LSU
Mathematics

# VIGRE@LSU: VIR Courses

### Documents

 VIGRE@LSU

## VIR (Vertically Integrated Research) Course Information

The Vertically Integrated Research or VIR courses are a new research experience for graduate and undergraduate students. These are the main educational instrument of the VIGRE activities. These courses run as Math 4997 and are intended to provide opportunities for students to learn about mathematical research in a vertically integrated learning and research community. Undergraduate students, graduate students, post-doctoral researchers and faculty may work together as a unit to learn and create new mathematics. Possible formats include group reading and exposition, group research projects, and written and oral presentations. Undergraduate students may have a research capstone experience or write an honors thesis as part of this course.

## Fall 2014

• MATH 4997-2: Vertically Integrated Research: Curves & Surfaces in 3D
• Instructor: Profs. Dasbach and Stoltzfus
• Prerequisites: Math 2057, Math 2085 (or Math 2090) or equivalent, or permission of instructors.
• Text:
• We will study the mathematical development of the geometry and manipulation of curves and surfaces in 3D with a blend of computer graphics. The course will develop the mathematical classification of surfaces by genus. As an application we will study embedded surfaces whose boundary is a given knotted circle.

The computer representation of curves and surfaces in Mathematica (STL "Standard Tesselation Language" files) will be covered as well as options for scanning and printing of surfaces.

## Spring 2014

• MATH 4997-2: Vertically Integrated Research: The A-Polynomial and its Relatives
• Instructor: Profs. Dasbach and Stoltzfus, Dr. Kearney.
• Prerequisites: Topology I (MATH 7510) or equivalent, or permission of instructors.
• Text:
• The A-polynomial of a knot roughly measures the eigenvalues of those representations of the fundamental group of the torus tube about a knot which extend to the entire knot complement. We will study different known ways to effectively compute using the various presentations of fundamental groups of knot complements. We will also have a visiting expert in the field during the semester.

## Fall 2013

• MATH 4997-6: Vertically Integrated Research: Diagrammatic Algebra and Representation Theory
• Instructor: Profs. Sage and Achar.
• Prerequisites: Familiarity with basic group theory and linear algebra, such as from Math 4200 and Math 4153
• Text:
• Ordinary "algebra" is essentially a language for keeping track of compositions of functions.  (Operations like addition and multiplication are, after all, themselves functions, and a complicated expression involving many of those operations is really a composition.)   "Diagrammatic algebra" is a general term that refers to various ways of working with function-like objects that can compose not just sequentially, but in a 2-dimensional way.  Diagrammatic algebra has its origins in category theory and has connections to and applications in representation theory, topology, mathematical physics, and other areas. In this course, we will look at many concrete, down-to-earth examples, which will involve drawing lots of pictures. We will then concentrate on how diagrammatic algebra can be used in representation theory to study reflection groups and quantum groups.
• MATH 4997-7: Vertically Integrated Research: The Alexander Polynomial and its Relatives
• Instructor: Profs. Dasbach and Stoltzfus, Dr. Kearney.
• Prerequisites: ElementaryTopology (MATH 4039) or equivalent, or permission of instructors.
• Text:
• One of the most studied knot invariants is the Alexander polynomial. There are many different ways known to effectively compute it. Some are combinatorial in nature, some are more closely related to the topology of the knot complement. We will discuss the Alexander polynomial as well as some of its relatives, such as knot signatures, the A-polynomial and Heegaard Floer knot homology.

