Pre-College
Undergraduate
Graduate
Postdoctoral
Administration
Travel
Documents
- Proposal
- Annual Report
VIGRE@LSU |
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.
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.
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.
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.
Total Enrolled - 7
Total Enrolled - 13
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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
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
For information on past VIR courses please click here.