Welcome to the webpage for Am I a Seminar?, a student-run (semi)-weekly informal seminar series at UVM featuring short talks by graduate and undergraduate students, snacks, and socializing. Am I a stands for All math is applied, the seminar being named in self-referential deference to the time-honored tradition of applying math to itself.
Participants brought problems relating to enumeration of certain families of trees, detecting structural properties of groups from associated graphs, and Erdős–Szekeres type problems. Many cookies were consumed.
The Hardy-Littlewood Maximal Function (Dylan Casey)
Maximal functions play a big role in analysis, specifically harmonic analysis. They are sublinear operators that provide information on the boundedness of certain L^p functions amongst other things. Simply, they tell us how bad a function can be. There are many, but the most important of these is the Hardy-Littlewood Maximal Function. We’ll explore its properties and many consequences.
Combinatorial Dominoes, (Sam Backlund)
This talk will introduce the combinatorial game of Domineering, in which
two players compete to arrange the most dominoes on a grid subject to a
simple constraint. Through the study of very small Domineering games we
will revisit the calculation of move advantage, a means of quantifying
ideal gameplay. Then we see how, on large Domineering boards, a game
naturally splits up into smaller subgames as moves are made. This
recursive style of board analysis should be interesting the next time
you sit down for a game.
When is a graph OK?, (Ada Morse)
Motivated by topological obstructions to the origami method of DNA nanostructure self-assembly design, we define and study origami knotted (OK) embedded graphs which cannot be used as targets for DNA origami self-assembly. In particular, we will present an intuitive, geometric algorithm whose input is an abstract 4-regular graph G and whose output is a surface-spatial embedding of G that is not OK. No prior knowledge of topological graph theory, knot theory, or the origami method is assumed.
Curvature Flow, (Ryan Gallagher)
At the heart of differential geometry lies the notion of curvature, a measure of how sharply a curve or surface turns at any given point. We will review the fundamentals of curvature for simple planar curves, and apply them to understanding the Curve-Shortening Flow, a well-studied continuous deformation of curves via their curvature vectors with tight connections to the heat equation. Our exploration of the Curve-Shortening Flow will wind through examples and visuals, culminating in the Gage-Hamilton and Grayson theorems classifying the Curve-Shortening Flow on simple planar curves. Time permitting, the extension of curvature and curvature flow to polygons will be discussed.
Hyperreal Numbers (Brandon Tries)
Selberg’s Sieve (Jack Dalton)
Expanding on the work of Eratosthenes and Legendre, Selberg developed a sieve method that achieves much more accurate estimates for sets of prime numbers. It was this sieve that Zhang modified as part of his prime gaps proof. We will discuss some of the basics ingredients, and work through a couple results.
Elliptic Curves, Trapdoors, and Public Key Cryptosystems (Garvin Gaston)
Attendees were invited to bring problems to share & discuss. Much of the seminar was spent on an amusing formulation of a special case of the Erdős Discrepancy Problem.
Nimbers and the Game of Nim (Sam Backlund)
In this talk, we will continue our analysis of Hackenbrush by adding a third type of edge. This addition, together with the machinery from previous sessions, will allow us to define the game of Nim and the very special Nimbers, a set of mathematical objects that behave like numbers in some ways but not in others. Nim and the Nimbers will be foundational in developing more advanced theorems in combinatorial game theory.
Linear Nonalgebra and the Theory of Ungraphs (Ada Morse)
This talk will begin with an exploration of what a theory of linear algebra without reference to algebraic operations might look like. Having found a reasonable contender, we will demonstrate its utility by constructing from it the subtheory of graphs without reference to vertex adjacency. Time permitting, we’ll look at a topological generalization of this subtheory.
Introduction to Fourier Series (Dylan Casey)
We’ll begin with Dirichlet’s Problem and derive the Fourier series for specific conditions. While giving us a unique solution to this problem, it raises more questions than it answers, some of which we will address.
Introduction to Zhang’s Bounded Prime Gaps Proof (Jack Dalton)
In this first talk, I’ll give some initial comments on bounded prime gaps, define some fun functions, and (time permitted) give an overview of the proof’s technique.
Elliptic Curves (Garvin Gaston)
An introduction to the arithmetic and geometry of elliptic curves, elliptic curves over C, and some discussion of how elliptic curves are being used in cryptography.
Beating Your Friends at Board Games: An Introduction to Combinatorial Game Theory (Sam Backlund)
In this first of a series of talks, we will introduce and play some simple combinatorial games and discuss the basics of their analysis. A combinatorial game is one in which play alternates over turns, and all information about the state of the game is public. For example, Tic-Tac-Toe and chess are combinatorial games while poker is not, as cards in a player’s hand cannot be seen by others. We will develop a working vocabulary and toolbox of theorems to determine when we can say a game is solved, what an ideal move is in a certain game state, whether there are advantages built in to the game’s design, etc, and in doing so develop potent mathematical methods to determine gaming strategies. Our goal throughout this series will be to heighten our appreciation of games of strategy rather than take the fun out of them. Come prepared to pit your wits against UVM’s mathematical community!
Curvature: The Fundamentals (Ryan Gallagher)
At the backbone of differential geometry sits the notion of curvature, a quantity measuring how much a curve or surface “curves” at a given point. In this talk, we will introduce the basics of planar differential geometry, including parameterizations of curves, various notions of the curvature vector, and the topology of curves. This talk aims to build a working understanding of curvature with the goal of later applying that knowledge to understanding the mean curvature flow, or curve-shortening flow, of planar curves.
Origami Knots in Graphs (Ada Morse)
Inspired by a topological obstruction to the origami method of DNA nanostructure self-assembly design, we define and study origami knots in Eulerian graphs embedded on orientable surfaces in 3-space. We present a complete characterization of origami knots in naturally-defined infinite families of triangular and rectangular toroidal grids, and discuss origami knots in composites of both types of grids embedded on higher-genus surfaces.