217 
Description: 
Constrained and unconstrained finite dimensional nonlinear programming, optimization and EulerLagrange equations, duality, and numerical methods. Optimization in Hilbert spaces and variational problems. EulerLagrange equations and theory of the second variation. Application to integral and differential equations. Additional Description: This course consists of two parts. The first part is concer ned with an introduction to Stochastic Linear Programming (SLP) and Dynamic Programming (DP). As far as DP is concerned, the course focuses on the theory and the appli cation of control problems for linear and nonlinear dynamic systems both in a deterministic and in a stochastic frame work. Applications aim at decision problems in finance. In the second part, we deal with continuoustime systems and optimal control problems in function space with em phasis on evolution equations. 
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MATH 6371 (2943)  Numerical Analysis


Prerequisites:  Graduate standing. 
Text(s):  Numerical Mathematics (Texts in Applied Mathematics), 2nd Ed., V.37, Springer, 2010. By A. Quarteroni, R. Sacco, F. Saleri. ISBN: 9783642071010 
Description:  Ability to do computer assignments. Topics selected from numerical linear algebra, nonlinear equations and optimization, interpolation and approximation, numerical differentiation and integration, numerical solution of ordinary and partial differential equations. 
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MATH 6383 (2944)  Probability Statistics


Prerequisites:  Graduate standing. MATH 3334, MATH 3338 and MATH 4378. 
Text(s): 
Recommended Text: John A. Rice : Mathematical Statistics and Data Analysis, 3^{rd} editionBrooks / Cole, 2007. ISBN13: 9780534399429. Reference Texts: P. MuCullagh and J.A. Nelder: Generealized Linear Models, 2^{nd} ed. 1999 Chapman Hall/CRC. ISBN: 9780412317606 Raymond H. Myers, Douglas C. Montgomery, G. Geoffrey Vining, Timothy J. Robinson, Generalized Linear Models: with Applications in Engineering and the Sciences, 2^{nd} ed. Wiley, 2010. ISBN: 9780470454633. 
Description: 
A survey of probability theory, probability models, and statistical inference. Includes basic probability theory, stochastic processes, parametric and nonparametric methods of statistics. Instructor's Description: This course is designed for graduate students who have been exposed to basic probability and statistics and would like to learn more advanced statistical theory and techniques in modelling data of various types, including continuous, binary, counts and others. The selected topics will include basic probability distributions, likelihood function and parameter estimation, hypothesis testing, regression models for continuous and categorical response variables, variable selection methods, model selection, large sample theory, shrinkage models, ANOVA and some recent advances. 
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MATH 6385 (12153)  ContinuousTime Models in Finance


Prerequisites:  Graduate Standing. MATH 6384 
Text(s):  Arbitrage Theory in Continuous Time, 3rd edition, by Tomas Bjork, Oxford University Press, 2009. (Primary) 
Description: 
Stochastic calculus, Brownian motion, change of measures, Martingale representation theorem, pricing financial derivatives whose underlying assets are equities, foreign exchanges, and fixed income securities, singlefactor and multifactor HJM models, and models involving jump diffusion and mean reversion. Additional Description: The course is an introduction to continuoustime models in finance. We first cover tools for pricing contingency claims. They include stochastic calculus, Brownian motion, change of measures, and martingale representation theorem. We then apply these ideas in pricing financial derivatives whose underlying assets are equities, foreign exchanges, and fixed income securities. In addition, we will study models involving jump diffusion and mean reversion and the use of levy processes in finance. 
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MATH 6397 (13924)  Introduction to Sobolev Calculus and Sobolev Spaces


