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Graduate Course Outlines, Summer 2020-Spring 2021
Contact
Summer 2020
- MATH 7999-1: Problem Lab in Algebra —practice for PhD Qualifying Exam in Algebra.
- Instructor:
- Prerequisite: Math 7210.
- Text: Online Test Bank.
- MATH 7999-2: Problem Lab in Real Analysis—practice for PhD Qualifying Exam in Analysis.
- Instructor:
- Prerequisite: Math 7311.
- Text: Online Test Bank.
- MATH 7999-3: Problem Lab in Topology—practice for PhD Qualifying Exam in Topology.
- Instructor:
- Prerequisite: Math 7510.
- Text: Online Test Bank.
Fall 2020
- MATH 4997-1: Vertically Integrated Research: Parallel Computational Math
- 9:00-10:20 TTh
- Instructor: Drs. Patrick Diehl and Hartmut Kaiser
- Prerequisites: None, but some basic knowledge about programming is beneficial.
- Text:
- This course will focus on the parallel implementation of computational mathematics problems using modern accelerated C++. The aim of this course is to learn how to quickly write useful efficient C++ programs.
The students will not learn low-level C/C++ instead they will learn how to use high-level data structures, iterators, generic strings, and streams (including interactive and file I/O) of the C++ ISO Standard library. In addition, highly-optimized linear algebra libraries are introduced since the course teaches to solve problems, instead of explaining low-level C++ and computer science algorithms, like sorting algorithms, which are provided in the C++ standard library.
The first part, provides a brief overview of the containers, strings, streams, input/output, and the numeric library of the C++ standard library. For linear algebra, we will look into Blaze which is an open-source, high-performance C++ math library for dense and sparse arithmetic.
The second part will solve computational mathematics problems based-on the the previous introduced features of the C++ standard library.
The third part will focus on the parallel features provided by the C++ standard library. Here, the implemented computational problems in the second part of the course will be parallized using the C++ standard library for parallelism and concurrency.
Since programming skills can only be improved by doing, there will be weekly programming exercises and a small project. After this course students have a basic overview of the C++ standard library to solve efficiently computational mathematics problems without using low-level C/C++.
- MATH 4997-2: Vertically Integrated Research: Categorification
- 12:00-1:20 TTh
- Instructor: Profs. Achar and Sage
- Prerequisites: Math 4200 and 4153
- Text:
- "Categorification" refers to the following principle: some rings and modules (or other kinds of sets with operations) are secretly just shadows of some more sophisticated mathematical objects (categories). Finding the appropriate category can reveal hidden structure that's invisible from a purely ring-theoretic perspective. For example, consider the semiring of natural numbers (including 0) with the operations of addition and multiplication. The category of finite-dimensional vector spaces is a categorification of this semiring. Indeed, the dimension gives a one-to-one correspondence between isomorphism classes of such vector spaces and the natural numbers. Moreover, addition and multiplication correspond to the direct and tensor product of vector spaces respectively. In this course, we will cover the foundational notions needed to understand what categorification is, and we will discuss a number of notable examples of categorification that have been discovered in the past two decades.
- MATH 7001: Communicating Mathematics I
- 3:00-4:50 TTh
- Instructor: Prof. Oxley.
- Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for one-semester-hour's credit.
- This course provides practical training in the teaching of mathematics at the pre-calculus level, how to write mathematics for publication, and treats other issues relating to mathematical exposition. Communicating Mathematics I and II are designed to provide training in all aspects of communicating mathematics. Their overall goal is to teach the students how to successfully teach, write, and talk about mathematics to a wide variety of audiences. In particular, the students will receive training in teaching both pre-calculus and calculus courses. They will also receive training in issues that relate to the presentation of research results by a professional mathematician. Classes tend to be structured to maximize discussion of the relevant issues. In particular, each student presentation is analyzed and evaluated by the class.
- MATH 7210: Algebra I
- 9:30-10:20 MWF
- Instructor: Prof. Hoffman.
- Prerequisites: A course in Linear Algebra. This means not only familiarity with matrix operations, but knowledge of vector spaces and linear transformations. An undergraduate course in abstract algebra is helpful, but not required. It is required that the student is able to read and write proofs.
- Text: Recommended:
D. Dummit and R. M. Foote, Abstract Algebra (3rd edition), John Wiley and Sons, ISBN: 978-0471433347
This book is available at Amazon and some cheap versions can be purchased, but it is not necessary to buy it; I will choose assignments from this text. - This covers basics on Groups, Rings and Modules.
