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
The content of the high school Statistical Reasoning course is approximately the first 75% of the content of the AP Statistics course. The technology taught and utilized in the course is Microsoft Excel.
MATH 6303 will model high school teaching. Class time will be used to go over the lessons that have been written. Students will also “teach” some of these lessons to their peers and will prepare additional materials for the high school classroom such as activities, homework and assessments. Additionally, students will provide editorial and feedback comments on some of the available lessons. Students will also create a few lessons on topics which are in the AP Statistics course but not in the Statistical Reasoning course. Some students may also choose to work on converting Microsoft Excel lessons into lessons using Google sheets.
Students need to purchase an AP Statistics preparation book (around $15).
Tests of the type encountered in Praxis II will count for 50% of the grade. The remaining 50% will be awarded for in-class presentations, and preparation, editing and writing of lessons and assessments. The amount of assigned work will depend on the number of credits the student has registered for.
We will mostly follow the book by Folland. We will also hand out notes as needed. There are several other very good books on analysis and measure theory:
The material in this course builds toward open problems in mathematical physics centered around wave dynamics and scattering in electromagnetics and acoustics. The mathematical topics form a coherent body of theory and techniques. The main components are the partial differential equations (PDE) of electromagnetics and acoustics and other derived phenomena in complex media; integral equations and boundary-integral representations of solutions to PDE; Fourier analysis and the residue calculus for the study of scattering and resonance; spectral theory of differential and integral operators illustrated and motivated by examples; and asymptotic analysis. The second semester of the course will concentrate on specific problems that are motivated by modern scientific and mathematical research. A goal is that students will be able to understand open problems in this area and be equipped with the basic mathematical tools to begin trying to solve them.
(1) Hyperbolic, Parabolic, and Elliptic Partial Differential Equations
(2) Examples: The Wave Equation, The Heat Equation, The Laplace Equation
(3) Sobelev spaces and existence and uniqueness of solution.
(4) Spectral theory
(5) Banach fixed point theorem
(6) Subdifferentials and nonlinear semigroups
(7) Hamilton Jacobi Equations
Outline:
Some special topics for student projects:
The goal of this course is to give a gentle introduction to some main ideas of the geometric Langlands program, with an emphasis on concrete examples. We will also discuss the number-theoretic motivation for the geometric conjectures. We will concentrate on GL(n) and indeed primarily on GL(2).
In the linear theory, one studies problems of spectral theory of differential operators, such as Schrödinger operators with different kinds of potentials on different domains. This invokes delicate properties of entire and meromorphic functions, moment problems, asymptotic analysis, and of course linear algebra. New problems in inverse spectral theory continue to be interesting--where we seek to characterize the differential operators that possess certain spectral data. I also want to include some more specialized topics, such as linear systems governed by indefinite quadratic forms--this is a part of linear algebra that seems to be new to most people.
The overarching concepts for the nonlinear theory are flows of vector fields and dynamical systems. Upon that basis, one studies diverse phenomena such as bifurcations, separation of time scales, bursting (such as in neurobiology), hysteresis, stability, control systems, chaos, strange attractors, and the Noether theorem on conservation laws corresponding to continuous symmetries. We will introduce these notions and study some of them in depth.
A semester course cannot do any justice to all of these topics. My goal is twofold: (1) to present the foundational rigorous theory of ODEs, and (2) to introduce a broad variety of topics in ODEs that highlight what makes the field interesting.
Machine Learning:
Basic tools: curve fitting, probability, loss functions, inference.
Regression: linear models, bayesian viewpoint, evidence approximation.
Classification: discriminants, generative models, bayesian viewpoint.
Neural Networks: basic architecture, optimization issues, back propagation, regularization, bayesian viewpoint.
Linear Algebra:
Basic tools: norms, projectors, spectral theorem, singular value decomposition.
Direct methods: LU factorization, Cholesky, least squares problem, QR factorization.
Iterative methods: Jacobi, Richardson, Gauss-Seidel, successive over-relaxation, steepest descent, conjugate gradient.
Eigenvalue problems: power methods, Rayleigh quotient iteration, deflation, QR algorithm.