Greenslopes Seminar

[Department of Mathematics]
Thursdays 11AM Weber 202
Co-Organizers: Cory Previte and Hilary Smallwood



Schedule:

For the seminar attendance sheet, click here.
Abstracts appear below.

Date Speaker Title Advisor
Aug 29 Liz Lane-Harvard Finite Geometry in Graph Theory Tim Penttila
Sept 5 CANCELLED!!! Please attend Math Education Seminar Weber 237 (11:30 - 12:30pm)
Sept 12 Ryan Becker Playing Games on Graphs None at this time
Sept 19 Eric Miles Treat your quivers to vector spaces and Bridgeland stability conditions Renzo Cavalieri
Sept 26 Brent Davis Numerical Algebraic Geoemetry Dan Bates and Chris Peterson
Oct 3 Anne Ho Benford’s Law     (*Special time: 12pm) Rachel Pries
Oct 10 Cory Previte Cycle Graphs Make the World Go Round Chris Peterson and Alexander Hulpke
Oct 17 Josh Maglione Isomorphism Testing of Modules James Wilson
Oct 24 Rachel Neville Persistent Homology of the Logistic Map Patrick Shipman
Oct 31 Tim Hodges Syzygies with Linear Algebra Dan Bates
Nov 7 Matt Niemerg Algorithms in NAG: 1) Using Monodromy to Avoid High Precision and 2) Decoupling Highly Structured Polynomial Systems Dan Bates
Nov 14 Steve Ihde My Research: a cornucopia of algebraic geometry and numerical linear algebra Dan Bates
Nov 21 Dr. Guang Lin Job Searching & Working at Pacific Northwest National Lab Special talk sponsored by SIAM
Nov 28 Fall Recess
Dec 5 Farrah Sadre-Marandi Build a Virus James Liu and Simon Tavener
Dec 12 Graduating Students Speed Talks There is a nice list, but the margin is too small

Abstracts:


Aug 29 -- Liz Lane-Harvard
While it appears that the major tool for constructing graphs or obtaining graph theoretic properties comes from linear algebra, Finite Geometry, in fact, can be an even handier tool. I will discuss how I use Finite Geometry to construct strongly regular graphs.



Sept 12 -- Ryan Becker
In this talk, we introduce a solitaire chip-firing game played on a finite connected graph and use it to define the critical group of a graph. We then describe a method of constructing large families of graphs whose critical groups are cyclic. In some cases, we are able to find formulae for the number of spanning trees on these graphs, which exactly determines the critical group. We conclude by posing some open questions that naturally arise from the content of the talk.



Sept 19 -- Eric Miles
Quivers are fun and combinatorial gadgets (as we saw last week!). We'll start there and play around with their quiver algebras, quiver representations, and then introduce Bridgeland stability conditions into the mix. Please don't be intimidated by anything here - I'll build everything from scratch and hope to make it highly accessible!



Sept 26 -- Brent Davis
I will give a crash course on the theory behind numerical algebraic geometry. I will skim over fundamentals like homotopy continuation, endgames, dimension counting, genericity, randomization, slices, witness sets, trace test, monodromy, and numerical irreducible decomposition.



Oct 3 -- Anne Ho
What do population sizes, death rates, Fibonacci numbers, and iPhone passcodes have in common? They all follow the phenomenon known as Benford’s Law, which states that 1 occurs as the first-digit in data sets about 30.1% of the time, 2 occurs as the first-digit about 17.6% of the time, and so on. In this talk, I’ll give an overview to Benford’s Law and its applications. Then, I’ll briefly discuss the project I worked on over the summer at a Sage development workshop involving Benford’s Law and its ties to number theory.



Oct 10 -- Cory Previte
Cycle graphs are full of nice properties. They are Eulerian, Hamiltonian, 2-regular, 3-colorable, and symmetric. In this talk, I will show that these simple (pun intended) graphs can provide a wealth of interesting examples. In particular, I will build a simplicial complex on these graphs and discuss how the corresponding homology is anything but trivial. All background definitions from graph theory and computational topology will be given to make this talk accessible to a general audience.



