# Information

- Thursday 11:00am, Weber 223
- Organizers: Johnathan Bush and Colin Roberts. Email us if you would like to give a talk!
- Seminar attendance sheet: PDF
- A useful collection of resources on giving mathematics talks.
- Schedule of talks.

## Comment Forms

Greenslopes is all about having an opportunity to improve as communicators of mathematics. Improvement requires feedback. This semester we will be handing out anonymous feedback forms for the audience to provide constructive commentary on Greenslopes talks. You can see the form here.

For audience members:

- Please fill out the forms! This is your opportunity to help someone else improve.
- Think about what you circle. Do not select "excellent" unless the talk really was exceptionally good in that area. (This is rare!) When you circle "needs improvement," try to give comments explaining what could be better.
- Think before giving solutions. It's better to point out something that could be improved than to say how to improve it.
- Be as specific as possible. "You were hard to understand" is less helpful than "you occasionally trail off into a mumble at the end of a sentence, which is hard to hear."
- Give the feedback you would want to receive!

For speakers:

- These forms are optional. If you would like to opt-out for your talk, let the organizers know!

# Schedule

Date | Speaker | Title | Advisor |

January 31 | No Greenslopes | Safety training with Cori Wong | |

February 7 | Vlad Sworski | Problem 21: An Exploration of Ring Dials | TBD |

February 14 | Vance Blankers | Alice and Bob: a Love Story | Renzo Cavalieri |

February 21 | Jeremy Buss | The Domino Effect | |

February 28 | Amie Bray & Shannon Golden | The Volume Bound for Nested Torus Links / Algebraically Defined Graphs and Their Girth | TBD |

March 7 | Liam Coulter | Synthetic Aperture Source Localization | Margaret Cheney |

March 14 | Colin Roberts | Tensors and Exterior Algebra | Clayton Shonkwiler |

March 21 | No Greenslopes | Spring break! | |

March 28 | Various | Recruitment Day speed talks | |

April 4 | Brady Tyburski | Abstract Mathematics is Less Abstract Than You Think: The Metaphors that Make Up Mathematics | Cameron Byerley |

April 11 | Naomi Fahrner | A MUSIC Variant Utilizing Hankel Tensors and Sliding QR Windows | Margaret Cheney |

April 18 | Andy Fry | Chip-Firing and some 'applications' | Renzo Cavalieri |

April 25 | Johnathan Bush | An Introduction to Nonstandard Analysis | Henry Adams |

May 2 | Brittany Carr | Generic Support Vector Machines and Radon's Theorem | Henry Adams |

May 9 | Catalina Camacho | Invariants of genus 4 curves over positive characteristic fields | Rachel Pries |

# Abstracts

## February 7: Vlad Sworski, *Problem 21: An Exploration of Ring Dials*

What do you get when a bored undergraduate studying group theory over the summer stumbles upon a puzzle game involving orientations and a specific type of structure? You get the beginnings of an independent research question that will occupy most of his free time. Imagine a game in which there are four dials that can each be adjusted to one of four positions: up, down, left, or right. However, turning a dial turns its neighbors as well. Given a specific starting orientation, can we reach any other orientation? Can we find the best solution for a given orientation? We explore the answers to these questions and more using parts of Module Theory, Linear Algebra, and more.

## February 14: Vance Blankers, *Alice and Bob: a Love Story*

Alice and Bob are a famous couple in mathematics, sending many a secret message and resolving an untold number of game-theoretical problems. On this Valentine's Day Greenslopes, we'll take a look back at their early relationship: from Alice's relentless pursuit of Bob, to their brief period of dynamic on-again-off-again couplehood, to their "going Dutch" first date. We'll laugh, we'll love, and we might just learn a bit of mildly interesting math along the way.

## February 21: Jeremy Buss, *The Domino Effect*

A brief history of mathematics education in America, and why you should build adding machines out of dominoes.

## February 28: Amie Bray & Shannon Golden, *The Volume Bound for Nested Torus Links / Algebraically Defined Graphs and Their Girth*

Part 1 (Amie): A knot is an embedding of the circle in three-dimensional Euclidean space. The classification of knots is aided by the use of invariants, such as the Jones and the Alexander Polynomials. Following Thurston, a knot is called hyperbolic if the points not on it have the structure of a hyperbolic geometry. In this talk, we will look at a class of hyperbolic knots called twisted torus knots. Using combinatorial methods, we improve on the volume bound of Champanerkar et al 2012 for the associated nested torus links. Our method replaces the ideal Tetraheda considered by Champanerkar et al. by ideal Octahedra. This research was conducted with Dr. Rolland Trapp and funded both by NSF grant DMS-1461286 and California State University San Bernardino.

Part 2 (Shannon): An algebraically defined graph \(\Gamma_\mathcal{R}(f(x, y))\) is constructed using a specific ring \(\mathcal{R}\) and function \(f(x, y).\) These graphs are bipartite with each partite set consisting of all coordinate pairs in \(\mathcal{R}^2\) . We denote the vertices of the first partite set by \((a_1, a_2)\) and of the second by \([x_1, x_2]\). Two vertices \((a_1, a_2)\) and \([x_1, x_2]\) are adjacent when their coordinates satisfy the equation \(a_2 + x_2 = f(a_1, x_1)\). The focus of our study is the girth, or length of a shortest cycle, of these graphs. In this talk, we will use incidence geometry to motivate our study of algebraically defined graphs. We will also discuss the effect that changing the ring \(\mathcal{R}\) and function \(f(x, y)\) has on the girth of the algebraically defined graph \(\Gamma_\mathcal{R}(f(x, y))\), with particular emphasis on the case \(\mathcal{R} = \mathbb{R}\).

