Student Topology Seminar
Friday April 17, 2015
JWB308 — 11:45  12:30
Radhika Gupta
Property (T)
Abstract:
We will define property (T) and exhibit some nonexamples. We will show that the integers and finite rank free groups do not have (T). Lastly, we will show that every countable group with property (T) is in fact finitely generated.
Friday April 10, 2015
JWB308 — 11:45  12:30
Matthew Smith
Piecewise Linear Morse Theory and Some Applications
Abstract:
I will give a brief introduction to piecewise linear morse theory, which is a generalization of morse theory applicable to locally finite simplicial complexes. We will discuss the Morse Lemma which enables one to reconstruct a space based only on local information gleaned from a Morse function. In the final part of this talk, we will apply this theory to prove that the commutator subgroup of the free group on two generators is a free group of infinite rank.
Friday April 3, 2015
JWB308 — 11:45  12:30
Hanna Astephan
SL(2,Z) as an Amalgam
Abstract:
First, we will introduce some basic notions of splittings of a group. Then we will use these concepts to prove that SL(2,Z) decomposes as a free product with amalgamation.
Friday March 13, 2015
JWB308 — 11:45  12:30
Dawei Wang
The BanachTarski Paradox and Amenable Groups
Abstract:
The BanachTarski paradox is nowadays stated as "given a ball in 3dimensional space, there is a way of decomposing it into finitely many disjoint pieces that can be rearranged to form two balls of the same size as the original one." It is believed to be the origin of amenable groups. In the talk, we will prove the BanachTarski paradox and introduce the idea of amenability and the von NeumannDay problem.
Friday March 6, 2015
JWB308 — 11:45  12:30
Radhika Gupta
Whitehead Graphs and Some Applications
Abstract:
In this talk, we will define the Whitehead graph for an element of F_n. We will use this graph to describe an algorithm to determine if the given word is contained in a proper free factor. This algorithm can be extended to determine if the given element is primitive.
Friday February 27, 2015
JWB308 — 11:45  12:30
Nick Cahill
The Hurwitz Automorphism Theorem
Abstract:
I use some elementary theory of discrete subgroups of PSL(2, R) to prove
that the classical result in Riemann surface theory that number of
isometries of a closed hyperbolic surface are bounded above by 84(g1).
Friday February 20, 2015
JWB308 — 11:45  12:30
Kishalaya Saha
Recurrent and Transient Random Walks
Abstract:
In this talk, we will prove the well known result that random walks on Z^n are recurrent for n=1,2 and transient for all n larger than 2.
Friday February 13, 2015
JWB308 — 11:45  12:30
James Farre
Comparison Geometries and Curvature
Abstract:
In this talk, we define notions of angle and curvature for an
arbitrary geodesic metric space X by comparing geodesic triangles in X to
triangles with analogous side lengths in euclidean space. We observe that
our new definitions coincide with the angle and curvature given by a
Riemannian metric on a smooth manifold. We give some examples of metric
spaces whose curvature is bounded above and see some consequences of
having bounded curvature.
Friday February 6, 2015
JWB308 — 11:45  12:30
Derrick Wigglesworth
The Grigorchuk Group
Abstract:
The Grigorchuk Group, introduced in 1980, serves as a key counterexample to many wouldbe theorems. In this talk, we will define the Grigorchuk group and prove that it is a finitely generated, infinite group in which every element has finite order. We will also mention several other interesting properties possesed by this group.
Friday January 30, 2015
JWB308 — 11:45  12:30
Morgan Cesa
BaumslagSolitare Groups and Dehn Functions
Abstract:
First, we will discuss the notion of a Dehn function on a 2complex. We will introduce the BaumslagSolitare groups, BS(m,n), and prove that they have an exponential Dehn function in the special case that m=1 and n is a prime.