- MATH 7230: Hypergeometric Functions
- 11:30-12:20 MWF
- Instructor: Prof. Tu.
- Prerequisites:
- Math 4201 and 7210 (Abstract Algebra), Math 4036 (complex analysis) or equivalent are very helpful. You should be comfortable with Galois theory, meromorphic and analytic functions.
- Know how to find resources.
- Text: There is no required text, but the following references and web sources are useful.
- Hypergeometric Functions over Finite Fields, by J. Fuselier, L. Long, R. Ramakrishna, H. Swisher, F.-T. Tu
- Notes on differential equations and hypergeometric functions, by F. Beukers.
- Hypergeometric abelian varieties, by N. Archinard
- Special functions, by G. E. Andrews, R. Askey, and R. Roy.
- Hypergeometric Functions, My Love, by M. Yoshida
- Gamma function, beta function, and classical hypergeometric functions are important special functions and have a wide range of applications in mathematics and Engineering. In number theory, hypergeometric functions enjoy tight connection with certain type of algebraic varieties and arithmetic properties of modular forms. Therefore, hypergeometric functions over complex field, finite fields, and their truncated version play important roles in the study of hypergeometric type varieties as they provide abundant arithmetic information regarding these varieties.
The goal of this course is to give a gentle introduction to some basic properties of classical hypergeometric functions, finite field analogues of hypergeometric functions, and their connection to hypergeometric type abelian varieties. In the later part of the semester, we will discuss some recent developments related to different versions of hypergeometric functions, and advanced topics, depending on audiences' interests.
- MATH 7240: Algebraic Geometry
- 1:30-2:50 TTh
- Instructor: Prof. Sage.
- Prerequisites: The first year algebra sequence.
- Text:
- Q. Liu, Algebraic Geometry and Arithmetic Curves
- D. Eisenbud and J. Harris, 3264 and All That
- Algebraic geometry has its origin in the study of solutions to systems of polynomial equations. It is of fundamental importance in a wide range of areas of mathematics such as number theory, representation theory, and mathematical physics and also has surprising applications to such fields as statistics, mathematical biology, control theory, and robotics.
Modern algebraic geometry is based on the fundamental notion of a scheme. This course will give an introduction to schemes and their geometry, with particular emphasis on motivating the definitions and constructions and providing many examples. Time permitting, we will use this machinery to discuss some concrete classical problems in intersection theory and enumerative geometry.
- MATH 7290: Reductive Groups
- 10:30-11:50 TTh
- Instructor: Prof. Achar
- Prerequisites: Math 7211, and some exposure to Lie theory and homological algebra
- Text: C. Jantzen, Representations of algebraic groups, 2nd ed
- This course is about algebraic representations of reductive algebraic groups over an algebraically closed field k. We will discuss the "easy" case where k is the field of complex numbers, but the main focus will be the case where k is a field of positive characteristic. Highlights will include Kempf's vanishing theorem, the Borel-Weil-Bott theorem, the Weyl character formula, the linkage principle, and the theory of translation functors. At the end of semester, we will cover some very recent developments in the theory of tilting modules. The first couple weeks will include a quick review of structure theory of reductive groups (maximal tori, Borel subgroups, root systems) but it will be helpful to have seen at least some of these concepts before.
- MATH 7311: Real Analysis I
- 9:00-10:20 TTh
- Instructor: Prof. Olafsson.
- Prerequisites: Math 4032 or 4035 or equivalent.
- Text: Real Analysis, Modern Techniques and Their Applications, by G. B. Folland and Lecture Notes by G. Olafsson.
- This is a standard introductory course on analysis based on measure theory and integration. We start by introducing sigma algebras and measures. We will then discuss measurable functions and integration of real and complex valued functions. As an example we discuss the Lebesgue integral on the line and n-dimensional Euclidean space. We also discuss the Lebesgue integral versus the Riemann integral. Important topics here are the convergence theorems, product measures and Fubini’s theorem and the Radon-Nikodym derivative. We give a short discussion of Banach spaces and then apply that to the Lp spaces. Further topics include functions of bounded variations Lebesgue differentiation theorems, Lp and its dual, and the Riesz-Markov-Saks-Kakutani theorem and Fourier series may be presented if there is sufficient time.
We will mostly follow the book by Folland. We will also hand out notes as needed. There are several other very good books on analysis and measure theory:
- L. Richardson: Measure and Integral: An Introduction to Real Analysis.
