Whitehead Graphs and Irreducible Half-clean HNN Extensions
Abstract: We describe given an irreducible half-clean HNN extension H of a finitely generated free group, how we can construct an ascending irreducible half-clean HNN extension G such that H is a subgroup of G. We first introduce the Whitehead automorphisms on free groups, followed by stating Stalling's theorem (1996) on Whitehead graphs. Then, we describe how to construct G by Stalling's theorem. It serves as an attempt for solving the converse of Feighn-Handel's theorem that finitely generated [f.g.] ascending HNN extensions of free group are half-clean HNN extension of (f.g. free groups).
The Nielsen-Thurston Classification of Mapping Classes of Surfaces and Pretrees
Abstract: We are going to do a joint expository talk on the Nielsen-Thurston classification of mapping classes of surfaces and definitions of pretrees, to learn some useful definitions in preparation for Jean-Pierre's talk the following day.
Contribution: I described three kinds of mapping classes of the torus as an example. They are analogous to the Nielsen-Thurston classification of mapping classes of hyperbolic surfaces (shown in the picture on the right).
π_1-Injection and Relative Hyperbolicity
Abstract: We will discuss a set of criteria for a subcomplex Y in a compact complex X satisfying π_1-injectivity. Then, we will discuss under a small-cancellation condition, π_1 X is hyperbolic relative to π_1 Y. The talk is going to be filled with examples and hence serves as a gentle introduction to disk diagrams, small cancellation theory and relative hyperbolicity. This is a joint work with Daniel Wise.
One-relator Product of Cyclic Groups
Abstract: Howie (2002) proved that any one-relator product of 3 cyclic groups is never a trivial group, via constructing a non-trivial representation to SO(3). After stating the conjecture, we first focus on S^1-equivariant homotopy. Then, we describe a sketch of the proof. We may talk about some applications or generalizations if time allows.
Holomorphic vector bundles
Given on Nov 11, 2021
At Geometric Working Group Seminar at McGill University
Organised by Brent Pym
Abstract: In the talk, we will introduce holomorphic vector bundles. We will present three equivalent definitions, followed by the property that short exact sequences of holomorphic vector bundles do not necessarily split. If time permits, we will introduce the definition and a basic example of Picard groups.
An uncountable family of finitely generated residually finite groups
Abstract: In the talk, we will introduce a family of finitely generated residually finite groups. These groups are doubles of a rank-2 free group along an infinitely generated subgroup H. Varying H yields uncountably many groups up to isomorphism. This is a joint work with Daniel Wise.
Small cancellation theory and word problems
Abstract: We will introduce the small cancellation theory, which is a convenient tool to generate examples. We first examine the essence of the proof about the word problem for the fundamental group of the orientable surface of genus 2. We then define a metric small cancellation condition and sketch a proof for the associated word problems with disk diagrams. Finally, we state other related small cancellation conditions and explore some of their applications.
Given on Nov 22, 2019
At Graduate Student Seminar at McGill University
Organised by Benoît Corsini
Abstract: Starting with the well-known parametrization of Pythagorean Triples, I will present a survey in number theory about solving equations, including ABC theorem for polynomials (Mason–Stothers theorem), Chevalley–Warning theorem, Hensel's lemma (partial) and (if time allows) the number of zeroes of diagonal quadratic and cubic forms in local fields. Emphasis will be put on elegance of proofs instead of the degree of generalisation.