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Student Topology SeminarFriday April 17, 2015JWB308 — 11:45 - 12:30 Property (T) Abstract: We will define property (T) and exhibit some non-examples. 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, 2015JWB308 — 11:45 - 12:30 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, 2015JWB308 — 11:45 - 12:30 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, 2015JWB308 — 11:45 - 12:30 The Banach-Tarski Paradox and Amenable Groups Abstract: The Banach-Tarski paradox is nowadays stated as "given a ball in 3-dimensional 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 Banach-Tarski paradox and introduce the idea of amenability and the von Neumann-Day problem. Friday March 6, 2015JWB308 — 11:45 - 12:30 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, 2015JWB308 — 11:45 - 12:30 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(g-1). Friday February 20, 2015JWB308 — 11:45 - 12:30 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, 2015JWB308 — 11:45 - 12:30 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, 2015JWB308 — 11:45 - 12:30 The Grigorchuk Group Abstract: The Grigorchuk Group, introduced in 1980, serves as a key counterexample to many would-be 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, 2015JWB308 — 11:45 - 12:30 Baumslag-Solitare Groups and Dehn Functions Abstract: First, we will discuss the notion of a Dehn function on a 2-complex. We will introduce the Baumslag-Solitare 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. |
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