Information

Thursday 11:00am, Weber 223
Organizers: Andy Fry and Pat Rosse.
Seminar attendance sheet: PDF
A useful collection of resources on giving mathematics talks.
Schedule of talks.

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Schedule

Date	Speaker	Title	Advisor(s)
August 29	No Greenslopes
September 5	Colin Roberts	Differential Forms and Stokes' Theorem in R^3	Clayton Shonkwiler
September 12	John Bush	Borsuk-Ulam Theorems in Various Guises and Generalizations	Henry Adams
September 19	Brittany Carr and Brenden Balch	Applications of Cellular Sheaf Theory to Skywave Signal Propagation and A Riemannian Distance on the space of Persistence Diagrams	Henry Adams and Clayton Shonkwiler
September 26	Naomi Fahrner	Summer at Ball Aerospace: Linear Arrays and GPS Anti-Jamming	Margaret Cheney
October 3	Shannon Golden and Justin O'Connor	Math Jam: A Paradigm of Inclusive Mathematics	Jess Ellis Hagman and Wolfgang Bangerth
October 10	Nate Mankovich	Geodesic Regression on the Grassmannian	Michael Kirby
October 17	Kelly Emmrich	Norm Euclidean Ideal Classes in Galois Cubic Fields	Rachel Pries
October 24	Lander Ver Hoef	Snarks! Two Infinite Families of Highly Symmetric Graphs	TBD
October 31	Dustin Story	Chicken Riddle Soup for the Soul	Alexander Hulpke
November 7	No Greenslopes	Math Day!
November 14	Graham Harper	You're still using matrices? An introduction to matrix-free supercomputing	Jiangguo (James) Liu
November 21	Scott Ziegler	Fermilab, CosmoSIS and Hamiltonian Monte Carlo (aka what I did last summer)	Jennifer Mueller
November 28	No Greenslopes	Thanksgiving break!
December 5	Dean Bisogno	Varieties, Algebraic Cycles, and Cohomology	Rachel Pries
December 12	Juanita Duque Rosero	Origami and Mathematics	Rachel Pries

Abstracts

Spetember 5 - Colin Roberts - Differential Forms and Stokes' Theorem in R^3

When learning multivariate calculus, we work with vectors and products of vectors to represent physical quantities. Of course, calculus then drives to see how these quantities can change from point to point. The main issue is that R^3 is especially nice and, due to this, it does not generalize without quite a bit more work. The goal is then to build a proper, and generalizable, notion of differential and integral calculus in R^3. Our key tool will be differential forms.

Think of forms as coordinate free measuring sticks that can be placed in not only R^3 but also any n-dimensional manifold. Forms have a rich algebraic structure and they can be differentiated and integrated. After defining forms somewhat heuristically, we will investigate the wedge product and exterior derivative. The exterior derivative will give us a cochain map on the de Rham cochain complex and we briefly mention the de Rham cohomology. Then, we can realize the different forms of the Fundamental Theorem of Calculus (divergence and Stokes’ theorem) under the guise of a more modern Stokes’ theorem formulated by Ellie Cartan.

September 12 - John Bush - Borsuk-Ulam Theorems in Various Guises and Generalizations

The Borsuk-Ulam theorem is a classic theorem in topology with many equivalent forms and a variety of interesting combinatorial and algebraic applications. It is typically stated as follows: given a continuous map $f$ from the $n$-sphere to $n$-dimensional euclidean space, some pair of antipodal points on the sphere must collide under $f$. Famously, the Borsuk-Ulam theorem implies the following: at any moment in time, there are two antipodal points on Earth with exactly the same temperature and pressure.

We'll consider a small subset of the theorem's implications and generalizations. Then, I'll describe and prove a new generalization of the theorem concerning maps from the $n$-sphere to higher dimensional space. No background in topology will be required.

September 19 - Brittany Carr - Applications of Cellular Sheaf Theory to Skywave Signal Propagation

Long distance communication is essential in today's interconnected world. When natural phenomenon interfere with advanced technological systems such as satellites and wired communication, skywave propagation can be used as an alternative method of communication. Skywave propagation relies on the ionosphere, a thick shell of ions which can reflect, refract, and absorb radio signals. But the ionosphere is a complex plasma which is extremely difficult to model.

Current methods use ionosodes to measure propagation through the ionosphere, which are expensive and sparsely distributed. Fortunately, amateur radio operators use skywave propagation to communicate across large distances on a regular basis. There is a large quantity of data and when used in the proper way, it can provide information about propagation conditions in the ionosphere.

The goal of this paper is to use a mathematical structure called a sheaf to combine the data in such a way that it provides information about the accuracy of current ionospheric models. This same structure could be used to construct a more accurate ionospheric model using the amateur radio data as its underlying information.

September 19 - Brenden Balch - A Riemannian Distance on the space of Persistence Diagrams

This talk will be about how we can define a Riemannian distance function for the space of persistence diagrams. We'll begin with an introduction to persistent homology, then later meet the Fisher metric. Following that, we'll see how this metric can provide us with a distance function on the space of diagrams. Other related distances will be introduced, time permitting.

