Course Rotation for Advanced Mathematics Courses

This is the expected course rotation for Advanced Mathematics Courses over the next few years. Graduate courses are not usually scheduled until, at the earliest, the end of the fall semester of the preceding academic year, for example, in December of 2015 for Fall 2016/Spring 2017. Thus, for graduate courses which are not cross-listed with a 400-level course, only those which have been scheduled or are associated to a qualifying exam or comprehensive exam are listed. Course offerings can change due to instructor availability, perceived demand, or other reasons, so even listed courses are not guaranteed to be offered.

An asterisk by a 800-level course number indicates it is not open to mathematics graduate students.

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Announcements for Fall 2017 courses

Math 450
Course Title: Math 450 -- Combinatorics
Time: MWF 9:30-10:20
Instructor: Tri Lai (
Textbook: ``A Walk through Combinatorics: An Introduction to Enumeration and Graph Theory" by Miklos Bona (Third or Fourth Edition)
Grades: will be based on homework and exams
Prerequisites: MATH 310 or 310H or 325
Intuitively speaking, Combinatorics is the branch of mathematics focusing on human's earliest mathematical action: counting. More precisely, Combinatorics studies enumeration and/or existence of arrangements of objects.
In this course we will explore classical topics of the field, including the pigeonhole principle, inclusion--exclusion, recurrence relations, partitions, permutations, and generating functions. We will roughly cover the first half of the excellent book ``A Walk through Combinatorics: An Introduction to Enumeration and Graph Theory" by Miklos Bona. The fourth edition of the book is recommended, however the third edition is also good. If time allows, we will investigate several modern topics at the end of the course, such that Enumeration of tilings and Bijective proofs. There will be weekly homework assignments and two exams (a midterm and a final).
Please contact the instructor, Tri Lai, if you have questions about the course.
Math 856
Course Announcement: Math 856 Differential Topology
Time: 1230-0145P TR Room: OLDH 303
Instructor: Brian Harbourne
The main text will be "Differential Topology", by Guillemin and Pollack, but we may draw on other sources too, such as Spivak's "Calculus on Manifolds", and Jack Lee's "Introduction to Smooth Manifolds", and maybe Frank Warner's "Foundations of Differentiable Manifolds and Lie Groups".
The main assessments will be periodic homework assignments.
The ideas explored in this course have infiltrated all areas of mathematics, including algebraic geometry and commutative algebra (my areas), topology (of course!) and many aspects of analysis. The basic objects of study are smooth manifolds, but this entails other things such as tangent bundles and vector bundles more generally, but also calculus (think Stokes' Theorem) and cohomology (in its de Rahm avatar). The exact topics we cover will be partly determined by the interests and backgrounds of the students taking the class.
For those who might be wondering what the differences are among various fields with overlapping names, here is a very abbreviated primer:
Algebraic geometry: the study of almost manifolds defined by polynomials (called varieties, which are smooth manifolds off a measure 0 subset), using both algebraic and geometric techniques (Emmy Noether played an important role here)
Algebraic topology: the study of topological spaces (often but not always manifolds, and often smooth manifolds at that) by assigning algebraic invariants (Emmy Noether played an important role here, too)
Differential geometry: the study of metric related properties (such as curvature) of manifolds with a metric (thus foundational for a lot of physics, where once again Emmy Noether played an important role!)
Differential topology: the study of smooth manifolds and their properties; the ideas and techniques here have guided thought and developments in the other areas mentioned above (I don't know of Emmy Noether having played a big role here, but she comes to mind since two weeks ago was the anniversary of her untimely death, in 1935, age 53)
Math 905
Course Title: Math 905 - Commutative Algebra
Time : MWF 10:30-11:20
Instructor: Alexandra Seceleanu
Textbook: Commutative Ring Theory by Matsumura (not required, but recommended)
Grades: will be based on homework and possibly a topic presentation
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Born out of the need for a rigorous framework in algebraic geometry and number theory, commutative algebra has evolved into a beautiful theory in its own right. Nowadays this is an active area of research, with no less that four important conjectures settled in the last calendar year.
In this course we will explore the technical foundations of the field, laid by masters like David Hilbert, Emmy Noether and Wolfgang Krull. I intend to cover the first 7 chapters in the textbook Commutative Ring Theory by Matsumura, with some additional material depending on the interest of the audience. Core topics include completions, integral extensions, valuation rings, dimension theory and the theory of regular sequences.
