Fall 2018 Schedule

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Conformal Tori in R^3: What do flat tori look like?

Steve Trettel | October 10

• You can make a torus in R^4 by identifying the opposite sides of a square (or rectangle) - and the resulting conformal structures are the same as those of the familiar rotation tori in R^3. Abstractly you can also form a torus by identifying opposite sides of a hexagon (exercise: convince yourself of this before the talk if you haven't seen this before!). But can you actually do this in R^4?
  
  In this talk, we will discuss the work of Pinkall, who in 1985 showed that all Euclidean tori metrically embed in the 3-sphere. Stereographically projecting to R^3 then allows us to "see" what the different Euclidean structures on a torus look like folded up (more precisely, we will realize all conformal structure on the torus as embedded surfaces in R^3). I've coded up some cool animations to show you!
  
  For graduate students who would like a refresher on some of the background mathematics that will be used in this talk, I will give a preparatory talk in the graduate student topology seminar, Tuesday 1-2 pm in 4607B.
  
  We will cover the definitions of (G,X) structure, conformal structure, Teichmuller Space, Deformation Space, and Moduli Space. Then we will compute the Teichmuller space of the torus and show it can be identified with the hyperbolic plane. Time-permitting we will talk about the construction of the Hopf fibration and its relation to the complex projective line and quaternions.
  
  

One Relator Artin Groups and Magnus's Freiheitssatz

Gordon Kirby | October 24

• In this talk, we will discuss Magnus's Freiheitssatz or Freedom Theorem for groups with one defining relation and the proof technique. Then, we'll define and explore a combinatorially defined cell complex used to study Artin groups, in which each two cell corresponds to a group with one defining relation. In this context, we'll present another proof of Freiheitssatz and discuss its relevance to constructing arbitrarily long words that reduce to the identity.

Hocus Locus - The Algebraic Geometry of the Canfield Joint

Christian Bueno | October 31

• The configuration spaces of mechanical linkages can provide physical models of complicated topological spaces. Many of these arise from polynomial constraints meaning they are real algebraic varieties. Conversely, geometry/topology can shed light on certain aspects of robotics. In this talk, I will sketch a picture of these connections and then focus on a fairly novel mechanism called the Canfield joint and present some results from my internship at NASA regarding its configuration space and kinematics.

Algebraic and combinatorial methods for real root counting

Ashlee Kalauli | November 7

• In this talk, we’ll discuss some basic computational algebraic geometry and combinatorics that were used to determine the number of positive, real points in an algebraic variety (assuming there are finitely many). These methods were used to find the number of central configurations in various cases of the n-body problem in celestial mechanics and the n-vortex problem in fluid dynamics. This talk will be accessible to anyone with a basic algebra background.

The Gauge Group of An Elliptic Surface

Nadir Hajouji | November 21

• Elliptic curves are great. Algebraic geometers and number theorists have been well aware of this for some time now, and more recently, people in fields like cryptography and string theory have seen the light. In particular, F-theorists have a way of associating a physical theory to a family of elliptic curves where many of the interesting physical properties are encoded in the geometry of the family. The goal of my talk is to explain how an F-theorist determines the gauge group of the physical system from the geometry of the family of elliptic curves. To that end, I will start by talking about the McKay correspondence, which will allow us to associate Lie groups to certain singularities. I will then give a quick introduction to elliptic curves and elliptic surfaces, and explain how to determine the gauge group associated to a 1-parameter family of elliptic curves.

Mathematics of Origami: Exploring Flat Vertex Folds

Jeffrey Stopple | November 28

• Origami that folds flat in the plane is simpler than three-dimensional constructions. The crease lines come together in vertices; a starting point is to consider the case when all folds meet in a single vertex. We will explore necessary and sufficient conditions on the number of folds and the angles between them. This will be an activity (not lecture) that’s appropriate for any liberal arts math appreciation class.
  
  This is an introduction for Jason Orozco’s talk on Friday at 1. He will define an equivalence relation on n fold single vertex flat origami, introduce a moduli space, and try to count the equivalence classes.

Folded States Of Single-Vertex Origami and their Corresponding Closed Meanders

Jason Orozco | November 30 (Friday 1:00 pm in SH 6635)

• This talk will explore one of the many mathematical rabbit holes that follows from the question "how many ways can a given crease pattern fold flat?" Emphasis is placed on the use of various programming tools for mathematical research including visualization as well as computation.
  
  The first half covers the discovery of equivalence classes of single-vertex origamis that share the same set of folded states, accompanied by animated diagrams created entirely using javascript and svg. Despite having an uncountably infinite number of ways to place n creases meeting at a vertex, it turns out that the space of possibilities can be partitioned into finitely many classes, for which the set of folded states is a class invariant. The classes can be enumerated programmatically by generating representative elements called M-points. We will take a brief look at the fractal pattern that emerges when plotting the number of completions for a range of partial M-points using Google Sheets.
  
  Next, we will investigate the solution set of folded states for a given single-vertex crease pattern using a Java program that draws all possible diagrams corresponding to solutions, i.e. ways to stack the layers of paper without self-intersections. The program only has to check a small subset of permutations of the stack due to a theorem proved earlier this year that every folded state is equivalent to one using what is called a "meandric permutation". The proof only constructs one such meander for a given folded state, however counting all such meanders remains open. We will end with ideas that may lead to a counting strategy and look at a GeoGebra project that helps visualize this.

A Transition of Complex Hyperbolic Space

Steve Trettel | December 12

• This is practice for a talk I am giving in the University of Minnesota geometry seminar next week. I will talk about some of my work in transitional geometry, and in particular, discuss an interesting degeneration of complex hyperbolic space and its relationship to real projective geometry. This talk should be accessible to graduate students of all levels.

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