All inquiries about our graduate program are warmly welcomed and answered daily:
grad@math.lsu.edu
All inquiries about our graduate program are warmly welcomed and answered daily:
grad@math.lsu.edu
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.
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.
This course provides practical training in the teaching of mathematics at the pre-calculus level, how to write mathematics for publication, and treats other issues relating to mathematical exposition. Communicating Mathematics I and II are designed to provide training in all aspects of communicating mathematics. Their overall goal is to teach the students how to successfully teach, write, and talk about mathematics to a wide variety of audiences. In particular, the students will receive training in teaching both pre-calculus and calculus courses. They will also receive training in issues that relate to the presentation of research results by a professional mathematician. Classes tend to be structured to maximize discussion of the relevant issues. In particular, each student presentation is analyzed and evaluated by the class.
This is a first course in algebraic number theory. We will cover classical material such as
We will treat measure theory and integration on measure spaces. The examples of the real line and of Euclidean space will be emphasized throughout. Topics will include the Hopf extension theorem, completion of the Borel measure space, Egoroff's theorem, Lusin's theorem, Lebesgue dominated convergence, Fatou's lemma, product measures, Fubini's theorem, absolute continuity, bounded variation, Vitali's covering theorem, Lebesgue differentiation theorems, and the Radon-Nykodim theorem. Applications to Lp and its dual, and the Riesz-Markov-Saks-Kakutani theorem may be presented if there is sufficient time.
This course is an introduction to the basic machinery of probability theory and leads to the study of stochastic processes. After reviewing measure theory and integration, we will study the fundamental mathematical structures and concepts used to model and understand random phenomena. These include random variables, distributions, sigma-algebras (which encode information), independence, and conditional expectations. Topics will include the laws of large numbers, central limit theorems, notions of convergence, infinitely divisible laws, and an initiation to stochastic processes.
In his Principia, Newton made a clear distinction between "practical" and "rational" mechanics, "[...] the science, expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever" (The Principia : Mathematical Principles of Natural Philosophy, translation by Cohen, Whitman & Budenz)
In this course, we will apply this approach to some of the most fundamental material behavior in continuum solid mechanics: Elasticity and Plasticity.
We will begin by a simple construction of the theory of non-linear elasticity. We will show how under proper hypothesis, it can be reduced to the simpler and more familiar linearized theory of elasticity, which we will study in detail. In particular, we will highlight how this construction leads to a variational model.
Then we will focus on plasticity, introduce some of the most classical models and present recent developments.
The course begins with an introduction to the theory and problems associated with Laplace's equation, the heat equation, and the wave equation. Then continues with a treatment of nonlinear first-order PDE including the Hamilton Jacobi equation and an introduction to conservation laws. The course finishes with an introduction to Sobolev spaces and the existence of weak solutions to second order elliptic equations.
What is the essence of the similarity between forests in a graph and linearly independent sets of columns in a matrix? Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph? Can one test in polynomial time whether a matrix is totally unimodular? Matroid theory examines and answers questions like these.
This course will provide a comprehensive introduction to the basic theory of matroids. This will include discussions of the basic definitions and examples, duality theory, and certain fundamental substructures of matroids. Particular emphasis will be given to matroids that arise from graphs and from matrices.
This course is a preparation course for the Core I examination in topology. It will cover general (point set) topology, the fundamental group, and covering spaces. A good supplementary reference is Chapter 1 of Algebraic Topology by Allen Hatcher, available online .
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.
This course provides practical training in the teaching of calculus, how to write mathematics for publication, how to give a mathematical talk, and treats other issues relating to mathematical exposition. Communicating Mathematics I and II are designed to provide training in all aspects of communicating mathematics. Their overall goal is to teach the students how to successfully teach, write, and talk about mathematics to a wide variety of audiences. In particular, the students will receive training in teaching both pre-calculus and calculus courses. They will also receive training in issues that relate to the presentation of research results by a professional mathematician. Classes tend to be structured to maximize discussion of the relevant issues. In particular, each student presentation is analyzed and evaluated by the class.
This will be a continuation of Professor Madden's course in Fall 2012. The text is A. Knapp's book: Basic Algebra. We will start with chapter VIII (Commutative rings and Modules) and chapter IX (Fields and Galois Theory). If time is left, we will do some topics from Knapp's book Advanced Algebra. This will depend on the interests of the students, but it could include Semisimple rings and modules, Representations of finite groups, Introduction to homological algebra.
Algebraic geometry began as the study of varieties - sets of solutions to systems of polynomial equations. The fundamental tool is the dictionary between the geometry of varieties and the algebra of their coordinate rings. Translating a problem from one perspective to the other often yields new techniques and intuition. Unfortunately, not every ring is the coordinate ring of a variety.
In this class, we will study schemes - a generalization of varieties which works with any commutative ring. The emphasis will be on the geometry and utility of schemes, as both can be lost in the technical details. Following the initial material, subsequent topics will be tailored to the skill and interests of the class. Homework will be assigned, with the possibility of student presentations in class.
