Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Dynamics, Geometry, & Groups Seminar

Dynamics, Geometry, & Groups - Merlin Incerti-Medici

Friday, March 6th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Merlin Incerti-Medici (Universität Zürich)

Title: Circumcenter extension maps for Hadamard manifolds.

Abstract: Given a geodesically complete CAT(-1) space, we can associate a boundary at infinity to it. This boundary is equipped with a geometric structure called cross ratio. While it was known for several decades that the boundary together with the cross ratio completely determines the interior space in some special cases, we recently learned that they always roughly determine the interior space. The key tool in this process is a construction called the circumcenter extension. In this talk, we survey known results about the circumcenter extension and show that its construction can be performed in a large class of CAT(0) spaces and still yields interesting results.

Dynamics, Geometry, & Groups - Sami Douba (McGill)

Friday, March 6th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Sami Douba (McGill)

Title: 2-Systems of arcs on spheres with prescribed endpoints.

Abstract: A 2-system of arcs on an n-punctured sphere S is a collection of (homotopy classes) of essential simple arcs on S joining punctures and pairwise intersecting at most twice. Bar-Natan proved that a 2-system of arcs on S beginning and ending at a fixed puncture has size at most n choose 3. In this talk, I will sketch a proof that the same holds for a 2-system of arcs on S joining a fixed pair of distinct puncture.

Dynamics, Geometry, & Groups - Abdullah Zafar (U of T)

Friday, February 28th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Abdullah Zafar (University of Toronto)

Title: Flows, Braids and Fractal Dynamics: Understanding Patterns of Play in Professional Football.

Abstract: Team movement analysis in football is critical in the preparation and evaluation of performance. Previous studies have analyzed team performance using reductionist approaches such as team centroid analysis with limited consideration of individual player movement and physical performance. In this talk we will look at how we can model the movement of a team as a flow field, derive metrics to quantify the team tempo, and then demonstrate the utility and importance of tempo to the physical training of players as well as team performance as a whole. We will then explore the tempo time series in more depth to better understand the dynamics of a football match: discovering the underlying Fractional Brownian Motion, the changing multi-fractal spectrum, and the difference between attractors around goal-scoring moments. We will then supplement the modelling of team movement with the use of algebraic braids by considering the space-time trajectories of the players. The prominence of different braid groups within the team and during different moments of the match demonstrate the varying movement strategies used by teams, and the computation of braid entropy is explored in order to determine the intra-team structure and coherence in team movement.

Dynamics, Geometry, & Groups - Francesco Cellarosi (Queen's)

Friday, February 14th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Francesco Cellarosi (Queen's University)

Title: Denjoy's non-transitive diffeomorphisms of the circle.

Abstract: H. Poincaré proved that an orientation-preserving homeomorphism f of the circle with irrational rotation number \alpha is semi-conjugate to the rotation by \alpha. Moreover, he proved that if the homeomorphism f is transitive, then the semi-conjugacy is a homeomorphism (and hence a conjugacy) and that if f is not transitive, then the semi-conjugacy is not invertible (and hence not a conjugacy). A. Denjoy constructed examples of non-transitive homeomorphisms (in fact, diffeomorphisms) of the circle with arbitrary irrational rotation number. I will review the history of the problem and explain Denjoy's construction.

Dynamics, Geometry, & Groups - Giusy Mazzone (Queen's University)

Friday, February 7th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Giusy Mazzone (Queen's University)

Title: On the stability of solutions to semilinear evolution equations.

Abstract: In this talk, we will discuss the stability and long-time behavior of solutions to nonlinear evolution equations in Banach spaces. We are particularly interested in semilinear parabolic systems of PDEs possessing a nontrivial manifold of equilibria, and whose linearization admits zero as a semi-simple eigenvalue. We will provide sufficient conditions that ensure asymptotic (exponential) stability of equilibria. We will also discuss an instability result for normally hyperbolic equilibria, and provide conditions for their attainability. Applications to fluid-solid interaction problems will be presented.

Dynamics, Geometry, & Groups - Rylee Lyman (Tufts University)

Friday, January 24th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Rylee Lyman (Tufts University)

Title: Train tracks and pseudo-Anosov braids in automorphisms of free products.

Abstract: The Nielsen–Thurston classification of surface homeomorphisms says that every homeomorphism of a surface either has a finite power isotopic to the identity, preserves the isotopy class of some essential multi-curve, or is isotopic to a pseudo-Anosov map, the most interesting kind. Bestvina and Handel introduced a similar classification for automorphisms of free groups. Here the analogue of a pseudo-Anosov homeomorphism is a train track map for an outer automorphism which is fully irreducible, a homotopy equivalence of a graph with extra structure. The analogy really is correct: pseudo-Anosov mapping classes of once-punctured surfaces induce fully irreducible outer automorphisms preserving a nontrivial conjugacy class and vice-versa. We discuss extensions of the train track theory to automorphisms of free products. Here the analogy is to mapping classes of punctured spheres. We show that fully irreducible automorphisms of free products of finite subgroups of SO(2) may be represented as pseudo-Anosov braids on orbifolds if and only if they preserve a non-peripheral conjugacy class.

Dynamics, Geometry, & Groups - Kasun Fernando (U of T)

Friday, January 17th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Kasun Fernando (University of Toronto)

Title: Edgeworth Expansions for (mostly) hyperbolic dynamical systems.

Abstract: Given a dynamical system which shows hyperbolicity on a large part of phase space, one would expect it to exhibit good statistical properties like rapid decay of correlations, the Central Limit Theorem (CLT), Large Deviation Principle and etc. In this talk, I will discuss sufficient conditions for such mostly hyperbolic dynamical systems to admit Edgeworth expansions in the CLT. Our focus is on systems that admit a Young tower with return times with an exponentially decaying tail. This is an on-going joint work with Françoise Pène.

Dynamics, Geometry, & Groups - Neil MacVicar (Queen's University)

Friday, November 22nd, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Neil MacVicar (Queen's University)

Title: Bratteli Diagrams and Cantor Minimal Systems.

Abstract: A Bratteli diagram is a kind of infinite graph for which a transformation can be defined on its path space. This talk will introduce the diagrams, their associated dynamical systems, and the relationship between these systems and systems described by a minimal homeomorphism acting on a Cantor space.

Dynamics, Geometry, & Groups - Elizabeth Field (UIUC)

Friday, November 15th, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Elizabeth Field (University of Illionois Urbana-Champagn)

Title: Trees, dendrites, and the Cannon-Thurston map.

Abstract: When 1 -> H -> G -> Q -> 1 is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an ``ending lamination'' on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(F_n), one can identify the resultant quotient space with a certain R-tree in the boundary of Culler-Vogtmann's Outer space.

Dynamics, Geometry, & Groups - Giulio Tiozzo (Queen's University)

Friday, November 1st, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Giulio Tiozzo (Queen's University)

Title: Entropy and drift for Gibbs measures on geometrically finite manifolds.

Abstract: The boundary of a simply connected, negatively curved manifold carries two natural types of measures: on one hand, Gibbs measures such as the Patterson-Sullivan measure and the SRB measure. On the other hand, harmonic measures arising from random walks. We prove that the absolute continuity between a harmonic measure and a Gibbs measure is equivalent to a relation between entropy, drift and critical exponent, extending the previous formulas of Guivarc’h, Ledrappier, and Blachere-Haissinsky-Mathieu. This shows that if the manifold (or more generally, a CAT(-1) space) is geometrically finite but not convex cocompact, harmonic measures are singular with respect to Gibbs measures.