Information
- Thursday 11:00am, Weber 223
- Organizers: Naomi Owens Fahrner and Scott Ziegler. Email us if you would like to give a talk!
- Seminar attendance sheet: PDF
- A useful collection of resources on giving mathematics talks.
Schedule
| Date | Speaker | Title | Advisor |
| January 30 | Colin Roberts | Riemannian Geometry for | Clayton Shonkwiler |
| February 6 | Johnathan Bush | Applications of Schur Polynomials to Barvinok-Novik Orbitopes | Henry Adams |
| February 13 | Andy Fry | Tropic(al) Thunder II: Electric Boogaloo: An Introduction: Tropical Geometry for Dummies | Renzo Cavalieri |
| February 20 | Joshua Mirth | A Study in Mediocrity | Henry Adams |
| February 27 | Kyle Salois | Lizards, Seagulls, and Turtles, oh my! | |
| March 5 | Emily Heavner | Introduction to Inverse Problems and Their Application in the Medical Field | Jennifer Mueller |
| March 12 | Justin O'Connor et al. | Sonya Kovalevsky Day | |
| March 19 | No Greenslopes | Spring Break | |
| March 26 | CSU Graduate Students | Lightning Research Talks | CANCELED |
| April 2 | Pat Rosse | Laplacian Eigenmaps for Time Series Analysis | Michael Kirby |
| April 9 | Naomi Owens Fahrner | Space Situational Awareness and Optimization | Margaret Cheney |
| April 16 | Harley Meade | Problems in Combinatorial Design Theory | |
| April 23 | Joshua Mirth | Algebraic Topology for | Henry Adams |
| April 30 | Elliot Krause | A quick introduction to elliptic curves and their applications | Jeff Achter |
| May 7 | Brian Collery | Understanding What You Mean by Sets | James Wilson |
Abstracts
January 30: Colin Roberts, Riemannian Geometry for Dummies Mathematicians with Little Background.
As graduate students, we have all worked with the calculus of flat spaces in Math 517 (whether we liked it or not!). There, we heavily abused the inner product structure in order to properly build a notion of a differential and haphazardly slid vectors around in space (without even mentioning so) to compute said differential. But, in a space that isn't flat, how do we do this? Initially, you may not care, but you do live on the surface of a ball!
Enter: Riemannian geometry. In the broadest generality, Riemannian geometry extends the inner product structure via the metric tensor g to manifolds whose geometry may not be flat, and whose topology may not be that of an open set of \(\mathbb{R}^n\). For free, one receives a unique notion of how to slide vectors around the manifold which, amazingly, gives one a lot of power. For example, we can compute the velocity and acceleration of a curve on a manifold or define physical/material operators such as the gradient, divergence, and Laplacian. Another great advantage is the ability to define a distance along a manifold (which someone doing some forms of machine learning may enjoy).
So, join me if you'd like to explore an intriguing generalization of Math 517. I'll try to include as many figures and as few convergent subsequences as possible (i.e., none)!
Prerequisites: A working knowledge of calculus in \(\mathbb{R}^n\) and basic knowledge of point set topology. Math 517 should be more than sufficient. A solid grasp of the basics finite dimensional linear algebra will be necessary. Differential geometry is much like parameterized linear algebra!
Helpful background: The more you know about smooth manifolds, the better; but, I will introduce the necessary background you will need. For me, Partial Differential Equations (PDEs) and physics are a large focus, so having a bit of background in this may be helpful, but is certainly not necessary. Some algebraic terminology around tensors will make certain sections easier to grasp.
February 6: Johnathan Bush, Applications of Schur Polynomials to Barvinok-Novik Orbitopes.
Let \(\mathrm{SM}_{2k}\colon S^1 \to\mathbb{R}^{2k}\) denote the map \(\theta \mapsto(\cos\theta,\sin\theta, \cos 3\theta, \sin3\theta, \dots, \cos(2k-1)\theta,\sin(2k-1)\theta)\). The convex hull of the image of \(\mathrm{SM}_{2k}\) is called the \(k\)-th Barvinok--Novik orbitope. These convex bodies are highly symmetric and their facial structure enjoys interesting combinatorial and geometric properties. On the other hand, a complete description of the faces of the Barvinok--Novik orbitopes is currently unknown for \(k>2\).
I will describe how the zero sets of a certain family of symmetric polynomials called Schur polynomials define (some, possibly all) faces of Barvinok--Novik orbitopes, suggesting a potentially new method to study the facial structure of these convex bodies.
February 13: Andy Fry, Tropic(al) Thunder II: Electric Boogaloo: An Introduction: Tropical Geometry for Dummies.
This is a retelling of a greenslopes talk I did a couple years ago, but now I actually know what I'm talking about. In this talk we will go over the basic definitions of tropical geometry and we will see what happens when algebraic geometry meets combinatorics! We will see that by tropicalizing algebraic curves we obtain combinatorial tools that help with certain computations.