The end goal is to gain a working knowledge of commutative algebra in order to give background for exploring recent developments. Feel free to contact the instructor either in person or by email with any questions you might have.
Math 923
Instructor: David Pitts
Class Time: M-W-F 12:30–1:20. (It may be possible to change this if necessary.)
Harmonic analysis is an extremely useful and important branch of analysis, which arose in the early 1800s with Fourier. Among other subjects, it is important in functional analysis, partial differential equations, applied mathematics, and engineering.
Harmonic analysis is concerned with representing functions utilizing other functions which have certain symmetries. Here are some examples of this: (1) Any function f : R → R can be written as a linear combination of an even and an odd function. (2) Let n ≥ 2 be a natural number, let T be the unit circle in the complex plane, and let λ = e2πi/n. Then any function f : TC can be represented as a linear combination of functions fk satisfying i>fk(λz) = λ kf(z) for every z ∈ T. (3) A possibly more familiar example is the fact that every integrablef : TC has a Fourier series associated to it.
In each of these examples, there are underlying abelian groups, and one attempts to write a given function as a linear combination of functions which have symmetries arising from the groups. There are two problems which need to be addressed: first finding the coefficients to be used in the linear combination (this is “analysis”) and then working out in what sense the linear combination represents the original function (this is “synthesis”).
Our course will begin with understanding the basics of what is commonly called “classical harmonic analysis”, that is harmonic analysis using the groups Z and T; we’ll then move on to use the group R, and finally if time permits, we’ll discuss what happens with a general abelian topological group. Prerequisites: Math 825-826 or equivalent. Some exposure to complex analysis (e.g. Math 423), measure theory (Math 921-922) along with general topology (either 871 or 922), is helpful but not essential. However, it is not necessary to have taken one or more of these courses.
Text: I will use a variety of sources, but the main one will probably be “An Introduction to Harmonic Analysis” by Y. Katznelson.
Please contact the instructor if you’d like further information about the course.
Math 928
Math 928 in the Fall will meet MWF 10:30-11:20 AM. Broadly speaking, this is linear algebra in infinite dimensions: infinite-dimensional vector spaces (e.g., spaces of functions sharing certain properties) and linear operators on them. The framework and methods to be discussed are fundamental to anyone having interest in analysis, operator algebras and partial differential equations (in many aspects, the modern theory of PDE's could be regarded as applied functional analysis).
Familiarity with real analysis is a necessary prerequisite, but the material does not focus on measure-theoretic proofs; rather it shares many aspects with topology and linear algebra. So if you have had Math 825-826, you can take this course simultaneously with Math 921.
Math 958
Description: Claude Shannon introduced information theory as "A Mathematical Theory of Communication" in his seminal 1948 paper. Information theory answers two important questions in communication: what is the fundamental limit of data compression, and what is the fundamental limit of communication? The first two-thirds of the course will focus on basic concepts in information theory including entropy, mutual information, prefix and Huffman codes, channel capacity, and the channel coding theorem. The last one-third of the course will focus on applications to other areas and/or more specific topics in information theory itself.
Information theory has applications to many areas. I would like students in the class to be interested in learning the material and ideally find connections to their own research program. To that end, if you are planning to take the class, please send me a brief description of the research you are doing (if you are already underway on your dissertation), or the direction of research you are currently considering. If you are really unsure, simply tell me some topics in math that you particularly like. If possible, I will try to find presentation topics that both tie in with your interests and fit the scope of the course. I hope to plan the course soon so if you can let me know earlier rather than later, it will be easier for me to accommodate you.
Prerequisites: A basic background in probability theory (measure theory not required) and a reasonable level of mathematical maturity and enthusiasm. It is not necessary to have taken 850-852 although an exposure to discrete math and proof techniques may be helpful.
Textbook: No required textbook. I will set some aside for reference for our class either in the library or in a shared area of the department. I am also likely to make a packet of relevant papers for each student.
Assignments: There will be presentations toward the end of the semester and possibly some other learning opportunities such as problem sets or in-class work. The amount may vary based on what you are hoping to get out of the class.
Please let me know if you have any questions.