- P. R. Halmos: Measure Theory (Graduate Text in Mathematics)
- A. Friedman: Foundations of Modern Analysis. Dover
- MATH 7350: Complex Analysis
- 12:30-1:20 MWF
- Instructor: Prof. Zhu.
- Prerequisites: Math 7311 or equivalent
- Text: Complex Analysis by Elias Stein and Rami Shakarchi, Princeton Lectures in Analysis II
- Topics: Theory of holomorphic functions of one complex variable; path integrals, Cauchy’s Theorem, power series, singularities, meromorphic functions, mapping properties, normal families, the Gamma functions and other topics.
- MATH 7360: Probability Theory
- 11:30-12:20 MWF
- Instructor: Prof. Sundar.
- Prerequisites: Math 7311 or its equivalent
- Text: Probability and Stochastics by Erhan Cinlar
- This is a self-contained introduction to modern probability theory. It starts from the concept of probability measures, and introduces random variables, and independence. After studying various modes of convergence, the Kolmogorov strong law of large numbers and related results will be established. Weak convergence of probability measures will be discussed in detail, which leads us to the central limit theorem and its applications. A main goal of the course is to develop the concept of conditional probability and its basic properties. Martingales will be introduced and several important results in martingale theory will be studied.
- MATH 7365: Applied Stochastic Analysis
- 10:30-11:50 TTh
- Instructor: Prof. Ganguly.
- Prerequisites: Graduate level Probability Theory (Math 7360). Math 7360 can be taken concurrently with this course.
- Text: The course will have no fixed text book. Some references will be mentioned in the class.
- Math 7365 is a course on stochastic process. A stochastic process can be thought of as a random function of time which originates in modeling temporal dynamics of many systems. Detailed modeling of such systems require incorporation of their inherent randomness which deterministic methods, for example, through differential equations fail to capture. Examples of such systems are numerous and wide ranging - from biological networks to financial markets. Markov processes, in particular, form one of the most important classes of stochastic processes that are ubiquitous in probabilistic modeling. They also lead to probabilistic interpretations of a large class of PDEs. For example, Brownian motion is the underlying Markov process whose probability distribution satisfies the heat equation. The course will cover theory of martingales and Markov processes in discrete-time. Some specific topics include Doob's decomposition theorem, Doob's inequalities, Burkholder-Davis-Gundy inequality, Kolmogorov's equations, generators, stationary measures and some elementary stability theory. The continuous-time case will be discussed if time permits. We will also discuss some stochastic algorithms like Markov Chain Monte Carlo, importance sampling, stochastic approximation methods which are instrumental in probabilistic approach to data-science. The course is also a gateway to the course on stochastic analysis (Math 7366).
- MATH 7380: Seminar in Functional Analysis
- 1:30-2:50 TTh
- Instructor: Prof. He.
- Prerequisites: Real Analysis (7311)
- Text: Harmonic Analysis in Phase Space by Gerald Folland, Princeton University Press.
- We will start with basic idea of quantization, CCR (canonical commutation relation) and discuss the Heisenberg uncertainty principle, Fourier-Wigner transform, Fock space, Weyl calculus, principal symbols, pseudo differential operators. This will be done with the introduction of Heisenberg group, perhaps the simplest noncommutative Lie group. Our main focus will be the first two chapters. If we have time, we will either get into the wave packets and Wave front set (Ch 3) involving more analysis, or the metaplectic representation (Ch 4) involving more group representation.
- MATH 7384: Topics in Material Science: Elliptic and Parabolic Regularization
- 2:30-3:20 MWF
- Instructor: Prof. Tarfulea.
- Prerequisites: The most important prerequisites are real and complex analysis. Experience with measure theory, Fourier analysis, and PDEs will be useful (as they occasionally appear in the proofs), but it will not be essential.
- Text: We will not use any one source as a text book. My lectures will draw from my notes and several references.
- Course Description:
The 20th century saw the development of many powerful tools of functional and harmonic analysis for building and analyzing solutions to partial differential equations (PDE). The focus shifted away from explicit classical solutions to large, carefully constructed spaces of rough functions, weak interpretations of derivatives, and topological/iterative arguments for finding so-called weak solutions to PDE. Weak solutions, though easier to find, are a priori not very regular. Much of this theory, including the resolution of the 19th Hilbert problem, then hinges on using the structure of the equation (in conjunction with various tools of function decomposition) to "bootstrap" the regularity of the solution, in many cases recovering differentiability, uniqueness, and physicality.These techniques form an important core skill set for research in elliptic and parabolic PDE, as well as a valuable springboard when studying many other evolution equations derived from physical models. We will focus on linear elliptic equations and equations of fluid mechanics (and possibly of viscoelasticity) as prototypes for this analysis.
