Capstone Courses
MTH 396 Prerequisites:
MTH 396 Prerequisites: Completion of Tier I Writing Requirement, MTH 309, MTH 310, and MTH 320 (or the honors equivalents, or approval of department) and approval of the department. Typically the department expects a cumulative GPA of at least 2.0 and an average of at least 2.0 across MTH 309, MTH 310 and MTH 320. Note: Email notification will be given once your override has been issued.
MTH 496 Prerequisites:
Completion of Tier I Writing Requirement and approval of the department. Typically the department expects students to have completed MTH 309, MTH 310, and MTH 320 (or the honors equivalents) with cumulative GPA of at least 2.0 and an average of at least 2.0 across MTH 309, MTH 310 and MTH 320. Additional prerequisite courses may be required and can be found in the descriptions below. Note: Email notification will be given once your override has been issued.
FS25 MTH 496 Section 001
M/W/F 9:10a – 10:00a
Instructor – Dr Casim Abbas
Title: Introduction to Differential Geometry
This course is an introduction to classical differential geometry focusing on surfaces in three-dimensional space. Using methods of Multivariable Calculus and Linear Algebra we will develop the notions of Gaussian Curvature and Mean Curvature. The aim is to cover some global results such as the Gauss-Bonnet Theorem and Rigidity results for the sphere. We will study what these notions of curvature tell us about the surface.
Prereq: MTH 320 Recommended : MTH 421
FS25 MTH 496 Section 002
T/Th: 10:20a – 11:40a
Instructor – Dr Sasha Volberg
Title: Harmonic Analysis on Hyper Cube
Recently the learning problems took the center stage in area of theoretical computer science. An amazing and beautiful thing is that they are harmonic analysis problems at heart. The lectures concern some natural and elementary question of learning theory and the approach to learning via harmonic analysis. Suppose you wish to find a N× N matrix by asking this matrix question that it honestly answers. For example you can ask question “What is your (1,1) element?” Obviously you will need N2 many questions like that. But if one knows some information on Fourier side one can ask only C(ϵ,δ) log log N questions if they are carefully randomly chosen. So, for example if matrix is of the size 256 × 256, the brute force requires approximately 64,000 queries, but good algorithm will need much less questions. How many? We will see in the course.
Can you recognize a function if you know it only on a part of its domain of definition? Of course not, but you have more chances if you know something about its spectral concentration. This is what we will try to study for functions on Hamming cube. We describe the basics of analysis of Boolean functions and its applications of the mathematics of learning theory and another theory called social choice, a topic studied by economists, political scientists, mathematicians, and computer scientists. Harmonic analysis on Hamming cube has many unusual and interesting features, and, recently started to play a role as one of mathematical foundation of big data and in Quantum computing.
Prereq: MTH 320
SS26 MTH 496 Section 001
M/W/F 10:20a – 11:10a
Instructor – Dr Mark Iwen
Title: Fourier Analysis with Applications
The two primary topics of the course will be Fourier Series and the Fourier Transform. Fourier series are used to express a periodic function in terms of infinite sums of sines and cosines. We will investigate the convergence of Fourier series, which turns out to be a subtle matter. The Fourier transform represents a given function in terms of a continuum of “frequencies”, and has various applications in areas such as Partial Differential Equations, Mathematical Physics, Signal Processing and Medical Imaging. We will develop the theory of the Fourier transform and illustrate its use in the aforementioned areas.
Additional Topics we might cover include (time permitting): Discrete Fourier Transforms, the Fast Fourier Transform (FFT) algorithm, Orthogonal Functions, Abstract Inner Product Spaces, Distributions, and Time-Frequency Analysis.
Prerequisite: MTH 320
SS26 MTH 496 Section 002
M/W/F 11:30a – 12:20p
Instructor – Dr Eric Roon
Title: An Introduction to Quantum Spin Systems
The Nobel Laureate Phillip W. Anderson is often quoted as saying “More is Different,” after his Science article of the same title. What Anderson is referencing is that the behaviors of solitary quantum particles are essentially understood, but in aggregate, emergent and unexpected properties of the whole appear. This course will discuss the theory of quantum spin systems from a mathematical point of view. A quantum spin system models the dynamics of quantized degrees of freedom (like electron spin) in solid-like materials where the particle positions are fixed. In the past two decades tremendous progress has been made in understanding the emergent properties of these models. I plan to discuss two main techniques which have been incredibly useful. The first is the matrix product state theory which allows one to transfer spatial information about the spin system to a family of linear dynamics. The second are Lieb-Robinson bounds which show that there is a fundamental limit to the speed of information propagation and can be used to estimate how persistent entanglement is across large distances. In this course I hope to share with you the sometimes challenging but often rewarding area of quantum spin system dynamics and to teach you that more can, in fact, be different.
Prereq: MTH 320