Announcements for Spring 2017 courses

Math 918
Math 918 : Topics in Algebra (Spring 2017) In this course we will explore cluster algebras and their connections to combinatorics, geometry, topology, representation theory, and string theory. Cluster algebras were discovered by Fomin and Zelevinsky in 2000. Since then, they have been shown to be related to a wide variety of branches of mathematics and mathematical physics. These fundamental objects can be constructed in an elementary manner. No background is required, as we will start from scratch. Almost all students interested in pure mathematics can benefit from attending this course. Roughly one-third of the lectures will be devoted to combinatorics including graph theory and algebraic combinatorics, and another one-third to geometry/topology including algebraic geometry, hyperbolic geometry, Teim\"uller theory and knot theory, and the rest to representation theory as well as homological algebra. There will be two types of assignments, from which each student can choose. One is for those who have a broad interest, and the other is for those who want to gain a deeper understanding of a specialized topic.
Math 934
Here is some disappointing news: Much of science relies on solving differential equations, and most differential equations cannot be solved by the human race. Now, here is some exciting news: if we allow for *approximate* solutions, we can solve almost all differential equations! This is one of the things that makes the Spring 2017 course:

Math 934 - Topics in Differential Equations

so great.

We will solve differential equations using numerical methods, but we will not leave mathematical rigor nor beauty behind! The subject of numerical analysis is full of clever ideas, elegant structures, dazzling schemes, subtle concepts, and a myriad of open problems. The subject of PDEs is incredibly vast, and one could spend several lifetimes studying just one PDE, and yet there are many thousands of PDEs out there. They are used to model phenomena such as weather, turbulence, blood flow, cancer growth, traffic, financial markets, ecology, acoustics, electricity, magnetism, star formation, and the bending of spacetime itself. Solving PDEs computationally not only unlocks new areas of science, but often leads to pretty pictures that can amaze people in poster sessions and astonish audiences at conferences.

To set the stage, we will begin with numerical solutions of ODEs (ordinary differential equations). We will quickly move on to numerical solutions of PDEs (partial differential equations). We will learn spectral/Fourier methods, and then move on to modern methods, including finite element methods, finite volume methods, saddle-point methods, and others. We will learn to program in Matlab, and we will also use a finite element library called FEniCS, which uses Python.

No programming background is necessary, and it is not necessary for you to have taken a course in PDEs. An undergraduate ODE course, such as Math 221, advanced knowledge of linear algebra, such as Math 415/815 (or anything beyond Math 314), and some analysis, such as Math 825/826 (could be taken concurrently), should be sufficient.

I hope you will join us this spring for Math 934!

Math 990

The study of knotted loops in three-dimensional space sits at the heart of low-dimensional topology. Knot theory is an eclectic topic with applications to biology, chemistry, and physics. Conjectures are easy to state, but proofs often involve borrowing heavily from other areas of mathematics, such as algebra, analysis, or combinatorics. This course surveys classical results in knot theory, as well as major developments that have taken place over the last 30 years. Topics to be covered include polynomial invariants, such as the Alexander, Jones, and Conway polynomials, homological invariants, like the powerful new Heegaard Floer and Khovanov Homology theories, and geometric tools, including Dehn surgery, bridge splittings, and thin position.

Grades will be computed from a minor amount of graded homework and student presentations over course materials. Although no prerequisites are required, it will be helpful to have taken the first semester of graduate topology.

Announcements for Fall 2016 courses

Math 911 Fall 2016
Course Announcement - Math 911 - Fall 2016 - Theory of Groups

Instructor: Susan Hermiller, 332 Avery Hall

The goal of this course will be an introduction to the theory of groups, with an emphasis on infinite groups.

Groups are a useful mathematical tool which originated in the study of symmetry of finite or infinite objects (eg tilings). Several methods for studying groups, including algebraic, combinatorial, computational, and topological/geometric techniques, will be discussed in this course, as well as connections and applications to other areas of mathematics.

