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Acylindrical Hyperbolicity, non-simplicity and SQ-universality of groups splitting over $\mathbb{Z}$
(De GruyterJournal of Group Theory, 2016-09-15)
We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\mathbb{Z}$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers ...
Bulk eigenvalue statistics for random regular graphs
(Institute of Mathematical StatisticsAnnals of Probability, 2016)
We consider the uniform random d-regular graph on N vertices, with d ∈ [N$^{\alpha}$,N$^{2/3−\alpha}$] for arbitrary α > 0. We prove that in the bulk of the spectrum the local eigenvalue correlation
functions and the ...
Uniform Bounds for Black--Scholes Implied Volatility
(Society for Industrial and Applied MathematicsSIAM Journal on Financial Mathematics, 2016-11-29)
In this note, Black--Scholes implied volatility is expressed in terms of various optimization problems. From these representations, upper and lower bounds are derived which hold uniformly across moneyness and call price. ...
On the number of non-zero elements of joint degree vectors
(Electronic Journal of CombinatoricsThe Electronic Journal of Combinatorics, 2017-03-31)
Joint degree vectors give the number of edges between vertices of degree i and degree j for 1 ≤ i ≤ j ≤ n-1 in an n-vertex graph. We find lower and upper bounds for the maximum number of nonzero elements in a joint degree ...
A Liouville hyperbolic souvlaki
(Institute of Mathematical StatisticsElectronic Journal of Probability, 2017-04-25)
We construct a transient bounded-degree graph no transient subgraph of which embeds in any surface of finite genus.
Moreover, we construct a transient, Liouville, bounded-degree, Gromov–hyperbolic graph with trivial ...
Markov numbers and Lagrangian cell complexes in the complex projective plane
(Mathematical sciences publishersGEOMETRY & TOPOLOGY, 2018)
We study Lagrangian embeddings of a class of two-dimensional cell complexes L_p,q into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles ...
Balanced semisimple filtrations for tilting modules
(American Mathematical SocietyRepresentation Theory, 2017-03-08)
Let $ U_l$ be a quantum group at an $ l$th root of unity, obtained via Lusztig's divided powers construction. Many indecomposable tilting modules for $ U_l$ have been shown to have what we call a balanced semisimple ...
The half plane UIPT is recurrent
(SpringerProbability Theory and Related Fields, 2017-03-07)
We prove that the half plane version of the uniform infinite planar triangulation (UIPT) is recurrent. The key ingredients of the proof are a construction of a new full plane extension of the half plane UIPT, based on a ...
Topological cycle matroids of infinite graphs
(ElsevierEuropean Journal of Combinatorics, 2017-02-01)
We prove that the topological cycles of an arbitrary infinite graph together with its topological ends form a matroid. This matroid is, in general, neither finitary nor cofinitary.
Infinitesimal Models of Algebraic Theories
(University of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsHomerton College, 2017-11-25)
Smooth manifolds have been always understood intuitively as spaces that are infinitesimally
linear at each point, and thus infinitesimally affine when forgetting about the base point. The
aim of this thesis is to develop ...