## Spring 2013

• MATH 4997-5: Vertically Integrated Research: Reflection Groups.
• Instructor: Profs. Sage and Achar.
• Prerequisites: Familiarity with basic group theory, such as from Math 4200
Reflection groups are a remarkable class of groups that benefit from a richly interwoven tapestry of combinatorial, geometric, and group-theoretic properties. They are connected to mathematical discoveries ranging from the very ancient (e.g., the classification of platonic solids) to the very modern (e.g, the Kazhdan-Lusztig conjectures on Lie algebra representations). In this seminar, we will study selected topics related to reflection groups. No prior familiarity with reflection groups is required.
• MATH 4997-6: Vertically Integrated Research: A Linear Algebra Based Approach to Quantum Computing and Quantum Information Theory
• Instructor: Prof. Lawson
• Prerequisites: A reasonably good background in linear algebra (e.g.Math 2085) should be considered minimal background for the course. Some very basic knowledge of quantum theory and elementary probability theory would also be helpful.
• Text: Quantum Computing, Nakahara and Ohmi, 2008, CRC Press
• Reference: Quantum Computing and Quantum Information, Nielsen and Chuang, 2000, Cambridge

This introductory course on the recently emerging topic of quantum computing and information theory will introduce students to major recent developments such as quantum encoding and cryptography, teleportation, error correction, and quantum computing. Basic concepts of quantum theory such as quantum states, qubits, entanglement, measurement, quantum gates etc. will be incorporated into the course. The mathematical content will center on a linear algebra approach to the subject through basic matrix theory (unitary and Hermitian matrices, positive and completely positive operators, Gram-Schmidt decomposition, etc.) together with some elementary probabilistic content.

• MATH 4997-7: Vertically Integrated Research: Algorithms and computations in knot theory.
• Instructor: Profs. Dasbach and Stoltzfus, Drs. Kearney and Tsvietkova.
• Prerequisites: Math 2057 (Calculus of Several Variables)

Description: We will explore computational methods in knot theory, particularly of hyperbolic knots, using open source tools: SnapPea, Snap and Bar Natan's Mathematica package KnotTheory.

We particularly welcome undergraduate participants for this VIR course as we will be developing geometric concepts of three-dimensional hyperbolic geometry related to the visualization and design of solid geometric objects which can now be printed on 3D printers.

• MATH 4997-8: Vertically Integrated Research: Representations of Algebras: Quivers
• Instructor: Profs. Yakimov and Muller
Quivers are simple diagrams which can encode linear algebraic data; specifically, vector spaces and linear maps between them. In this informal seminar aimed at advanced undergraduates, we will study quivers, and how to build moduli spaces which parametrize all possible choices of data. This seminar will expose students to a casual research atmosphere and teach them concepts on the cutting edge of modern mathematics. Out-of-class obligations will be minimal.

## Fall 2012

• MATH 4997-1: Vertically Integrated Research: Representations of Algebras
• Instructor: Profs. Yakimov and Muller
• Total Enrolled - 7

In recent years representation theory of algebras has played a key role in many developments in cluster algebras, mathematical physics, and noncommutative geometry. The goal of this seminar-type course is to learn the fundamental constructions in this field and then study applications in the above mentioned areas. The course will naturally be split into two parts, "building fundamentals" and "investigating applications". Students will have the opportunity to give lectures in both parts of the course. The first part will cover the general representation theoretic and homological background, in particular quivers with relations and the Auslander-Reiten quivers of artin algebras. The second part of the course will cover applications to categorification of cluster algebras, Jacobian algebras, and noncommutative projective geometry.
• MATH 4997-2: Vertically Integrated Research: Reflection Groups.
• Instructor: Profs Sage and Achar.
• Prerequisites: Familiarity with basic group theory, such as from Math 4200
• Total Enrolled - 13

Reflection groups are a remarkable class of groups that benefit from a richly interwoven tapestry of combinatorial, geometric, and group-theoretic properties. They are connected to mathematical discoveries ranging from the very ancient (e.g., the classification of platonic solids) to the very modern (e.g, the Kazhdan-Lusztig conjectures on Lie algebra representations). In this seminar, we will study selected topics related to reflection groups. No prior familiarity with reflection groups is required.
• MATH 4997-3: Vertically Integrated Research: Theory of distributions.
• Instructor: Profs. Rubin and Olafsson.
• Prerequisites: 2057 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.
• Total Enrolled - 11