There will be three main topics (with the ratio of time spent on them depending upon the interest of the students):

  1. Algebraic properties of groups: Nilpotence and solvability are algebraic measures of how close a non-abelian group is to being abelian. They're special cases of a more general construction, known as poly properties, which make a group tractable because the poly structure allows for many proofs by induction. We'll discuss relationships among the finitely generated nilpotent, polycyclic, and solvable groups, as well as the geometry of these groups.
  2. Combinatorial and computational group theory: Combinatorial properties of a presentation or Cayley graph of a group determine the subgroup structure and give algorithms to solve a variety of problems. We'll talk about combinatorial properties of "van Kampen diagrams" - planar graphs built from a group presentation - and computational algorithms and software for groups that are constructed from these properties.
  3. Geometric group theory: Turning a finitely generated group into a metric space and studying the geometry or topology of that space also gives algebraic and computational information about the group. This part of the course will include a discussion of what geodesics (= shortest paths between points) and growth functions (= "Hilbert functions" for those of you in commutative algebra!) can tell you about a group.

Prerequisites: Math 872, or permission of instructor.

Text: There will be no formal text for the course, but much of the material we will cover can be found in "A Course in the Theory of Groups" by D.J.S. Robinson, "Combinatorial group theory" by R.C. Lyndon and P.E. Schupp, and "Word Processing in Groups" by D.B.A. Epstein, J.W.Cannon, D. Holt, S. Levy, M. Paterson, and W. Thurston.

If you'd like more information, please feel free to write me or stop by.

Math 924 Fall 2016

The introductory material in the theory of analytic functions is one of the most complete, beautiful and useful theories in all of mathematics. We will study complex-valued functions of a complex variable. For example, the appropriate generalization of ex in our setting is ex+iy = ex(cos x + isin y). We say that a function is analytic in a domain Ω in the complex plane if for every z ∈ Ω, f0(z) = limh→0(f(z + h) − f(z))/h exists. Due to the nature of this complex limit, this condition is much stronger than differentiability of a function of a real variable. Analyticity of f has some remarkable consequences. For instance, if f is analytic in Ω, then f has infinitely many derivatives in Ω, and can in fact be represented by a convergent power series. Here’s another consequence of analyticity: If f(z) is known on a simple closed curve γ ⊂ Ω, then f(z) is completely known inside of Ω. The heart of this theory is the interplay between line integration in the complex plane and Taylor series. We also explore topological, geometric, algebraic and computational properties of analytic functions. All aspects of the theory tie together nicely, and almost everything that you want to be true is in fact true.

The prerequisites for the course are Math 825 and 826. We will be using the book “Real and Complex Analysis” by Walter Rudin.

Math 934 Fall 2016
Some Topics of Interest:
  1. Cone Theory in a Banach Space
  2. Normal cones
  3. Generating cones
  4. Regular and fully regular cones
  5. Solid cones
  6. Reproducing cones
  7. Applications of the Schauder Fixed Point Theorem to solve integral equations and boundary value problems.
  8. Use of degree theory to solve boundary value problems.
  9. Use of the Mountain Pass Theorem to solve boundary value problems.
  10. Application of Krasnoselki’s fixed point theorem in a cone.
  11. Use of the Leray-Schauder Alternative in differential equations.
  12. Use of Index Theory in differential equations.
Homework will be due at the end of each three week period.
Math 953 Fall 2016

Instructor: Brian Harbourne

Fundamentally, Algebraic Geometry is the study of solution sets of polynomial equations. It's rare that one can find exact solutions, but one can ask questions like: Is the solution set empty or non-empty? What is the dimension of the set of solutions? Is the set of solutions irreducible? What can be said about the topology of the solution set?

All of these questions can be reinterpreted in terms of commutative algebra, so one aspect of algebraic geometry is the interplay between geometry and commutative algebra. Not only can it be helpful to consider geometric questions from an algebraic perspective, it can also be useful to consider algebraic questions from a geometric point of view.

This course will give you the foundation for being able to take both points of view. It will cover the main theorems of both affine and projective algebraic geometry and their connections to commutative algebra. Additional topics will be selected based on the interests of the class.

There will be no formal text, but course lectures will be supplemented by on-line resources, such as these notes.

