VIGRE@LSU |

## VIR (Vertically Integrated Research) Course Information

## (Past Information)

**Fall 2010**

- 4997-1
**Vertically Integrated Research: Equivariant cohomology---Algebra and the shape of space.**Prof. Sage. - Instructor: Prof. Sage. .
- Prerequisites: For graduate students: 7210 and 7510. For undergraduates: 4200 and 2085, or permission of the instructor.
- References: .

*Total Enrolled - 6, Graduate - 6, Undergraduate - 0*

On a rotating hollow sphere, there are exactly two points that don't move at all (we might call them the "north and south poles"). The sphere belongs to a large class of important topological spaces with two key features: (1) they have some (perhaps more than one) kind of rotational symmetry, and (2) they have finitely many points that don't move when the space is rotated. "Equivariant cohomology," the subject of this seminar, is a powerful tool for studying the geometry of such spaces. Using equivariant cohomology, we will explore remarkable connections to combinatorics and group theory.

This semester, we will discuss an important result from current research in representation theory called the geometric Satake isomorphism, restricting ourselves to the simplest possible case of the 2 by 2 invertible complex matrices GL2(C). Roughly speaking, this result states that representations of GL2(C) (i.e., group homomorphisms from GL2(C) to GLn(C) for any n) can be understood in terms of the equivariant cohomology of a topological space called the affine Grassmannian. The goal of this course is to make sense of this isomorphism as explicitly as possible and to come up with a new simple proof in this case. (No background in representation theory is assumed.)

- 4997-2
**Vertically Integrated Research: Lyapunov Functions, Stabilization, and Engineering Applications.**Prof. Malisoff. - Instructor: Prof. Malisoff with Prof. de Queiroz, and Prof. Wolenski .
- Prerequisite: For graduate students: 7320, 7386, or permission of the instructor. For undergraduates: 4027, 4340, or permission of the instructor.
- Text: Notes and recommended references provided by the instructors.
- References: M.S. de Queiroz, D.M. Dawson, S. Nagarkatti, and F. Zhang, Lyapunov-Based Control of Mechanical Systems. Control Engineering Series, Birkhauser, Cambridge, MA, 2000. ISBN: 0-8176-4086-X
- M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions. Communications and Control Engineering Series, Springer-Verlag London Ltd., London, UK, 2009. ISBN: 978-1-84882-534-5

*Total Enrolled - 3, Graduate - 3, Undergraduate - 0*

Mathematical control theory is one of the most central and fast growing areas of applied mathematics. This course will help prepare students for research at the interface of engineering and applied mathematics. The first part provides a self-contained introduction to the mathematics of control systems, focusing on feedback stabilization and Lyapunov functions. The second part will be a series of lectures by faculty from the LSU College of Engineering about open problems in control. The third part will explore ways of solving the problems. The only prerequisite is a graduate or advanced undergraduate course on the theory of differential equations. Students from engineering or mathematics are encouraged to enroll.

- 4997-3
**Vertically Integrated Research: Cluster Algebras.**Prof. Yakimov and Dr. Muller.
- Instructor: Prof. Yakimov and Dr. Muller.
- Prerequisite:
- References:

*Total Enrolled - 11, Graduate - 6, Undergraduate - 5*

Cluster Algebras are a topic of great interest in current mathematics. They were defined by Sergey Fomin and Andrei Zelevinsky in 2001 in relation to problems in combinatorics and Lie groups. Only a few years later they started playing a key role in a number of developments in representation theory, topology, combinatorics and algebraic geometry.

The beauty of the subject is that a great deal of it requires almost no prerequisites. Thus undergraduate students who register will be able to understand and lecture on a number of topics.

One of the main goals of the course is to go over applications and relations to various areas of mathematics. Graduate students specializing in representation theory, topology, combinatorics and algebraic geometry will see relations to each of these areas and will be asked to make presentations on their area of expertise.

