Alejandro Galvan
Drawing (some) Infinite Graphs
|
|
The free group \(F\langle A\rangle\) over a set \(A\) is the group generated by \(A\) and such that elements of \(A\) are not related to each other. Any group can be expressed as a quotient of a free group; for this reason, understanding the subgroups of \(F\langle A\rangle\) is a fundamental part of group theory. These subgroups can be represented as a certain type of directed graphs, called "Stallings automata". These graphical tools allow us to operate with subgroups and study their properties, as well as tackle some questions in algorithmic group theory such as the Membership Problem (given a subgroup and an element, does that element belong to the subgroup?) or the Conjugacy Problem (given two subgroups, are they conjugate?).
|
|
Brian Mintz
How to Pick the Pope, or a Pizza -- The Mathematics of Democracy
|
|
Groups need to make decisions, and there are a wide variety of ways this can be done, each maximizing different notions of fairness. Social Choice Theory provides a mathematical framework to investigate these possibilities rigorously. Infamous for its many impossibility results, this topic reveals some fundamental limits to democracy. Beyond this, we'll discuss potential resolutions to these problems, as well as their real world implications. Come and see how mathematics, and voting, can make the world a better place!
|
|
Jonathan Lindbloom
Priorconditioned generalized Krylov subspaces for sparsity-promoting inversion: a bleeding edge hybrid projection technique
|
|
Hybrid projection methods are highly effective at efficiently solving large-scale inverse problems with \(\ell_2\)-norm ("smoothing") regularization penalties, requiring only a low-dimensional subspace to obtain the solution. However, attempts to adapt these methods for sparsity-promoting regularization penalties lead to high computational cost due to the requirement of a high-dimensional subspace to represent the solution. In this talk, we present a bleeding edge technique (so new that it is still under development) for efficiently solving sparsity-promoting inverse problems. The technique relies on preconditioned generalized Krylov subspaces and learns a natural low-dimensional subspace for representing sparse solutions. If you've attended my GSS talks in the past, you will see some familiar characters!
|
|
|
Brian Mintz
Math Department Check-in
|
|
Over the past few years, we've made a tradition of meeting to discuss our experiences at Dartmouth so we can brainstorm / share potential solutions. If there have been things that didn't work well for you, even small issues, this is a dedicated time to ask around and see how other people have handled this or organize for departmental change. And if everything is great, please come to share your secrets, and support your fellow grad students!
|
|
|
Matthew Ellison
Universal Quadratic Forms and the 15 Theorem
|
|
We will take a look at the representation problem for integral quadratic forms, following a 2014 lecture of John Conway at the Princeton Mathematics Colloquium. We'll touch on a bunch of different topics -- including genera, the p-adics, quadratic reciprocity, a computational mistake of Ramanujan, and the recent 15 and 290 theorems.
|
|
|
Rohan Kapoor
A computability-theoretic approach to the Kakeya conjecture
|
|
The Kakeya conjecture is an old open problem in geometric measure theory and harmonic analysis. The first half of my talk will introduce the problem and its history, as well as some current research in the field. The second half of my talk will introduce the idea of Kolmogorov complexity and give a new alternate proof of the Kakeya conjecture in 2 dimensions using these computability-theoretic techniques.
|
|
|
Mark Lovett
Zero-Sum Resource Games: An Influence Model
|
|
From the ecology of fishing to the economics of stock trading and the voting dynamics of politics, where there are resources there is competition; this is a fact of life. I am going to present how I am studying these games using a model of expected returns and player influence from my research and present examples from politics.
|
|