Prerequisites: It would be good to have taken 817-818 and probably 901-902.
Math 415/815 Spring 2016
This is a course about the structure of vector spaces and linear transformations. Linear Transformations are a fundamental tool in many parts of mathematics. Understanding their structure motivates many topics, such as modules in algebra or Banach spaces in analysis.
We will define general vector spaces and look at infinite dimensional ones, such as polynomials or continuous functions, as well as more familiar finite-dimensional ones. For linear transformations, topics include kernel and range, diagonalisation, and duality. Highlights of the course will be the spectral theorem and the Jordan canonical form (and what it means geometrically). Depending on time available and the class's preferences, we may cover various factorizations, quadratic and bilinear forms, and maybe even rings of linear transformations. Particularly in these later sections, the goal is to see how linear algebra is used across mathematics.
Prerequisites: Math 314 (a first course in linear algebra) and either Math 310 or Math 325 (in general, some experience in constructing proofs).
Please contact the instructor, Allan Donsig, if you have questions about the course.

Announcements for Spring 2016 courses

Math 831 Spring 2016
This course is intended as a prerequisite-free intro to problems involving functions dependent on more than one variable, for example, on several space coordinates and time.
The syllabus will include a fair amount of background material: a revision of multivariable calculus and geometry (vector fields, partial derivatives, differentials, divergence theorem, implicit function theorem), some ordinary differential equations (so Math 830 is helpful, but not required), the concept of integral curves and surfaces, spaces of continuous and differentiable functions, equations describing physical processes (transport of matter, vibrations, acoustics, heat transfer), and foundational results on existence and uniqueness of solutions to such problems.
This course may also be thought of as a review of calculus and analytic geometry with applications to differential equations. Even if you have already taken the Math 830--842 sequence, you may find Math 831of interest.
Math 433/833 Spring 2016
This course studies nonlinear optimization, both the mathematical theory and its effective application to solve practical problems. We will begin by reviewing the optimization methods of calculus, and then consider numerical iterative methods, including Newton's method and least squares. We then proceed to the major topic of the semester: Convex optimization. This topic is highly important in economics and finance, where the functions are often not differentiable but are convex. Fortunately, many Calculus-based methods can be extended to this setting. A major highlight here are the Karush-Kuhn-Tucker conditions for optimality of a convex program.
Many of the techniques in the course are not just of interest in mathematics, but also in science and industry. We will also do a fair amount of programming (in a language of your choice, but Matlab or Python are natural choices). No prior knowledge of programming is assumed, so this may be a good place to learn programming, or broaden your programming experience.
Math 911 Spring 2016
Groups are a Mathematical tool developed in the 19th century and originally designed to study symmetries of other Mathematical objects. Starting in the 20th century Mathematicians began studying groups themselves using topological and geometric methods culminating in the birth of Geometric Group Theory in the 1980’s when Gromov formalized the field by considering the group itself as metric space and connecting its large scale geometry to algebraic properties. Today, geometric group theorists use and formulate connections between a wide variety of fields to study algebraic, geometric and topological phenomena.
This course is designed to give students a well-rounded introduction to a variety of important topics in the field. It will be broken up roughly into thirds by general topics, though the precise material will depend on the interests of attendees.
First, the course will begin by covering some basic topics in the field, including: presentations of groups and basic constructions (free groups, free products, amalgamated free products and HNN-extensions); Cayley graphs of finitely generated groups; the word problem for groups; geometric “sameness” (quasi-isometry); topology at infinity of Cayley graphs (ends compactification of groups); hyperbolic geometry.
Next we will delve deeper into the geometry of the Cayley graph. Possible topics include groups with interesting geometry (hyperbolic and relatively hyperbolic groups, right-angled Artin and Coxeter groups), classes of groups with rigid geometry (quasi-isometric rigidity), decompositions of a group into simpler groups (Bass-Serre theory), and asymptotic geometry of groups.