- 4997-4
**Vertically Integrated Research: The colored Jones polynomial.**Profs. Dasbach, Stoltzfus, and Russell. - Instructor: Prof. Dasbach. .
- Prerequisite: Undergraduate Topology or permission of instructor.
- References: .

*Total Enrolled - 13, Graduate - 13, Undergraduate - 0*

The colored Jones polynomial is one of the more mysterious objects in knot theory. We will start with various definitions of it and will try to develop some of its properties. The methods will be elementary.

**Spring 2010**

**MATH 4997-1: Vertically Integrated Research: Vertically integrated research: Equivariant cohomology---Algebra and the shape of space**.- Instructor:
**Prof. Sage**. - Prerequisites: For graduate students: 7210 and 7510. For undergraduates: 4200 and 2085, or permission of the instructor.
- References:

*Total Enrolled - 9, Graduate - 8, Undergraduate - 1*

On a rotating hollow sphere, there are exactly two points that don't move at all (we might call them the "north and south poles"). The sphere belongs to a large class of important topological spaces with two key features: (1) they have some (perhaps more than one) kind of rotational symmetry, and (2) they have finitely many points that don't move when the space is rotated. "Equivariant cohomology," the subject of this seminar, is a powerful tool for studying the geometry of such spaces. Using equivariant cohomology, we will explore remarkable connections to combinatorics and group theory.

This semester, we will discuss an important result from current research in representation theory called the geometric Satake isomorphism, restricting ourselves to the simplest possible case of the 2 by 2 invertible complex matrices GL2(**C**). Roughly speaking, this result states that representations of GL2(**C**) (i.e., group homomorphisms from GL2(**C**) to GLn(**C**) for any n) can be understood in terms of the equivariant cohomology of a topological space called the affine Grassmannian. The goal of this course is to make sense of this isomorphism as explicitly as possible and to come up with a new simple proof in this case. (No background in representation theory is assumed.)

**MATH 4997-2: Vertically Integrated Research: Combinatorics.**- Instructor:
**Profs. Ding and Bilinski.** - Prerequisite:

*Total Enrolled - 9, Graduate - 9, Undergraduate - 0*

The theme this semester will be graph minors (so basic graph theory will be required). In particular, we will work on special cases for excluding Petersen graph and/or excluding K6. We will also work on problems that are directly related to the current research of our faculty members in combinatorics.

**MATH 4997-3: Vertically Integrated Research: Dimer models and knot theory.**- Instructor:
**Profs. Dasbach, Stoltzfus, Russell** - Prerequisite: Math 2057 and Math 2085 for undergraduates.

*Total Enrolled - 9, Graduate - 8, Undergraduate - 1*

A dimer covering of a graph is a subset of edges that covers every vertex exactly once. Dimer models are important objects of study in statistical mechanics, probability theory and more recently also in topology. We will introduce the basic concepts and we will discuss open problems for dimer models of graphs on surfaces.

**MATH 4997-4: Vertically Integrated Research: Discontinuous Galerkin Methods.**Wednesday and Friday 9:30-11:00.- Instructor:
**Prof. Sung.**. - Prerequisite: Math 2057, a numerical analysis course, and an ability to program.

*Total Enrolled - 9, Graduate - 8, Undergraduate - 1*

Discontinuous Galerkin (DG) methods are numerical methods for solving differential equations. We will review DG methods developed in recent years and discuss open problems. Graduate students can develop DG methods for various applications. Undergraduate students with background in multivariable calculus, linear algebra and programming will have the opportunity to participate in computational projects.

**MATH 4997-5: Vertically Integrated Research: Group Representations and Physics.**- Instructor:
**Profs. Olafsson and He.**. - Prerequisites: The prerequisite for undergraduate students is Math 4032 and for graduate students is 7000 level analysis.

*Total Enrolled - 6, Graduate - 6, Undergraduate - 0*

In this course, we will discuss the connections between analysis, group theory and physics. The students will read articles and give presentations. The selection of topics will be discussed at the organizational meeting at the beginning of the semester. Topics can include, but not limited to, geometric quantization, uncertainty principle, lowest-energy representations, reflection positive and duality theory.