This course is intended to provide opportunities for undergraduate and graduate students to work in a research community, learn and create new mathematics. Possible formats include group reading and exposition, research projects, written and oral presentations. The subject of the course is introduction to the theory of distributions (or the generalized functions). This theory, created by S.L. Sobolev and L. Schwartz, is fundamental in the 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 4997-4: Vertically Integrated Research: Algorithms and computations in knot theory.
• Instructor: Profs. Dasbach and Stoltzfus, Drs. Kearney and Tsvietkova.
• Prerequisites: Math 2057 (Calculus of Several Variables)
• Total Enrolled - 16

Description: We will explore computational methods in knot theory, particularly of hyperbolic knots, using open source tools: SnapPea, Snap and Bar Natan's Mathematica package KnotTheory.

We particularly welcome undergraduate participants for this VIR course as we will be developing geometric concepts of three-dimensional hyperbolic geometry related to the visualization and design of solid geometric objects which can now be printed on 3D printers.

• MATH 4997-5: Vertically Integrated Research: Resonance in wave scattering.
• Instructor: Prof. Shipman.
• Prerequisites: : For undergraduate students: 1-dimensional differential and integral calculus. For graduate students: A complex variables course and graduate real analysis (Math 7311).
• Text: Notes by Maciej Zworski: Lectures on Scattering Resonances.

The phenomenon of resonance is familiar in popular and scientific tradition and commonly lies behind acoustic, electromagnetic, and mechanical processes and devices. We witness it in events such as the collapse of the Tacoma Narrows bridge, the shattering of a glass by acoustic resonance, anomalous absorption by the noble gases at specific energies, and super-sensitive frequency-dependence of light reflection from periodic surfaces. The topic of this course will be a mathematical theory that applies to a great variety of problems of resonance in wave scattering by objects in quantum and classical wave mechanics.

The primary literature reference will be the lecture notes on scattering resonances by Maciej Zworski. The classes will be run like a seminar, in which students and faculty will take turns presenting portions of the notes, related examples, and supporting material.

The mathematical theory of resonance involves sophisticated methods of complex variables and operator theory of differential equations. Even so, there is a rich array of simple and interesting models whose analysis is accessible to undergraduate students. The role of graduate students and faculty will be to learn and present the notes and supporting material. Undergraduate students will gain exposure to advanced mathematical techniques but will not be expected to grasp all the mathematics in the notes or lectures. Their role will be to work out and present models that illuminate specific resonant scattering phenomena.