September 26 - Naomi Fahrner - Summer at Ball Aerospace: Linear Arrays and GPS Anti-Jamming

I spent this summer at Ball Aerospace in Westminster, CO. I will discuss the various projects I worked on while there. My two main projects included angle of arrival algorithms for impinging signals on linear arrays and computationally cheaper algorithms for GPS anti-jamming. Most current methods for GPS anti-jamming (and beamforming in general) include taking an inverse of the covariance matrix, which is computationally expensive. Instead, I worked on a gradient descent algorithm outlined by Otis Frost in his paper 'An Algorithm for Linearly Constrained Adaptive Array Processing'. I compared this algorithm with other current methods in use in order to understand its viability.

October 3 - Shannon Golden and Justin O'Connor - Math Jam: A Paradigm of Inclusive Mathematics

Math Jam is the re-branding of a math outreach program run by graduate students and the Mathematics Department for the past 11 years. This name change represents our shifted focus towards increasing access to under-served populations, unlike typical STEM programs, which tend to focus on talented youth. Over the past year, through targeted recruiting efforts as well as a program in Sterling, CO, we have increased outreach and engagement towards our new vision: to spread the joy of mathematics through non-curricular activities. This talk aims to highlight successes and failures of our new paradigm, as well as discuss challenges faced by graduate student-run programs.

October 10 - Nate Mankovich - Geodesic Regression on the Grassmannian

Given a time series (or similar) dataset whose elements are naturally represented as matrices, we desire to find the geodesic which best fits these data. This is best understood as the cousin of linear regression. We will review the basics of linear regression, introduce the Grassmannian and set up an optimization problem for geodesic regression on the Grassmannian. This talk contains a few hypothetical real world examples and is mainly expository.

October 17 - Kelly Emmrich - Norm Euclidean Ideal Classes in Galois Cubic Fields

Lenstra introduced the notion of a norm-Euclidean ideal class as a generalization of norm-Euclideanity of a number field. He classified all quadratic number fields possessing a norm-Euclidean ideal class. We investigate the Galois cubic case. We show that up to discriminant 10^{11} at most two such number fields possess a nontrivial norm-Euclidean ideal class, and we conjecture no more exist. In an attempt to settle our conjecture, we prove explicit bounds on the first few non-residues of cubic characters under the generalized Riemann hypothesis.

October 24 - Lander Ver Hoef - Snarks! Two Infinite Families of Highly Symmetric Graphs

Snarks are a particular type of 3-regular graphs that serve as important sources of counterexamples in graph theory. In this talk, we will present an introduction to the relevant basic graph theory, discuss the history of snarks including some early infinite families of snarks, then introduce two new infinite families of snarks that can be drawn with rotational symmetry and have extremely large automorphism groups.

October 31 - Dustin Story - Chicken Riddle Soup for the (departed) Soul

Join us for a special Halloween edition of Greenslopes! Bring a team of 3-4 for a spooky riddle/puzzle solving competition with your emcee D-Sizzle! There will be real life prizes!

November 7 - Math Day!

November 14 - Graham Harper - You're Still Using Matrices? An Introduction to Matrix-Free Supercomputing

This nontechnical (and hopefully it will be slightly silly) presentation will give an overview of how computers think and provide frameworks for how we can think to make computer-based mathematics research more efficient. I'll start by talking about matrices and the progress that has been made with matrices from a computing standpoint. After that, I will talk about how parallel computing works and explain a little computer architecture. Lastly, I'll show why matrices aren't good enough still, and finally I will (hopefully) end with some fun demonstrations.

November 21 - Scott Ziegler - Fermilab, CosmoSIS and Hamiltonian Monte Carlo

This past summer I had the opportunity to work at Fermi National Accelerator Laboratory under the NSF Mathematical Sciences Graduate Internship (NSF-MSGI). My task was to investigate the viability of using Hamiltonian Monte Carlo to sample from posterior distributions arising from the solution of an inverse problem in the field of cosmology. In this talk we’ll take a look at some widely used Markov Chain Monte Carlo methods and see the ways in which Hamiltonian Monte Carlo differs from these techniques. I’ll also talk about the internship experience and provide information for those interested in NSF-MSGI or other internships.

November 28 - Thanksgiving!

December 5 - Dean Bisogno - Varieties, Algebraic Cycles, and Cohomology

Let C be a variety defined over a field K. Arithmetic geometers are interested in understanding how the field of definition of C changes the set of points on C. The section conjecture concerns the splitting of a group homomorphism between the fundamental group of C and the automorphism group of C. If the section conjecture were true it would imply the existence of an algorithm for finding points on varieties of genus bigger than 1. In this talk I will provide some background for this topic and conclude with some results I worked on over the summer at an AMS math research community.

December 12 - Juanita Duque Rosero - Origami and Mathematics

We will see how to trisect an angle by just folding paper! I will explain why is this possible and some other constructions that can be done with origami. We will also show that this method helps solving cubic equations. Come and fold some paper! If you get bored, here are the instructions on how to make a Christmas star: http://www.origami-instructions.com/origami-8-pointed-star.html

Past Semesters