Proposed Course Topics:
- We begin with a treatment of the Laplace equation in the first week just to set the tone of the course.
- We will then work with Lebesgue spaces and weak derivatives, proving the fundamental inequalities for basic functional analysis (Holder, Gagliardo-Nirenberg-Sobolev, Morrey), and the trace theorem.
- We then transition to weak solutions for partial differential equations (via Lax-Milgram theorem), which then leads into the formulation and resolution of the 19th Hilbert problem (methods of de Giorgi, Nash, and Moser, and also Harnack inequalities) on smoothness of solutions in the calculus of variations.
- We will then see this theory develop further in the context of viscous fluid equations (leading to the Prodi-Serrin conditions and weak-strong uniqueness) that exploits the divergence structure of the problem, as well as the Krylov-Safonov approach adapted for non-divergence equations.
- We will then add Fourier analysis into the mix with theorems of basic harmonic analysis (Littlewood-Paley decompositions, Besov spaces, Bernstein inequality) and see these implemented in the resolution of the first half of the Onsager conjecture for the Euler equations.
Your grade is largely based on the homework assignments. If the class is not too large, we will also have student presentations for a few of the easier and self-contained results.
- MATH 7386: Partial Differential Equations
- 12:00-1:20 TTh
- Instructor: Prof. Bulut.
- Prerequisites: Math 7311 (Real Analysis I) or equivalent, or consent of department.
- Text: Primary Text: Partial Differential Equations by Lawrence C. Evans
Additional optional references:
Elliptic partial differential equations by Qing Han and Fanghua Lin
Semilinear Schrödinger Equations by Thierry Cazenave. - This course provides an introduction to the theory of partial differential equations. Topics to be covered include:
- Introduction to elliptic, parabolic, and hyperbolic partial differential equations
- Examples: Laplace's equation, the heat equation, and the wave equation
- Introduction to Sobolev spaces, weak derivatives, existence and uniqueness of solutions.
- Elliptic equations: existence, regularity and the maximum principle.
- Selected additional topics (chosen according to time and class interest).
a) Nonlinear dispersive PDE with an emphasis on the nonlinear Schrodinger equation.
b) Introduction to Calculus of Variations: Euler-Lagrange equations, existence of minimizers, eigenvalues of self-adjoint elliptic operators.
- MATH 7390-1: Topics in Numerical Analysis
- 1:30-2:20 MWF
- Instructor: Prof. Wan.
- Prerequisites: Numerical Linear Algebra: Math 4064, Probability Math 3355, or equivalent
- Text: Lecture notes from the instructor.
- This is an introductory course on topics in uncertainty quantification and machine learning that are closely related to scientific computing. After clarifying some classical problems in statistical learning such as classification and regression, we will discuss some topics closely related to deep learning such as deep neural networks, Bayesian regression, high-dimensional density estimation, etc. We will pay particular attention to the relation between machine learning techniques and classical theories or problems in scientific computing such as approximation theory, inverse problems, model reduction, etc. The algorithms will be implemented in Python based on some open libraries such as Tensorflow from Google.
- MATH 7390-2: Theory of Distributions
- 8:30-9:20 MWF
- Instructor: Prof. Rubin.
- Prerequisites: : Multidimensional Calculus (2057, 4035 or equivalent).
- Text: V.S. Vladimirov, Methods of the Theory of Generalized Functions, Taylor & Francis, 2002. Some other texts will be also used.
- The subject of the course is introduction to the theory of distributions (or generalized functions). This theory, created by S.L. Sobolev and L. Schwartz, is a background of every educated mathematician. It enables one to differentiate non-differentiable functions, evaluate divergent integrals, solve differential equations of mathematical physics, and do many other useful things in analysis and applications.
- MATH 7490: Matroid Theory
- 8:30-9:30 MWF
- Instructor: Prof. Oxley.
- Prerequisites: Permission of the Department
- Text: J. Oxley, Matroid Theory, Second edition, Oxford, 2011.
- What is the essence of the similarity between forests in a graph and linearly independent sets of columns in a matrix? Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph? Can one test in polynomial time whether a matrix is totally unimodular? Matroid theory examines and answers questions like these.