**Fall 2009**

**MATH 4999-1: Vertically Integrated Research: Physics and Group Representation.**- Instructor:
**Prof. He and Prof. Olafsson.**. - Prerequisite:
- References: Unitary Group Representations in Physics, Probability and Number theory by George Mackey, 1978; "Symmetric and Unitary Group Representations: I Duality Theory" R.W. Hasse and P. H. Butler, Journal of Physics; A: Math and Theoretical (17) 1984, 61-74; The Principles of Quantum Mechanics by P. Dirac, 1981; Dynamical Symmetries of Nuclear Collective Models, D. Rowe, Prog. Part. Nucl. Phys (37) 265-348, 1996.

*Total Enrolled - 8, Graduate - 7, Undergraduate - 1*

This VIR class will concentrate on the interplay between group actions, group representations and physics. Symmetries in the physical world are often described by abstract groups in mathematics. For example, electrons on an orbit observe a certain symmetry defined by permutation group. We will focus on finite groups and continuous groups like the symplectic groups. It will cover topics that are interesting for undergraduate students, like the groups of permutations, and how one can determine their representations. For graduate students then there will be more advance topics like bounded symmetric domains, Geometric quantization, unitary representations of the symplectic groups and why those are interesting for questions in physics.

We will start with some introductory topics in physics and representations of the symmetric group and then slowly move into unsolved problems in mathematical physics. The topics are relevant in fields like analysis, number theory, and physics.

**MATH 4999-3: Vertically Integrated Research: Visualization of knots on surfaces.**- Instructor:
**Prof. Dasbach**with Profs. Heather Russell and Neal Stoltzfus

*Total Enrolled - 8, Graduate - 6, Undergraduate - 2*

Over the last 20 years knot theory became one of the central areas in mathematics. One studies properties of knots, like the unknotting number which measures the easiest way to unknot a knot. Our interest will be in the surfaces on which knots project in some nice way, and what those projections tell us about the knot.

We will learn and apply methods in computer graphics, differential geometry, knot theory and other areas of mathematics. The course is intended for both undergraduate and graduate students.

**MATH 4999-4: Vertically Integrated Research: Equivariant cohomology: Algebra and the Shape of Space.**- Instructor:
**Prof. Sage.**. - Prerequisite:
- Text: .

*Total Enrolled - 8, Graduate - 5, Undergraduate - 3*

**MATH 4999-5: Vertically Integrated Research: Complex Materials and Fluids.**- Instructor:
**Prof. Lipton with S. Armstrong.**. - Prerequisite: Math 2057 for undergraduate students; Math 7311 for graduate students.
- Text: .

*Total Enrolled - 8, Graduate - 4, Undergraduate - 4*

The course provides a self contained and hands on introduction to the field of homogenization theory as well as a guide to the current research literature useful for understanding the mathematics and physics of complex heterogeneous media. The first part of the course introduces the variational tools and useful asymptotic techniques necessary for characterizing the macroscopic behavior of heterogeneous media. Next we explore methods for constructing solutions of field equations inside extreme microstructures such as the the space filling coated spheres construction of Hashin and Shtrikman and the confocal ellipsoid construction of Milton and Tartar. The third part of the course shows how to apply these tools and field constructions to recover new theorems that characterize extreme field behavior inside complex materials

in terms of the statistics of the random medium and the applied incident fields.

**Spring 2009**

**MATH 4999-1: Vertically Integrated Research: Mathematical Problems in Relativity Theory**- Instructor:
**Prof. Lawson**. - Prerequisite for graduate students who will enroll: Math 7311. (For undergraduates: Math 4031 would be nice, but Math 2057 and some linear algebra are sufficient.)
- References: 1.
*Special Relativity*(M.I.T. Introductory Physics Series) by A. P. French. Paperback: 304 pages Publisher: W. W. Norton; 1 edition (August 19, 1968), ISBN-10: 0393097935 or ISBN-13: 978-0393097931.