## Spring 2012

• MATH 4997-1: Vertically Integrated Research: All you ever need to know about SL(2,R).
• Instructor: Dr. Harris and Profs. He and Olafsson
• Prerequisites:
• Text: Chapter V in Unitary Representations and Harmonic Analysis - An Introduction, M. Sugiura and An Introduction to Harmonic Analysis on Semisimple Lie groups by V. S. Varadarajan
Representations of semisimple Lie groups play a central role in several parts of mathematics and physics, including number theory, geometry, and quantum physics. In this course we will discuss several aspects of this broad theory but concentrate on the simples example SL(2,R). In this course our main goal is to prove the Plancherel theorem for SL(2,R). We will cover the theory of invariant eigendistributions as well as some results on orbital integrals.
• MATH 4997-2: Vertically Integrated Research: The fundamental lemma and affine Springer fibers.
• Instructor: Dr. Bremer, Prof. Achar and Prof. Sage
• Prerequisites: 4200 and 2085, or permission of the instructor.
• Text: None
We will study topics related to a major and very recent advance, the proof of the so-called Fundamental Lemma in the Langlands program, for which Ngo Bau Chao received the Fields Medal in 2010. Specifically, we will study the approach via affine Springer fibers developed in the work of Goresky-Kottwitz-MacPherson. Affine Springer fibers are geometric objects that can often be described very concretely--for instance, in one class of important examples, an affine Springer fiber is just a collection of 2-spheres touching at points. These spaces can be studied in a very combinatorial way that lends itself to hands-on computations. No prior knowledge is presumed beyond basic familiarity with abstract and linear algebra. Attendance in this seminar in Fall 2011 is not required.
• MATH 4997-6: Vertically Integrated Research: Cluster Algebras.
• Instructor: Dr. Muller, Prof. Yakimov and Prof. Dan-Cohen
• Prerequisites:
• Text:
Cluster Algebras are a new and exciting intersection between a wide array of mathematical fields. 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 the fundamentals require almost no prerequisites. We will begin with an introduction presented by students familiar with the topic, and there will be a heavy emphasis on examples and simplicity throughout. Thus new undergraduate students who register will be able to understand and lecture on a number of topics. Graduate students specializing in representation theory, topology, combinatorics and algebraic geometry will see relations and applications of cluster algebras. This semester of this course will begin to develop the categorical and homological aspects of category theory, as we see how elementary manipulations of diagrams lead to operations on certain categories. Concrete, computable examples will be emphasized throughout. The class will also transition from exposition of known material to research-oriented learning and exploration. Students will not be required to present or do homework, but participation is strongly encouraged to help follow the material.
• MATH 4997-4: Vertically Integrated Research: Knot invariants coming from Lie algebras.
• Instructor: Prof. Dasbach, Prof. Stoltzfus, and Dr. Kearney
• Prerequisites: Math 7510: Topology I
• Text: Research Literature

We will study knot invariants arising from Lie algebras including the proof of the MMR conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander-Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the "colored" Jones polynomial). encapsulated in the paper of Dror Bar-Natan and Stavros Garoufalidis

In addition, we will read the paper of Saleur & Kauffman relating the Lie algebra GL(1|1) to the Alexander Polynomial and other approaches to the MMR conjecture by XS Lin and A. Vaintrob.

• MATH 4997-5: Vertically Integrated Research: Domain Decomposition Methods.
• Instructor: Dr. Barker, Prof. Brenner and Dr. Park
• Prerequisites: Permission of Instructor
• Text: None required
Domain decomposition methods are numerical methods for partial differential equations that provide a natural approach to solving large scale problems on parallel computers. We will survey the field and discuss new developments. Implementation issues will also be addressed.

## Fall 2011

• MATH 4997-1: Vertically Integrated Research: Electromagnetic Waves in Heterogeneous Structures.
• Instructor: Dr. Welters, Prof. Lipton and Prof. Shipman
• Prerequisites:
• Audience: Graduate and undergraduate students; those interested in partial differential equations, spectral theory, and applications.

This course centers upon the partial differential equations of electrodynamics in periodic media. The spectral theory for fields in periodic structures is called Floquet theory. It is associated with the Floquet transform, which is the Fourier transform of the integer lattice subgroup of Euclidean space. Even the one-dimensional case is mathematically interesting, being naturally connected to differential-algebraic equations and the spectral theory thereof. An intriguing application is to "slow light", whose analysis requires the analytic perturbation theory of non-selfadjoint operators and in particular the perturbation of matrices with nontrivial Jordan blocks.

The course will be in the style of an interactive seminar, in which undergraduate and graduate students and faculty will present related topics or problems. Participants will learn about directions and open problems in current mathematical research.

• MATH 4997-2: Vertically Integrated Research: The fundamental lemma and affine Springer fibers.
• Instructor: Profs Sage and Achar.
• Prerequisites: Math 4200 and 2085, or permission of the instructor.

We will study topics related to a major and very recent advance, the proof of the so-called "Fundamental Lemma" in the Langlands program, for which Ngo Bau Chao received the Fields Medal in 2010. Specifically, we will study the approach via "affine Springer fibers" developed in the work of Goresky-Kottwitz-MacPherson. Affine Springer fibers are geometric objects that can often be described very concretely–for instance, in one class of important examples, an affine Springer fiber is just a collection of 2-spheres touching at points. These spaces can be studied in a very combinatorial way that lends itself to hands-on computations. No prior knowledge is presumed beyond basic familiarity with abstract and linear algebra.