February 20: Joshua Mirth, A Study in Mediocrity.
What does it mean to compute an average? As simple as the question sounds, it gives rise to some surprisingly rich geometric problems. We will start by trying to find the correct definition of the average of a collection of points, then we will explore Hermann Karcher's work on center of mass problems in non-Euclidean geometries and discuss some recent results on computing averages in the Wasserstein space of probability measures.
February 27: Kyle Salois, Lizards, Seagulls, and Turtles, oh my!
We define a \(k\)-lizard \(P\) to be a polygon whose sides are allowed to take one of \(k\) directions. A \(k\)-scale graph is any graph that can be formed as the intersection graph corresponding to the maximal convex \(k\)-lizards contained in some \(P\). We will examine some simple families of graphs that can be made in this construction, develop some intuition for whether a graph is \(k\)-scale or not, and then show that no class \(L_k\) of \(k\)-scale graphs is contained in any other class \(L_j\).
March 5: Emily Heavner, Introduction to Inverse Problems and Their Application in the Medical Field
Although inverse problems is not usually a field in mathematics that is formalized until the graduate level, many undergraduate students do work with inverse problems. In this talk, we will first introduce the concept of inverse problems by building intuition through applications and examples. After we establish what an inverse problem is, we will discuss the method of least squares and use it to solve a common inverse problem: given a blurry image, construct the original (unblurred) image. Finally, we will look at how inverse problems play a role in the medical field through data collection and medical imagining.
March 12: Justin O'Connor et al., Sonya Kovalevsky Day.
In honor of Sonya Kovalevsky Day, Greenslopes on March 12 will take place in the North Ballroom of the LSC from 10:20-11:10. Contact Justin O'Connor, Kelly Emmrich, or Harley Meade for more information.
March 19: Spring Break!
March 26: CSU Graduate Students, Lightning Research Talks.
Due to COVID-19, Recruitment Day was canceled. We will use this day to test online options for continuing Greenslopes throughout the Spring semester.
April 2: Pat Rosse, Laplacian Eigenmaps for Time Series Analysis.
With "Big Data" becoming more available in our day-to-day lives, it becomes necessary to make meaning of it. We seek to understand the structure of high-dimensional data in which we are unable to easily plot. What shape is it? What points are "related" to each other? One primary goal is to simplify our understanding of the data both numerically and visually. First introduced by M. Belkin, and P. Niyogi in 2002, Laplacian Eigenmaps (LE) is a non-linear dimensional reduction tool that relies on the basic assumption that the raw data lies in a low-dimensional manifold in a high-dimensional space. Once constructed, the graph Laplacian is used to compute a low-dimensional representation of the data set that optimally preserves local neighborhood information. In this talk, we present a detailed analysis of the method, the optimization problem it solves, and we put it to work on various time series data sets.
April 9: Naomi Owens Fahrner, Space Situational Awareness and Optimization.
This talk will be an introduction to Space Situational Awareness (SSA) using the greedy optimization algorithm. Space Situational Awareness is the process of tracking objects in orbit. Additionally, SSA aims to predict the future location of these objects. We call these objects resident space objects (RSO). Resident space objects consist of active and inactive satellites and space debris. We will look at the greedy algorithm as a tool to optimize the total number of objects detected at a given time.
April 16: Harley Meade, Problems in Combinatorial Design Theory.
In this talk, I will give an introduction to combinatorial design theory and how it connects to geometries, as well as connections to frame theory. This talk will also explore some open or recently resolved problems in combinatorial design theory such as the existence of Steiner \(t\)-designs, Hall's conjecture, the Hadamard conjecture, and bounds on sizes of covering designs.
April 23: Joshua Mirth, Algebraic Topology for Chemists Applied Mathematicians.
Many of the physical properties of a molecule are determined by the structure of its energy landscape. Without assuming any background knowledge about homology or chemistry, I will discuss how tools from algebraic topology can be used to study the structure of energy landscapes. In particular, I will introduce the Morse complex, explain its relation to persistent homology, and show how a Künneth-type formula can be used to exactly compute the persistent homology of the n-alkane energy landscape. This is part of ongoing work with the NSF-DELTA project.
April 30: Elliot Krause, A quick introduction to elliptic curves and their applications.
The study of elliptic curves has been a hotly studied topic amongst number theorists and arithmetic geometers for a few decades now. But why? In this talk we'll take about 5 minutes to learn some key properties of elliptic curves, and then look at applications like integer factorization, cryptography and Fermat's last theorem.
May 7: Brian Collery, Understanding What You Mean by Sets.
Re-examining sets with the aim of doing mathematics on a computer and seeing some unexpected consequences.
Past Semesters