2.*Special Relativity*(Springer Undergraduate Mathematics Series) by N.M.J. Woodhouse. Paperback: 192 pages, Publisher: Springer, ISBN-10: 1852334266 or ISBN-13: 978-1852334260

*Total Enrolled - 5, Graduate - 5, Undergraduate - 0*

This is an LSU VIR course, or vertically integrated research course, for graduate and undergraduate students. It will be taught as Math. 4999-1, to allow both undergraduate and graduate students to participate for credit. The topic is the foundations of special relativity and the related mathematics, including algebra, analysis, and geometry. Topics to be covered include 4--dimensional space--time, reference frames and coordinates, Lorentz transformations, simultaneity, time dilation and length contraction, causality and prohibition of motion faster than light, and composition of velocities. Geometric ideas will be stressed, and there will be an attempt to make connections with hyperbolic geometry. Students will have the opportunity to discuss selected topics in smaller groups and then present them to the others. The course serves as an introduction to research. The expectation is therefore, that the smaller groups will discuss open problems and work on them as we get more familiar with the topic.

**MATH 4999-2: Vertically Integrated Research: Computing the Bandwidth of Graphs.**- Instructor:
**Prof. Ding**and**and Dr. Bilinski**. - Prerequisite: Math 2020.

*Total Enrolled - 7, Graduate - 7, Undergraduate - 0*

This is a 3-hour research course with graduate credit. Students will read research papers and solve open problems. More details on the topic, Computing the bandwidth of graphs, can be found in **Prof. Ding's website**.

**MATH 4999-3: Vertically Integrated Research: Differential geometry of curves and surfaces in space.**- Instructor:
**Prof. Dasbach**and**Prof. Stoltzfus**. - Prerequisite:

*Total Enrolled - 9, Graduate - 2, Undergraduate - 7*

We will discuss the differential geometry of curves and surfaces in space. The course will also incorporate the use of Mathematica in the computation and visualization. As applications we will offer projects in Computer Graphics and 3-dimensional positioning systems (location detection in space with the help of two cameras).

Project: Each student would complete a project (possibly in groups) and make a presentation on a geometric topic.

**MATH 4999-4: Vertically Integrated Research: Equivariant cohomology: Algebra and the Shape of Space.**- Instructor:
**Prof. Sage**. - Prerequisite: For graduate students: 7210 and 7510. For undergraduates: 4200 and 2085, or permission of the instructor.

*Total Enrolled - 8, Graduate - 7, Undergraduate - 1*

On a rotating hollow sphere, there are exactly two points that don't move at all (we might call them the "north and south poles"). The sphere belongs to a large class of important topological spaces with two key features: (1) they have some (perhaps more than one) kind of rotational symmetry, and (2) they have finitely many points that don't move when the space is rotated. "Equivariant cohomology," the subject of this seminar, is a powerful tool for studying the geometry of such spaces. Using equivariant cohomology, we will explore remarkable connections to combinatorics and group theory.

**MATH 4999-5: Vertically Integrated Research: Complex Materials and Fluids.**- Instructor:
**Prof. Shipman**. - Prerequisite: 2057 for undergraduate students; 7311 for graduate students.

*Total Enrolled - 11, Graduate - 3, Undergraduate - 8*

This seminar will investigate the dynamical behavior of complex materials and fluids and the behavior of fields that they host. One of the topics will be the interaction of electromagnetic waves with composite materials consisting of natural and artificial components. This includes the creation and characterization of meta-materials and the phenomena of resonance. The mathematics will involve a wide range of topics in analysis and partial differential equations, including homogenization, scattering theory, special functions, and integral equations. For more information, please see Professor Shipmans' website.

### Fall 2008

**Math 4999-3:**Computing bandwidth of graphs.**Instructors:**M. Bilinski (VIGRE postdoc) and G. Ding.

*Total Enrolled - 7, Graduate - 5, Undergraduate - 2*