This course will provide a comprehensive introduction to the basic theory of matroids. This will include discussions of the basic definitions and examples, duality theory, and certain fundamental substructures of matroids. Particular emphasis will be given to matroids that arise from graphs and from matrices.
- MATH 7510: Topology I
- 10:30-11:20 MWF
- Instructor: Prof. Dani.
- Prerequisites: Advanced Calculus (Math 4031)
- Text: Topology (2nd ed.) by James R. Munkres.
- This course is a preparation course for the Core I examination in topology. It will cover general (point set) topology, the fundamental group, and covering spaces. We will also introduce simplicial complexes and manifolds, using them often as examples. A good supplementary reference is Chapter 1 of Algebraic Topology by Allen Hatcher, available online .
- MATH 7590-1: Riemannian Geometry
- 10:30-11:50 TTh
- Instructor: Prof. Baldridge.
- Prerequisites: Math 7510 (Topology I) or equivalent
- Text: Manfredo do Carmo, Riemannian Geometry (Birkhauser, ISBN: 0-8176-3490-8)
- Riemannian Geometry. This course is an introduction to Riemannian geometry, the study of smooth manifolds endowed with Riemannian metrics. We will cover the basics: Riemannian metrics, connections, geodesics, curvature, Jacobi fields, completeness, and spaces of constant curvature. We will then proceed to some topics that relate curvature, topology, and analysis, such as the Hopf--Rinow Theorem, Bonnet--Myers Theorem, Preissman's Theorem and the Rauch Comparison Theorem.
This is an excellent class to take if you are interested in topics like symplectic geometry, algebraic geometry from a Kahler manifold perspective, hyperbolic geometry and geometric group theory, symplectic and contact topology, Lagrangian and Legendrian sub manifolds, mirror symmetry, analysis on manifolds, gauge theory, path integrals and Chern-Simons theory, representation varieties, L^2-cohomology, and, of course, the mathematics behind general relativity. We won’t cover these topics, but this class will make them significantly easier to learn and put them into a larger context if you are currently learning them.
- MATH 7590-2: Characteristic Classes
- 9:00-10:20 TTh
- Instructor: Prof. Cohen.
- Prerequisites: Exposure to cohomology theory, e.g., as provided by MATH 7512 (and expanded on in MATH 7520)
- Text: Characteristic Classes, by J. Milnor and J. Stasheff
- Characteristic classes are cohomology classes associated to a vector bundle, providing measures of twisting in the bundle. This course will focus largely on vector bundles over smooth and complex manifolds, and the problem of classifying these bundles. Specific topics include Stiefel-Whitney classes, Grassmannians, Chern classes, the Euler class, and Pontryagin classes.
Spring 2021
- MATH 4997-1: Vertically Integrated Research: Categorification
- 12:00-1:20 TTh Onlline
- Instructor: Profs. Achar and Sage
- Prerequisites: Math 4200 and 4153
- Text: None
- "Categorification" refers to the following principle: some rings and modules (or other kinds of sets with operations) are secretly just shadows of some more sophisticated mathematical objects (categories). Finding the appropriate category can reveal hidden structure that's invisible from a purely ring-theoretic perspective. For example, consider the semiring of natural numbers (including 0) with the operations of addition and multiplication. The category of finite-dimensional vector spaces is a categorification of this semiring. Indeed, the dimension gives a one-to-one correspondence between isomorphism classes of such vector spaces and the natural numbers. Moreover, addition and multiplication correspond to the direct and tensor product of vector spaces respectively. In this course, we will cover the foundational notions needed to understand what categorification is, and we will discuss a number of notable examples of categorification that have been discovered in the past two decades. (This course continues the theme of the Fall 2020 semester, but that semester is not a prerequisite for this course, and new participants are welcome.)
- MATH 4997-2: Vertically Integrated Research:
- 1:30-2:50 TTh
- Instructor: Prof. Vela-Vick
- Prerequisites:
- Text:
- MATH 4997-3: Vertically Integrated Research:
- 1:30-2:50 TTh
- Instructor: Profs. Wolenski and Madden
- Prerequisites:
- Text:
- MATH 7002: Communicating Mathematics II
- 3:00-4:50 T Th Online
- Instructor: Prof. Oxley .
- Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for one-semester-hour's credit.
- MATH 7211: Algebra II
- 9:30-10:50 MW
- Instructor: Prof. Ng.