Finally we will explore interactions between topology and group theory. Possible topics include groups of homeomorphisms and automorphisms (mapping class groups & Out(F_n)), three dimensional manifolds, and optimization problems in topology (stable commutator length and quasimorphisms).
Prerequisites: MATH 872, or permission of the instructor. Text: There will be no formal textbook for the course, though a good references for most of the material in the course are: “Metric Spaces of Non-positive Curvature” by M. Bridson and A. Haefliger, “Topics in Geometric Group Theory” by P. De la Harpe, and “scl” by D. Calegari.
For more information, feel free to contact the instructor.
Math 918 Spring 2016
Instructor:Alexandra Seceleanu Time: TR 12:30 - 1:45 pm Book: David Eisenbud’s “The geometry of syzygies” Topic: The aim of this course is to give a rather concrete taste of the interplay between homological algebra and algebraic geometry. While these are broad areas of study in their own right, we will focus on some aspects that are fundamental, yet do not require very sophisticated machinery. In particular, we’ll learn about invariants of (graded) ideals such as Hilbert functions, projective dimension, Castelnuovo-Mumford regularity and apply them to study sets of points and curves in projective space. We will aim to work through the text book mentioned above, developing any necessary background material as needed. Additional topics may be presented based on audience’s interest. And of course, there will be plenty of computations in Macaulay 2. Pre-requisites: at the very minimum 817-818 and 901-902 (taking 902 simultaneously may also work). You will get the most out of this class if you have seen some commutative algebra before, however you should be able to get the gist of things without prior exposure as well. If in doubt regarding prerequisites, please come talk to me as individual circumstances may vary.
Math 937 Spring 2015
This is the second semester introduction to modern methods used in the study of partial differential equations. This follows the material covered in Math 941, but Math 941 is NOT a prerequisite for this class. The material covered will include: Sobolev Spaces, conservation laws, weak solutions and Gallerkin methods for the main PDEs, and nonlinear PDEs. The main problem for every PDE is to establish its wellposedness, i.e. existence, uniqueness, and continuity of solutions with respect to initial data. We will see that each of these questions can be answered in more than one way depending on the topology in which we are working. The prerequisites are Ordinary Differential Equations and some Real Analysis. Some introductory material from Measure Theory, Lebesgue Integration, and Functional Analysis will be offered throughout the course.
Math 958 Spring 2015
Math 958: Extremal Graph and Hypergraph Theory This course will focus on results and methods in extremal graph theory, extremal hypergraph theory, and Ramsey theory. Questions in this area include the following. How many edges can a graph have if it has no complete subgraph on 3 vertices? Among any 5 points in the plane in general position, must some 4 determine a convex quadrilateral? How many edges are needed to guarantee that a graph on 3k vertices contains k triangles with no vertices in common?
The course will begin with extremal results for forbidden subgraph problems, including Turán's Theorem and other classical results. We will then move on to the much more challenging realm of extremal questions for hypergraphs.
We will also discuss results in Ramsey theory, as well as the related van der Waerden and Hales-Jewett theorems. We will touch on randomized versions of these questions: for example, when is it true that with high probability, every 2-coloring of an Erdos-Renyi random graph contains a monochromatic triangle?
Along the way, we will study important methods in extremal graph theory, including stability, Szemerédi's regularity lemma and its applications, and the recently developed hypergraph container method.
In the second part of the course, students will present recent papers on the topics covered in the course.
Prerequisites: A previous course in discrete mathematics is preferred, and a fairly basic knowledge of probability is necessary, but mathematical maturity is most important. If you're at all interested in taking this course but aren't sure about your background, please talk to me or e-mail the instructor.
There is no formal textbook for this course. However, a good part of the material is contained in the following survey articles: D. Conlon, J. Fox, and B. Sudakov, Recent developments in graph Ramsey theory. Surveys in Combinatorics 2015, 49-118. P. Keevash, Hypergraph Turán problems. Surveys in Combinatorics 2011, 83-140. J. Komlós, A. Shokoufandeh, M. Simonovits, and E. Szemerédi, The regularity lemma and its applications in graph theory. Theoretical Aspects of Computer Science, 84-112. J. Komlós and M. Simonovits, Szemerédi's regularity lemma and its applications in graph theory. Combinatorics, Paul Erdös is Eighty, Vol. 2, 295-352.