Prerequisites: 
Graduate standing. 
Text(s): 
There is no specific text for the course and the instructor will provide references for different parts of the material and some of his own notes. Three good references are specific chapters in the texts of: 
Description: 
This course is an introduction to the theory of weak derivatives and Sobolev spaces as used currently in the analysis of partial differential equations and numerical analysis. The rules of calculus change substantially when derivatives are defined in a weak sense. Conditions for product or chain rules to hold are quite different from those in the classical theorems. Many functions with singularities and corners may have weak derivatives with nice properties and various formulae hold with extra terms or different interpretations. In many engineering models and physical problems the analysis using weak derivatives produces results that better describe the observed behavior. The prerequisites for this course are classical multivariate calculus, and knowledge of Lebesgue and Borel measure on R^{N} and elementary Banach and Hilbert space theory as in graduate Real Analysis M6320 or equivalent. First weak derivatives of L^{1}_{loc}− functions on open subsets Ω ⊂ R^{N} will be defined. The definition generalizes the classical definition in some ways and is not a pointwise definition. These definitions enable the statement and proof of weak versions of the basic theorems of both 1dimensional and multivariate calculus. These include the product rule, the chain rule, the fundamental theorem of calculus and the GaussGreen (divergence) theorem. Then some results that only hold for weak derivatives will be proved starting with results on commutativity of weak derivatives the derivatives of convolutions, of infs and sups of pairs of functions and the approximation of measurable functions using mollifiers. Sobolev spaces such as H^{1}(Ω), W^{1,p}(Ω), H(div, Ω) and others will be defined and their properties described. These included completeness and imbedding theorems, the Poincar’e and Friedrich’s inequalities and Rellich type theorems. Also some results about trace operators, spaces and equivalent inner products and norms. If time permits, some results about W^{1,p}(R^{N} ) and the Sobolev and Morrey embedding theorems will be treated. 
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MATH 6397 (13925)  Time Series Analysis


Prerequisites: 
Graduate standing. 
Text(s):  An Introduction to Analysis of Financial Data with R, by Ruey S. Tsay, Wiley, 2013. 
Description: 
May be repeated with approval of chair. Additional Description: The course is about time analysis with special emphases on financial and energy data. The course covers ARIMA models, ARCH/GARCH models, nonlinear models, high frequency data analysis, parameter estimation for diffusion and related processes, multivariate time series, extreme value analysis, Copulas, Levy processes, and an introduction of Markov chain Monte Carlo Methods. We will use R for computing. 
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MATH 7321 (13926)  Functional Analysis II


Prerequisites:  Graduate standing and MATH 7320 or instructor consent 
Text(s):  Instructor Notes: Xeroxed set of lecture notes will be available. Recommended texts:Pedersen's "Analysis Now" or Conway's "A Course in Functional Analysis". 
Description: 
Catalog Description: This course is part of a two semester sequence covering the main results in functional analysis, including Hilbert spaces, Banach spaces, topological vector spaces such as distributions, and linear operators on these spaces. 
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MATH 7350 (13927)  Geometry Manifolds


Prerequisites: 
Graduate standing. Instructor's Prerequisite: Math 4331 and familiarity with multivariable calculus (at least at the level of Math 2433Calculus III) or consent of the instructor 
Text(s): 
Recommended: John M. Lee, Introduction to Smooth Manifolds, 2nd edition other relevant books will be placed on reserve in the library

Description:  This course describes the basic notions and constructions of differential geometry, and some of the more advanced results. It includes: manifolds, the inverse and implicit function theorems, submanifolds, partitions of unity; tangent bundles, vector fields, the Frobenius theorem, Lie derivatives, vector bundles; differential forms, tensors and tensor fieldson manifolds; exterior algebra, orientation, integration on manifolds, Stokes' theorem; Lie groups. A few additional topics might be also covered, depending on the interest of the audience. 
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MATH 7394 (13929)  Ergodic Theory & Thermodynamic Formalism


Prerequisites:  Graduate standing. 
Text(s):  The primary text is “An Introduction to Ergodic Theory”, by Peter Walters. Another useful textbook is “Foundations of Ergodic Theory”, by Marcelo Viana and Krerley Oliveira. I will also refer to “Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms”, by Rufus Bowen, as well as various other primary sources from the research literature. 
Description: 
Ergodic theory is a central part of the theory of dynamical systems, studying the asymptotic statistical properties of systems evolving in time that preserve an invariant measure. Systems with chaotic behavior generally possess many invariant measures, and thermodynamic formalism borrows tools from statistical mechanics to select a distinguished measure that is physically relevant. The first part of the class will cover topics in classical ergodic theory, including Birkhoff’s ergodic theorem, entropy, and the classification of Bernoulli automorphisms. The remainder of the course will discuss thermodynamic formalism, including the description of SinaiRuelleBowen measure via absolute continuity, the description of Parry measure via a variational principle, and the connection between the two via the general theory of equilibrium states. Some time will be spent describing the different approaches to thermodynamic formalism and SRB measures in uniform hyperbolicity: RuellePerronFrobenius operators indirectly via symbolic dynamics or directly via anisotropic Banach spaces; specification and expansivity; and the geometric approach via averaged pushforwards. Time permitting, we will discuss connections to dimension theory and geometric measure theory, and will conclude with a discussion of the nonuniformly hyperbolic setting. 
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MATH 7396 (13927)  Multigrid Methods


Prerequisites:  Graduate standing. 
Text(s):  TBA 
Description:  TBA 