- Prerequisites: Math 7210 or equivalent.
- Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003.
- This is the second part of our graduate algebra sequence. Topics will include field theory, Galois theory, basics of commutative algebra and algebras over a field, Wedderburn’s theorem, Maschke’s theorem, tensor products and Hom for modules, possibly some introduction to homological algebra or linear representations of finite groups if time permitted.
- MATH 7230: Algebraic Number Theory
- 2:30-3:20 MWF
- Instructor: Prof. Long.
- Prerequisites: 7210 Abstract Algebra I
- Text: A Course In Algebraic Number Theory by Robert Ash
- Basic concepts and results in algebraic number theory including number fields, Dedekind domains, and local fields.
- MATH 7280: Étale Cohomology
- 9:30-10:20 MWF Online
- Instructor: Prof. Hoffman.
- Prerequisites: The minimum is the material covered in Math 7210 (Algebra) and Math 7510 (Topology), but some familiarity with Algebraic Geometry, Math 7240 is very helpful.
- Text: No required text, but we will take material from various places, such as:
- J. Milne, Étale Cohomology (PMS-33), Volume 33 (Princeton Mathematical Series)
- Lei Fu, Etale Cohomology Theory (Revised Edition) (Nankai Tracts In Mathematics Book 14)
- Eberhard Freitag , Reinhardt Kiehl, et al, Etale Cohomology and the Weil Conjecture (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge
- Günter Tamme and M. Kolster, Introduction to Étale Cohomology (Universitext)
- SGA 4, 5, 7, available online.
- This will be more like a seminar or topics course. Each student will pick a topic to present. I and/or the student will lecture about these themes:
- Cohomology of varieties.
- Motives.
- Grothendieck topologies and topoi.
- Smooth, unramified and étale morphisms.
- The fundamental group.
- Cohomology of curves.
- Basic theorems of étale cohomology.
- Zeta functions of varieties.
- Deligne’s proof.
- MATH 7290: Quiver Varieties
- 1:30-2:50 TTh Online
- Instructor: Prof. Sage.
- Prerequisites: The first year algebra sequence and some familiarity with algebraic geometry
- Text: Quiver Representations and Quiver Varieties by A. Kirillov Jr
- Quiver varieties are geometric objects which encode certain data from linear algebra. A quiver is a directed graph, and a representation of a quiver is an assignment of a vector space to each vertex and a linear map to each edge subject to compatibility relations determined by the graph. Quiver varieties parameterize equivalence classes of quiver representations. They are symplectic manifolds with a particularly rich structure, and their geometry has deep connections with symplectic geometry and Lie theory. In this course, we will first develop the basic theory of quiver representations and quiver varieties. We will then discuss various applications, including
- geometric constructions of the universal enveloping algebra of Kac-Moody Lie algebras;
- Crawley-Boevey's solution to the Deligne-Simpson problem: the existence problem for irreducible Fuchsian differential equations on the Riemann sphere with specified singular points and specified residues at the singularities.
- MATH 7320: Ordinary Differential Equations
- 12:00-1:20 TTh
- Instructor: Prof. Shipman.
- Prerequisites: Undergraduate advanced calculus, undergraduate complex variables, and core graduate analysis
- Text: My notes; and Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by J. Guckenheimer and Ph. Holmes (Springer, Applied Mathematical Sciences 42).
- This course will include the theories of linear and nonlinear ordinary differential equations. These are genuinely different areas of mathematics. Linear ODEs fall within the area of spectral theory, while nonlinear ODEs are the core of the study of dynamical systems.
The overarching concepts in solving linear ODEs are (1) for initial-value problems: the matrix exponential and how it reflects the canonical forms of matrices and (2) for boundary-value problems: the solution operator, or Green function. I will introduce the spectral theory of linear differential operators and its intimate connection with complex analysis.
The overarching concepts for the nonlinear theory are flows of vector fields and dynamical systems. Upon that basis, one studies diverse phenomena such as bifurcations, separation of time scales, bursting (such as in neurobiology), hysteresis, stability (this is the connection between linear and nonlinear), control systems, chaos, and strange attractors.
Certainly, a course cannot come close to doing justice to all of these topics, and neither can one person. My goal is twofold: (1) to present the foundational rigorous theory of ODEs, and (2) to introduce a broad variety of topics in ODEs that highlight what makes the field interesting.
- MATH 7330: Functional Analysis
- 1:30-2:50 Online
- Instructor: Prof. Estrada.