## March 7: Liam Coulter, *Synthetic Aperture Source Localization*

In recent years the subject of localizing sources of electromagnetic radiation has become a rich research topic in the mathematics and signal processing communities. In this talk I will present a method for localizing point sources in a target scene using a technique from Synthetic Aperture Radar (SAR) image processing. Specifically, the method makes use of Time Difference of Arrival (TDOA) data from two or more receivers, at least one of which is in motion, and a filtered backprojection operator, to reconstruct an image of the target scene.

## March 14: Colin Roberts, *Tensors and Exterior Algebra*

Tensors provide a natural generalization of coordinate independent transformations in linear algebra. In fact, many natural concepts in linear algebra such as endomorphisms, inner products, and determinants turn out to be specific kinds of tensors. In the talk, we will define tensors and the graded algebras in which they live. Also, we will take note of some special types of tensors, namely the symmetric and alternating tensors. Along the way we will stop and do some key examples and short proofs. Time permitting, I will wave my hands a bunch and introduce the notion of a tensor field on a manifold and the exterior algebra of differential forms.

## March 28 *Recruitment Day speed talks*

Speed talks by various graduate students in the department.

## April 4: Brady Tyburski, *Abstract Mathematics is Less Abstract Than You Think: The Metaphors that Make Up Mathematics*

Intuitive metaphors give us a means to make sense of mathematics by likening abstract mathematical concepts to our everyday experiences. But what if I told you that mathematics actually comes from these basic metaphors which are grounded in how our bodies interact with the physical world? In this expository talk, I'll share some of the cognitive science and linguistics research that backs up this claim and then we'll explore the conceptual metaphors that underlie various fields of mathematics. This will all be based on what I've learned from reading the book Where Mathematics Comes From by George Lakoff and Raphael Nuņez.

## April 11: Naomi Fahrner, *A MUSIC Variant Utilizing Hankel Tensors and Sliding QR Windows*

Though much work has been done in the area of Synthetic Aperture Radar (SAR) imaging, there is still much more to be done. Current SAR images contain a great deal of noise, yielding a less than desirable image clarity. Although many noise-reducing methods currently exist, there is much noise they are unable to eliminate. In an effort to maximally reduce noise in a SAR image, we create a new noise-reducing method by combining two current methods. These two methods are the MUSIC algorithm and the superset selection and pruning method. This new method gains benefits from both of these individual methods. The SAR data we will work with is the Fourier transform of a sparse scene with noise. The spatial invariance of the Fourier transform enables us to find the support of sparse vectors or matrices. This is done by creating Hankel tensors and comparing the range space of these Hankel tensors with the Fourier matrices from the SAR data. We use QR decomposition, along with the MUSIC algorithm, to create sliding windows for the pixels in an image. In this paper, we detail both the one dimensional and two dimensional versions of this new algorithm.

## April 18: Andy Fry, *Chip-Firing and some 'applications'*

Originally developed by a Bak, Tang and Wiesenfeld in 1987, and later generalized by Dhar in 1990, the abelian sandpile model (a form of chip-firing) is the first example of a dynamical system displaying self-organized criticality (SOC). In physics, SOC is a property of dynamical systems that are naturally attracted to their critical point. In this talk, I will give examples of chip-firing in combinatorics and also in algebraic (tropical) geometry.

## April 25: Johnathan Bush, *An Introduction to Nonstandard Analysis*

Potentially contrary to your intuition, there exists an ordered field extension of the real numbers that contains both infinite and infinitesimal numbers. We'll discuss a construction of this field extension, and we'll consider the resulting infinite and infinitesimal numbers and their relationship with the standard real numbers. Last, we'll use infinitesimals to do basic calculus without ever taking a limit.

## April 30: Brittany Carr, * Generic Support Vector Machines and Radon's Theorem*

A support vector machine, (SVM), is an algorithm which finds a hyperplane that optimally separates labeled data points in \(\mathbb{R}^n\) into positive and negative classes. The data points on the margin of this separating hyperplane are called support vectors. We study the possible configurations of support vectors for points in general position. In particular, we connect the possible configurations to Radon's theorem, which provides guarantees for when a set of points can be divided into two classes (positive and negative) whose convex hulls intersect. If the positive and negative support vectors in a generic SVM configuration are projected to the separating hyperplane, then these projected points will form a Radon configuration.

## May 9: Catalina Camacho, * Invariants of genus 4 curves over positive characteristic fields*

Curves over fields of positive characteristic (eg. finite fields) can be classify by some invariant, probably the most well known of which is the genus. Over these curves we can also define a special kind of point, called Cartier point. It happens that the number of these points on a curve depends on the invariants that we (that is, I) care about. I will define the a-number and the p-rank and talk about the case where the genus of the curve is 4. I will try to make this a more down to earth version of my prelim talk.