Announcements for Fall 2015 courses

Math 487/887 Fall 2015
Math 487 / 887: Probability Theory Time: Tuesday-Thursday 2:00 - 3:15 Instructor: Steve Cohn Prerequisites: Math 314, Math 325 Text: Lectures in Elementary Probability Theory and Stochastic Processes, by Jean-Claude Falmagne
We'll start with the concepts and methods of elementary probability: Sample space and event, conditioning and independence, Bayes' theorem, sampling and counting. From there we'll move on to random variables and their accessories: Probability density, mass and distribution functions, joint and marginal distributions, expectation, variance and covariance, conditional expectation, moment-generating and characteristic functions. We'll use these tools to (among other things) establish two of the great classical results on the limiting distributions of sums of independent random variables -- the law of Large Numbers and the Central Limit Theorem. Time permitting, we will also cover the Borel-Cantelli lemmas and 0-1 laws. The last part of the course will be an introduction to stochastic processes, in particular, to random walks and Poisson processes.
We will also examine the measure-theoretic foundation of modern probability theory: Probability as measure, events as measurable subsets of a sample space, random variables as measurable functions, moments as Lebesgue integrals of random variables, etc. (Measure theory is not a prerequisite for the course.)
Probability theory, interesting and beautiful in its own right, should also appeal to students of analysis, applied or discrete mathematics, physics, biology, computing and finance. The course provides a good foundation for Math 489/ 889 (Stochastic Processes).
Math 856 Fall 2015
This fall we will be offering a course on the elements of smooth manifolds. When it last ran (fall, 2013) it met at MWF 10:30-11:20; I do not know yet when it will be scheduled for this fall.
The goal of the course is to provide a survey of the fundamental concepts of differential topology. Essentially, differential topology adds the methods of calculus - differentiability, tangency, and integration - to the study of topological spaces. The collection of spaces for which such an addition makes the most sense are called manifolds. Quoting the first words of the text:
"Manifolds are everywhere. These generalizations of curves and surfaces to arbitrarily many dimensions provide the mathematical context for understanding `space' in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and outside of pure mathematics they are becoming increasingly important to scientists in such diverse fields as genetics, robotics, econometrics, computer graphics, biomedical imaging, and, of course, theoretical physics."
The overall goal of the course is to provide an introduction to the objects, concepts and some of the techniques of this field. The specific topics covered will depend partly on those attending. The first several chapters of Lee's "Introduction to Smooth Manifolds" (which will serve as the primary text) will be our guide in the beginning.
The prerequisites for 856 will be some knowledge of point-set topology, enough to be comfortable with topological spaces, open covers and compactness (Math 871 is more than sufficient) plus undergraduate-level linear algebra and multivariable calculus.
For a beginning student this course can serve as an alternative to Math 871 (point-set topology); students who have seen topology might choose to use Math 856/Math 872 as a year-long sequence.
Mark Brittenham
Math 918 Fall 2015
Prerequisite: (multi)-linear algebra Instructor: Thanh Vu MWF: 12:30-1:20
Course description: Our goal is to introduce a beautiful area of algebraic geometry that have a lot of applications, namely the secant varieties of Segre-Veronese varieties. Our main focus is in their defining equations. Though the title is very algebraic geometry, we don't require prior knowledge of algebraic geometry or commutative algebra and is suitable to students in either algebra, geometry, or discrete math. The first two weeks we introduce some general results about secant varieties. Third week, we will introduce the representation theory of general linear groups and symmetric groups which will then bring the algebraic geometry problems to a problem in multilinear algebra. So by week four, you can just work with vector spaces, matrices and graphs! Many problems can be stated as finding rank of certain explicitly description 0-1 matrices, linear relations among explicitly described vectors or some invariants of certain graphs.
Notes will be uploaded on my website for the course a week prior to lectures. People are encouraged to work on a group of 3-4 on finding equations of certain secant varieties of certain Segre-Veronese. If successful, this will turn to be a good research papers! We will use macaulay2, maple or sage to do computation. So if you have used some programming language, it will be useful for the course. Based on demand, I might hold a seminar accompanying the course.
Math 928 Fall 2015
Math 928--929, Functional Analysis, Fall 2015 INSTRUCTOR: David Pitts TIME: M-W-F 12:30--1:20, 110 Avery Hall TEXT: John Conway, "A course in functional analysis, 2nd ed.". Secondary sources (not required) are Edwards, "Functional Analysis: theory and applications" and Rudin, "Functional Analysis." PREREQUISITES: Math 921-2 is preferred, but concurrent enrollment in Math 921 should be fine. Some background with basic complex variables (e.g. Cauchy's integral formula) is helpful, but is not essential.