• MATH 4997-3: Vertically Integrated Research: Cluster Algebras.
• Instructor: Prof. Yakimov and Dr. Muller.
• Prerequisites: Permission of Instructor.

Cluster Algebras is 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.

During the second semester of the course we will be able to fully explore relations with other subjects such as topology, combinatorics, and group theory. There will be more guest lectures by professors in those fields and more student presentations.

For all students who decide to join the class from the second semester, we will arrange for introductory lectures by current students and the instructors.

• MATH 4997-4: Vertically Integrated Research: Links on a Torus and Dimer Invariants
• Instructor: Profs. Dasbach and Stoltzfus.
• Prerequisites:

The knot theory VIR course will study the properties of links with diagrams which project to the torus fiber of the Hopf link. Specifically, we will study the dimer models for the zig-zag links introduced by Stienstra, their Kasteleyn matrices and work to develop related link invariants for these links.

• MATH 4997-5: Vertically Integrated Research: All you ever need to know about SL(2,R).
• Instructor: Profs. Harris, He and Olafsson.
• Prerequisites: For graduate students: 7311 or equivalent. For undergraduate students 1550 and 1552 or equivalent.

Representations of semisimple Lie groups are play a central role in several parts of mathematics and physics, including number theory, geometry, and quantum physics. In this course we will discuss several aspects of this broad theory but concentrate on the simples example SL(2,R). We discuss the classification of representations, harmonic analysis on the upper half plane, orbital integrals and other topics. We will partially follow the book by V. S. Varadarajan: An Introduction to Harmonic Analysis on Semisimple Lie groups.

## Spring 2011

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

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.)

Total Enrolled - 11, Graduate - 11, Undergraduate - 0

• 4997-2: Vertically Integrated Research: Mathematical Problems in Quantum Information Theory.
• Instructor: Prof. Lawson.
• Prerequisite: A reasonablly good background in linear algebra (e.g.Math 2085) and some basic knowledge of probability should be sufficient background for the course.
• Text: Quantum Information by Stephen Barnett, Oxford Press, 2009.

This introductory course on the recently emerging topic of quantum information theory will introduce students to major recent developments such as quantum cryptography, teleportation, error correction, and quantum computing. Basic concepts of quantum theory such as quantum states, entanglement, measurement, etc. will be incorporated into the course, as well as a few other basic background ideas such as elementary information theory. The mathematical content will center on matrix theory (unitary and Hermitian matrices, positive and completely positive operators, Gram-Schmidt decomposition, etc.) together with some probabilistic content.

Total Enrolled - 14, Graduate - 6, Undergraduate - 8

• 4997-3: Vertically Integrated Research: Cluster Algebras.
• Instructor: Prof. Yakimov and Dr. Muller.
• Prerequisite: Permission of Instructor.

Cluster Algebras is 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.

During the second semester of the course we will be able to fully explore relations with other subjects such as topology, combinatorics, and group theory. There will be more guest lectures by professors in those fields and more student presentations.

For all students who decide to join the class for the second semester, we will arrange for introductory lectures by current students and the instructors.

Total Enrolled - 11, Graduate - 10, Undergraduate - 1

• 4997-4: Vertically Integrated Research: Representations of knot groups.
• Instructor: Dasbach, Stoltzfus, and Dr. Russell.
• Prerequisite: a fundamental understanding of abstract algebra.

We will study representations of fundamental groups of knot complements and their combinatorics. Topics will be: The A-polynomial of knots, representations of knot groups into SU(n), combinatorial interpretations of certain knot group representations.

Total Enrolled - 12, Graduate - 10, Undergraduate - 2