- Prerequisites: Math 7311 or its equivalent.
- Text: Trves,F., Topological Vector Spaces, Distributions, and Kernels; Dover, 2006.
- A standard first course in functional analysis. Topics include Banach spaces, Hilbert spaces, Banach algebras, topological vector spaces, spectral theory of operators and the study of the topology of the spaces of distributions.
- MATH 7366: Stochastic Analysis
- 10:30-11:20 MWF
- Instructor: Prof. Kuo.
- Prerequisites:
- Text: There are three texts:
- Kuo, H.-H.: Gaussian Measures in Banach Spaces. Lecture Notes in Math.,Vol. 463, Springer, 1975. (Reprinted by Amazon, 2006)
- Kuo, H.-H.: White Noise Distribution Theory, CRC Press, 1996
- Kuo, H.-H.: Introduction to Stochastic Integration. Universitext, Springer, 2006
- Course description: In this course we will cover the following three topics in stochastic analysis:
- Abstract Wiener space: Gasuss measures, measurable norms, Gross-Sazonov theorem, transformation formula, Gaussian processes, heat equation.
- White noise theory: Theory of generalized functions, Minlos theorem, white noise functionals, characterization theorems, Hitsuda-Skorokhod integral.
- Stochastic integration: Brownian motion, Wiener integral, Ito integral, Ito's formula, Girsanov theorem, multiple Wiener-Ito integrals, extension of Ito's theory to anticipating stochastic processes.
- MATH 7380: Singular Integrals
- 11:30-12:20 MWF Online
- Instructor: Prof. Nguyen.
- Prerequisites: MATH 7311 and MATH 7350.
- Text: Loukas Grafakos, Classical Fourier Analysis, Third Edition, GTM 249, Springer, New York, 2014. xviii+638 pp. ISBN: 978-1-4939-1193-6.
Recommended Reference: Elias Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. - This is an introductory course in Fourier Analysis with an emphasis on the boundedness of Singular Integral Operators of convolution type. Such a boundedness property plays a fundamental role in various applications in pure and applied analysis. The course also covers such classical topics as Interpolation, Maximal Functions, Fourier Series, and possibly Littlewood-Paley Theory if time permits.
- MATH 7384: Topics in Material Science: Riemann surfaces and RH problems
- 12:30-1:20 MWF Online
- Instructor: Prof. Antipov.
- Prerequisites: one of 7311, 7350, 4036.
- Text:
- The whole field of computational complex analysis has grown rapidly during the past several decades. Partly, this is due to discovering new methods in the theory of conformal mappings, integral equations and boundary-value problems of the theory of analytic functions which have helped to solve long-standing problems in physics and engineering sciences. This course focuses on the theory and computational aspects of conformal mappings for multiply connected regions and the Riemann-Hilbert problem (RHP) of the theory of automorphic functions and analytic functions on Riemann surfaces of algebraic functions.
s
First this course will give an introduction to the classical RHP of the theory of analytic functions. Then more advanced topics will be considered. They include elements of the theory of Schottky groups, quasiautomorphic analogues of the Cauchy kernel, the Schottky-Klein prime function, and the RHP of the theory of symmetric automorphic functions. We will also discuss elements of the theory of Riemann surfaces of algebraic functions including abelian integrals and differentials, the Riemann theta function, the Jacobi inversion problem, the RHP on Riemann surfaces, and analogues of the Cauchy kernel (the Weierstrass and Hensel-Landsberg kernel). This course will give an introduction into the modern theory of circular and slit conformal mapping of canonical domains into multiply connected free boundary domains. Applications of the conformal mappings and the RHPs for automorphic functions and analytic functions on Riemann surfaces to fracture mechanics, inverse problems of materials science (multiple neutral inclusions), fluid mechanics (the hollow vortices problem) and canonical diffraction theory will be discussed.
- MATH 7390: Topics in Nonlinear Optimization Theory and Algorithms
- 1:30-2:50 TTh Online
- Instructor: Prof. Zhang.
- Prerequisites: Math4032 or equivalent
- Text: Class Notes
- This class will cover classical nonlinear optimization theory and algorithms. Tentative topics include but not limited to Line search methods, Newton and quasi-Newton methods, Conjugate gradient methods, KKT optimality conditions, Penalty methods, Sequential quadratic programming, Trust region methods, nonsmooth optimization.
- MATH 7410: Graph Theory
- 1:30-2:20 MWF Online
- Instructor: Prof. Oporowski.