ABOUT FUNCTIONAL ANALYSIS: Functional analysis has its roots in solving systems of equations which have infinitely many equations and infinitely many variables. Such equations arise naturally in differential and integral equations. The subject of functional analysis is now vast and growing rapidly. It is a central subject in modern analysis, and its techniques are used throughout analysis. In addition to differential and integral equations, applications can be found in ergodic theory, harmonic analysis, operator theory, representation theory, and the theory of topological groups. More recently, there has been considerable interest in non-commutative geometry and topology.
Vector spaces of functions are usually infinite dimensional. Functional analysis provides techniques for the study of such spaces and linear transformations between them. So in some sense, functional analysis might be thought of as infinite dimensional linear algebra. The infinite nature of the objects under study means that it is necessary to use approximations to study them, which is where the analysis arises.
ABOUT THE COURSE: We will begin with a thorough study of Hilbert spaces, and then move to the study of compact operators on Hilbert space. We will see how to approximate compact operators with finite rank operators. We will prove the spectral theorem for compact operators (this is a theorem about diagonalization) and apply the theory to the solution of a differential equation.
Following a brief discussion of more general operators and Banach spaces, we will discuss the Hahn-Banach Theorem and its uses. This leads to the subject of convexity theory and duality for Banach spaces, which in turn leads to the theory of locally convex topological vector spaces. Locally convex spaces are fundamental to understanding distribution theory, which is used in a variety of contexts, particularly weak solutions to differential equations.
Math 928 finishes with a discussion and some applications of the principle of uniform boundedness.
Math 929 will discuss linear operators on Banach spaces and their spectra (the spectrum of a linear operator in finite dimensions is the set of eigenvalues), and we will then move back to a discussion of operators on Hilbert spaces, which culminates in the very powerful and elegant general spectral theorem for normal operators. Along the way we will develop the analytic functional calculus, which enables one to define functions of an operator A, e.g. exp(A) or sin(A).
The course sequence will include applications, which can be tailored to student interest. For example, I will include some material on distributions, which are extremely useful for differential equations; we will see how locally convex spaces can be used to give a firm foundation for understanding distributions.
Math 941 Fall 2015
Instructor: Petronela Radu, Avery Hall 239, 472-9130, Textbook: Partial Differential Equations by Lawrence Craig Evans (Graduate Studies in Mathematics), second edition
Course information: This graduate course offers an introduction to modern methods used in the study of partial differential equations. The material will deal with the main linear PDEs (the transport, Laplace, heat, and wave equations) and some of the nonlinear PDEs associated with them. We will also cover topics such as distributions, weak solutions, Sobolev spaces, and obtaining estimates of solutions to PDEs. The main problem for every PDE is to establish its wellposedness, i.e. existence, uniqueness, and continuity of solutions with respect to initial data; we will see that each of these questions can be answered in more than one way depending on the topology in which we are working.
The course will cover material from Chapters 1- 5 from the textbook and some additional topics (such as distributions).
The prerequisites are Ordinary Differential Equations and some Real Analysis. Some introductory material from Measure Theory, Lebesgue Integration, and Functional Analysis will be offered throughout the course.
Evaluation: The grade is computed with the following formula: 50% Homeworks (3–5 homeworks) 50% Final Exam
Additional references: 1. Partial Differential Equations by F. John 2. An Introduction to Partial Differential Equations by M. Renardy and R. Rogers 3. Lecture notes written by L. Tartar available at
Math 958 Fall 2015
Math 958: Codes and Curves, TR 9:30-10:45, Prerequisites: Math 817-818, Instructor: Judy Walker

Coding theory deals with the design and performance of error-correcting codes for reliable transmission over noisy channels. Algebraic geometry first made its way into the subject in about 1980, when a construction using algebraic curves was used to prove the existence of families of codes that were better than any previously known families. Since that time, so-called algebraic geometry codes have found other uses in coding theory, and other, very different, applications of algebraic geometry to various problems in coding theory have proven quite fruitful. This course will be centered on applications of algebraic geometry in coding theory. We will address the basics of linear codes and of algebraic geometry codes and then proceed to explore a variety of other areas within communications in which algebraic geometric techniques have been applied to get good results, such as convolutional codes, network coding, polar coding, MIMO interference, and compressed sensing.

This course should appeal to a broad group of students with interests in algebra or discrete math. No specific background in algebraic geometry or coding theory will be assumed. Students will also have the opportunity to tailor the course to their particular interests as the latter half of the course will incorporate student presentations of recent research papers.