The theory of D-modules provides an algebraic approach to the study of linear partial differential equations. It has important applications to many fields of mathematics, including representation theory, singularity theory, and the Langlands program. In this theory, one studies solutions to systems of partial differential equations in terms of modules over rings of differential operators. For example, when the system involves n variables and has polynomial coefficients, the appropriate ring of differential operators is the Weyl algebra: the noncommutative algebra generated by the linear functions x_{1},...,x_{n} and the corresponding partial derivatives ∂_{1},...,∂_{n}.
In the first part of the course, we will study the Weyl algebra and its modules, i.e. D-modules on C^{n}. We will then discuss D-modules on smooth complex algebraic varieties; here, it is necessary to introduce sheaf-theoretic methods. A main goal of the course is to discuss the Riemann-Hilbert correspondence--a vast generalization of Hilbert's 21st problem on the existence of linear differential equations with a prescribed monodromy group--in some special cases. Time permitting, we will discuss some applications to representation theory and to the geometric Langlands program.
This course is an introduction to elliptic curves and modular forms, which underlie many of the notable results in modern number theory, including Fermat's Last Theorem and Catalan's Conjecture. Topics will include: elliptic curves, elliptic functions, elliptic curves over finite fields, L-functions, modular forms, theta functions, Eisenstein series, Hecke operators, Shimura correspondence, arithmetic applications, integer partitions
The course will cover the qualitative theory of Ordinary Differential Equations. This includes the usual existence and uniqueness theorems, linear systems, stability theory, hyperbolic systems (the Grobman-Hartman Theorem), and if time permits, an introduction to Control Theory.
A standard first course in functional analysis. Topics include Banach spaces, Hilbert spaces, Banach algebras, topological vector spaces, spectral theory of operators and the study of the topology of the spaces of distributions.
Tempered distributions are an important class of generalized functions which are well-adapted for integral transformations of functions and distributions in several variables arising in mathematics, physics, statistics, and many other areas. The main topics of this course-seminar are the theory of tempered distributions, convolution operators, and the Fourier transform. We also plan to consider applications to several important classes of operators in Analysis. This course is a nice addition to 4997-3 in Fall 2013, however, it does not assume any familiarity with generalized functions.
The study of heterogeneous media has a distinguished history involving the fundamental contributions of J.C. Maxwell and A. Einstein. Over the last thirty years there has been an explosion of activity in applied science and mathematics delivering new methods for the design of heterogeneous media with novel properties. The course provides a self contained and hands on introduction to the theory as well as a guide to the current research literature useful for understanding the mathematics and physics of complex heterogeneous media. The course begins with an introduction of variational tools and asymptotic techniques necessary for characterizing macroscopic behavior of multi-scale 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 to recover fundamental theorems that characterize extreme field behavior inside complex materials in terms of the statistics of the random medium.
We will study stochastic processes, with special emphasis on Brownian motion. After an examination of the nature of Brownian motion paths we will study the beautiful relationship between stochastic processes and partial differential equations. Our exploration will include the Feynman-Kac formula, which stands at the juncture of probability theory, differential equations, and functional analysis. We will also study stochastic integrals and Ito's formula for stochastic differentials. At a more abstract level the course will include an introduction to abstract Wiener spaces and analysis on such spaces. The mathematics developed in this course finds a wide range of applications, ranging from finance to quantum physics.
The main theme of this course will be graph theory. We will discuss a wide range of topics, including spanning trees, eulerian trails, matching theory, connectivity, hamiltonian cycles, coloring, planarity, integer flows, surface embeddings, and graph minors. For more information see Math 7400.
This is an introduction to the theory of graph minors. We will discuss many problems of the following two types: determine all minor-minimal graphs that have a prescribed property; determine the structure of graphs that do not contain a specific graph as a minor. We will focus on connectivity and planarity.
This course will introduce the homology theory of topological spaces. To each space there is assigned a sequence of abelian groups, its homology groups. We will discuss methods for calculating these groups, and use them to prove results such as the Brouwer fixed-point theorem (in all dimensions), the Ham Sandwich Theorem, and generalizations of the Jordan curve theorem.
This course gives an introduction to the theory of manifolds. Topics to be covered include: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields, differential forms, Frobenius's theorem, integration on manifolds, Stokes's theorem, de Rham cohomology, Lie groups and Lie group actions.
The course is an introduction to advanced knot theory. Topics that are covered are: Basic properties of knots, links, braids and 3-manifolds, state and Seifert surfaces for links, quantum invariants, knot (co-) homology theories.
A TQFT is a functor from a cobordism category to a category of vector spaces (or more generally a category of modules). A cobordism category is (roughly speaking) a category whose objects are manifolds of a certain dimension ( say d) and whose morphisms are manifolds of dimension d+1 which mediate between two manifolds of dimension d. We will discuss TQFTs from an axiomatic point of view. We will discus various examples of TQFTs. We will discuss a method that constructs a TQFT starting with invariants of closed (d+1)-dimensional manifolds. We will focus mainly on the case d=2. Along the way, we will discuss invariants of 3-dimesnional manifolds from a skein theory point of view. Skeins are linearizations of the set of links in a given 3- manifold.