Oct 17 -- Josh Maglione
Modules arise in many different contexts as we have seen in previous Greenslopes talks. A fundamental problem in (computational) representation theory is the ability to differentiate two modules. We look at the state of the art in isomorphism testing of (finitely generated) modules over finite dimensional F-algebras, for an arbitrary field F. I will briefly discuss the algorithm's predecessor, and outline how the current algorithm runs. At the heart of this algorithm are very deep algebraic results, so naturally, we turn to Jacobson for answers. [I will only assume basic algebraic definitions: abelian groups, fields, rings, vector spaces, and homomorphisms between them.]



Oct 24 -- Rachel Neville
Understanding the underlying structure of point cloud data is a problem that intersects combinatorics, topology and algebra. Persistent homology is a fairly recently developed tool used to “see” major features in this data. I will give a big picture intro and then will talk about a surprising pattern that arises when computing the persistent homology of a 1-dimensional chaotic system.



Oct 31 -- Tim Hodges
Given a module generated by polynomials $f_{1},...,f_{n}$ in $\mathbb{C}[x_{1},...,x_{m}]$ a syzygy is a n-tuple $(\alpha_{1},..,\alpha_{n})$ such that $\sum\limits_{i=1}^n \alpha_{i} \cdot f_{i} = 0$. Syzygies can be computed by Buchberger's algorithm for computing Groebner Bases. Unfortunately with time Groebner bases have been computationally impossible as the number of variables and number of polynomials increase. The aim of this research is to give a way to compute syzygies without the need for Groebner bases and possibly still give the same information as Groebner bases gives. The approach is to use the monomial structure of each polynomial in our generating set to build syzygies using nullspace computations.



Nov 7 -- Matt Niemerg
1) When solving polynomial systems with homotopy continuation, the fundamental numerical linear algebra computations become inaccurate when two paths are in close proximity. The current best defense against this ill-conditioning is the use of adaptive precision. While sufficiently high precision indeed overcomes any such loss of accuracy, high precision can be very expensive. In this article, we describe a simple heuristic rooted in monodromy that can be used to try to avoid the use of high precision.
2) An efficient technique for solving polynomial systems with a particular structure is presented. This structure is very specific but arises naturally when computing the critical points of a symmetric polynomial energy function. This novel numerical solution method is based on homotopy continuation and may apply more broadly than this specific setting. An illustrative example from magnetism is presented, along with some timing information and open problems.



Nov 14 -- Steve Ihde
We will walk through the brightly colored forest of preconditioning polynomials systems. You will get some festive treats like dual spaces, macaulay matrices, and h-bases. There will also be a bonecrushing mathup between multiplicity and the closedness subspace. This is a warm-up for my Prelim next month. I will try to treat everything carefully and make sure it is very understandable.



Nov 21 -- Dr. Guang Lin
Dr. Lin will give a brief presentation covering how to search for and apply to positions at PNNL. He will provide an overview of the PNNL’s mission, vision, and history, and describe the working environment, research areas, and abilities that PNNL scientists are looking for. The presentation will be followed by a question and answer period with Dr. Lin for those interested in pursuing a career at PNNL.



Dec 5 -- Farrah Sadre-Marandi
The complex arrangements of macromolecules in a virus shell, called the capsid, are marvels of molecular architecture. Specific requirements of each type of virus have resulted in a fascinating apparent diversity of organization and geometrical design. Nevertheless, there are certain common features and general principles of architecture that apply to all viruses. This talk will focus on virions with an icosahedral structure and its classification number, called the triangulation number. Given a hexagonal lattice, we can construct 3D models (build our own virus) by geometric folding. Lastly, even though the capsid shape of a large majority of viruses do conform to these structures, I will mention a few important exceptions, such as HIV-1, which still pose as problems for today’s mathematicians.











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