UCSB Discrete Geometry & Combinatorics Seminar

Steve Trettel | October 11

This talk will be an introductory (aimed at first and second year graduate students!) talk on configuration spaces, starting from simple examples involving the circle and leading to a really neat description of the Poincare Homology sphere. No particular background is assumed, so come on by and check out the discrete geometry seminar! Configuration spaces are topological spaces meant to parameterize the ways that one object (say a collection of points) can occur within another object (like a manifold). We will look at the space of equilateral triangles inscribed in a circle, the space of chords in a disk, and the space of all triangles inscribed in a circle as some warm-ups, and then will consider the space of ways to tile the Euclidean plane with different patterns. This will lead us on an adventure to understand coset spaces of some Lie groups, and manifolds given by polyhedra with face identifications. With this new knowledge we will try to understand the space of ways to inscribe different platonic solids in a sphere, and construct from this Poincare's counter example to his original conjecture.

Gordon Kirby | October 18

This talk will aim to give an introduction to visualizing Coxeter groups of low rank. An abstract definition of Coxeter groups will be given as well as an explicit geometric realization in terms of a linear representation. The focus will be on explicitly computing examples and how to draw nice pictures to better understand the geometry in examples of low rank coxeter groups. No particular background is assumed first and second year graduate students are encouraged to attend

Hao Chen | October 25

In 1982, Maxwell defined the notion of "level" for Coxeter groups, and proved that those of level 2 correspond to infinite ball packings, generalizing the famous Apollonian packing. Motivated by studies on "limit roots", we revisit Maxwell's work and renew the definition of "level". The new definition is more geometric and yields many more infinite ball packings. Behind the scene, this work is actually a generalization of classical hyperbolic Coxeter group. More specifically, Coxeter groups of level 1 are hyperbolic Coxeter groups with finite fundamental domains, as considered by Bourbaki and Vinberg. Hyperbolic Coxeter groups of higher levels, on the contrary, have hyperideal fundamental domains. This talk is partially based on joint works with Labbe.

Steve Trettel | Nov 1 (1:30 pm start time)

Two years ago I began trying to understand the intrinsic geometry of positively and negatively curved space while reading Thurston's book "Three Dimensional Geometry and Topology." Many of you have heard me speak on this topic before - and I'm excited to share with you the next installment! For the past five months, I have been collaborating with graduate students in the Media Arts and Technology department to use VR technology to accurately visualize the three-sphere with its natural constant-curvature geometry. This talk will begin with a short discussion of positively curved geometry and how to render it in virtual reality - then we will walk over together to Elings hall and look at some regular 4-polytopes and the Hopf fibration inside of the Allosphere. The Allosphere is a ~30ft diameter spherical screen that groups of people can go inside of, don 3D glasses, and all experience three-dimensional graphics together! I am very excited about using virtual reality to explore three-dimensional geometry, and can't wait to give my first talk (of hopefully many) which will actually "take place" inside of the geometry we are discussing!

Jon McCammond | Nov 8

Every simple undirected graph corresponds to a real symmetric matrix called its adjacency matrix. Since such matrices have only real eigenvalues it makes sense to bound the eigenvalues above or below and then to investigate what this bound implies for the structure of the original graph.

In this talk I will discuss a famous result from 1970s algebraic graph theory due to Cameron, Goethals, Seidel and Schulte which completely classifies those graphs whose adjacency matrices have eigenvalues bounded below by -2. Their method of proof came as a complete surprise at the time.

Nadir Hajouji | Nov 29

Fun fact: over any finite field, there are exactly 7 isomorphism classes of del Pezzo surface of degree 5.

After explaining what those words mean (e.g. surfaces will mostly be finite collections of points), I will prove the result and do my best to show that it is indeed a fun and geometric fact.