- Prerequisites:
- Text:
- The main theme of this course will be graph theory. We will discuss a wide range of topics, including spanning trees, eulerian trails, matching theory, connectivity, hamiltonian cycles, coloring, planarity, integer flows, surface embeddings, and graph minors. For more information see Math 7410.
- MATH 7490: Graph Minors.
- 10:30-11:50 TTh Online
- Instructor: Prof. Ding.
- Prerequisites: Math 4171 or equivalent
- Text: None (lecture notes will be distributed)
- This is an introduction to the theory of graph minors. We will discuss many problems of the following two types: determine all minor-minimal graphs that have a prescribed property; determine the structure of graphs that do not contain a specific graph as a minor. We will focus on connectivity and planarity.
- MATH 7512: Topology II
- 9:00-10:20 TTh
- Instructor: Prof. Vela-Vick.
- Prerequisites: Math 7510
- Text: Algebraic Topology, by Allen Hatcher
- This course covers the basics of homology and cohomology theory. Topics discussed including singular and cellular (co)homology, Brouwer fixed point theorem, cup and cap products, universal coefficient theorems, Poincare duality, Alexander duality, Kunneth theorems, and the Lefschetz fixed point theorem.
- MATH 7550: Differential Geometry
- 10:30-11:50 TTh
- Instructor: Prof. Vela-Vick.
- Prerequisites: Math 4031 (or equivalent) and Math 7510
- Text: Differential Topology, by Victor Guillemin and Alan Pollack
- This course gives an introduction to the theory of manifolds. Topics to be covered include: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields, differential forms, Frobenius's theorem, integration on manifolds, Stokes's theorem, de Rham cohomology, Lie groups and Lie group actions.
- MATH 7590-1: Geometric Topology: Khovanov homology and ribbon graphs.
- 9:00-10:20 TTh
- Instructor: Prof. Baldridge.
- Prerequisites:
- Text:
- In this course we will cover: the Jones polynomial, Khovanov homology, which is the categorification of the Jones polynomial, virtual links, the relationship between virtual links and ribbon graphs with perfect matchings, and combinatorial invariants of ribbon graphs that can be defined via this relationship. If you are interested in Topological Quantum Field Theories (TQFTs), categorification, graph theory, and/or classical (virtual) knot cobordisms, then you should consider this course.
About Khovanov homology: The Jones polynomial (1980s) was one of the first invariants of knots that was not geometrically defined. To this day, its precise geometric meaning is still clouded in mystery. Khovanov homology lifts the Jones polynomial to a homology theory that makes surprising connections between representation theory and knot theory, and more recently, graph theory.
You should be familiar with the basics of graph theory and topology, some knot theory, and the homological algebra of chain complexes. The rest of the material we will discuss as it is needed. The topics in this course are on the frontier of mathematics—this will be an excellent opportunity for you to learn many modern mathematical techniques that cross a wide range of fields.
- MATH 7590-2: Geometric Group Theory
- 10:30-11:20 MWF Online
- Instructor: Prof. Dani.
- Prerequisites: 7210, 7510, 7512.
- Text: None
- The guiding principle in geometric group theory is to try to glean information about infinite groups via their actions on metric or topological spaces. An important class of spaces in the field consists of "CAT(0) cube complexes". These are CW complexes with cubical cells satisfying a certain "non-positive curvature" condition. This course will focus on the study of groups which act properly and cocompactly on CAT(0) cube complexes. In particular, we will see how the non-positive curvature of the spaces forces certain algebraic properties of the group. Along the way we will introduce all of the basic geometric group theory concepts that we need.
- MATH 7710: Advanced Numerical Linear Algebra.
- 11:00-12:20 MW Online
- Instructor: Prof. Walker.
- Prerequisites: linear algebra, advanced calculus, some programming experience (but not required).
- Text: Fundamentals of Matrix Computations, 3rd edition, by D. S. Watkins.
- Description: This course will develop and analyze linear algebra algorithms. Fundamentals will be discussed, i.e.:
- Basic tools: norms, projectors, spectral theorem, singular value decomposition.
- Direct methods: LU factorization, Cholesky, least squares problem, QR factorization.
- Iterative methods: Jacobi, Richardson, Gauss-Seidel, successive over-relaxation, steepest descent, conjugate gradient.
- Eigenvalue problems: power methods, Rayleigh quotient iteration, deflation, QR algorithm.