Graduate Student Mini-Conference

SIAM and the Graduate Seminar series will host the following speakers on Thursday, April 9th in Thackeray 704. We have guest speaker Leo Rebholz from Clemson University who will talk about mathematics careers in industry:

1:00 - 1:20 pm - Scott Zimmerman - Geodesics in the Heisenberg Group

1:25 - 1:45 pm - Jake Mirra - Counterexample to Gromov's Conjecture

1:50 - 2:10 pm - Jon Holland - Intuitionistic logic, negative-dimensional tensors, and angular momentum

2:15 - 2:35 pm -Ian Martiny - A Lower Bound on Non-Trivial Cycle Length for the 3n+1 Problem

2:40 - 3:10 pm - Leo G. Rebholz - Clemson University

Speaker: Scott Zimmerman

Title: Geodesics in the Heisenberg Group

Abstract: In this talk, I will discuss the explicit formulation of geodesics in the Heisenberg group H^1. If time permits, I will discuss a new proof of the parametric equations for these geodesics in higher dimensional Heisenberg Groups (from a paper published with Dr. Hajlasz) and a proof of the analyticity of the distance function in the Heisenberg group.

Speaker: Jake Mirra

Title: Counterexample to Gromov's Conjecture

Abstract: The Heisenberg Group is the simplest non-trivial example of a Subriemannian Manifold.  Despite its simplicity, surprisingly little is known about the structure of this metric space.  I will provide an overview of my construction technique for a counterexample to a long-standing conjecture about the Heisenberg Group due to Gromov.  The proof is in progress and will rely on computer assistance. 

Speaker: Jon Holland

Title: Intuitionistic logic, negative-dimensional tensors, and angular momentum

Abstract:  In the 1970s, Roger Penrose proposed a kind of object that he called a "negative-dimensional tensor". These could be used to decompose the structure constants of the Lie algebra of angular momentum in three dimensions into a simple product of tensors of dimension -2. I shall give a construction of such negative-dimensional objects, that rests on an intuitionistic interpretation of the famous Euler "identity" 2+4+8+16+... = -2.

Speaker: Ian Martiny

Title: A Lower Bound on Non-Trivial Cycle Length for the 3n+1 Problem.

Abstract: The 3n+1 problem, or Collatz problem, is defined by a function on the positive integers: T(n) = n/2 if n is even and T(n) = (3n+1)/2 if n is odd.

There are two behaviors this function can exhibit: either iterations of T become unbounded, or the iterations of T will enter a cyclic pattern. The 3n+1 conjecture states the former never happens and that T always enters a cycle after reaching the number 1.We demonstrate a method for providing a minimum on the length of a non-trivial cycle (if one exists); using this method we prove if a cycle of T exists it must have at least 10,439,860,591 elements.