The talk concerns critical behavior of percolation on finite random networks with heavy-tailed degree distribution. In a seminal paper, Aldous (1997) identified the scaling limit for the component sizes in the critical window of phase transition for the Erdos-Renyi random graph. Subsequently, there has been a surge in the literature identifying two universality classes for the critical behavior depending on whether the asymptotic degree distribution has a finite or infinite third moment.
In this talk, we will present a completely new universality class that arises in the context of degrees having infinite second moment. Specifically, the scaling limit of the rescaled component sizes is different from the general description of multiplicative coalescent given by Aldous and Limic (1998). Moreover, the study of critical behavior in this regime exhibits several surprising features that have never been observed in any other universality classes so far.
While contact geometry is often described as the odd-dimensional analogue of symplectic geometry, Arnol'd pointed out that the relation between these two geometries is analogous to that between affine and projective geometries. A natural (vague!) question is to investigate the extent to which this analogy extends to the degenerate versions of symplectic and contact structures, namely Poisson and Jacobi geometries.
The aim of this talk is to attempt to formalise the above question and to illustrate the intimate relation between these six geometries through a few specific problems, including isotropic realisations and Poisson/Jacobi manifolds of compact types. The talk is based on joint work with María Amelia Salazar, and on ongoing joint work with Camilo Angulo and María Amelia Salazar.
Written under supervisionof Dr. Lennart Meier
Richard Brauer introducedthe concept of the Brauer group of a field K to classify central simplealgebras over K. An important example of a central simple algebra is a cyclicalgebra which we can associate to a cyclic Galois extension over K and anelement of the multiplicative group of K.
In this thesis, we study local fields and determine which elements of theBrauer group can be reached if we restrict ourselves to the subgroup of K* ofdiscriminants of elliptic curves over K. We show that essentially any expectedelement of the Brauer group can be reached. Using local class field theory wealso show that the answer differs when we look at elliptic curves over the ringof integers of a local field over the 2- and 3-adic numbers.
Written under the supervision of Dr. Karma Dajani, with Dr. SjoerdDirksen as second reader.
We revisit the idea of the random beta-transformation fornon-integral beta>1. This is a measurable dynamical system motivated by thestudy of number expansions in non-integer bases, which behave very differentlyto the usual expansions in integer bases. During this presentation we willdiscuss these expansions, which motivate the definition of the randombeta-transformation, which we provide and discuss. We then provide a basicexposition of the theory of g-measures, by first motivating their definition,and then discussing some of their unexpected links to thermodynamical formalism,culminating in Ruelle's Operator Theorem. Finally we use the machinery ofg-measures to construct a new family of strongly mixing invariant measures forthe random beta-transformation for certain algebraic beta.
The team can be joined using the following link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_MDQzZDUxYTktM2U1Zi00YzFjLWE1NDktMzM1OGYwYzFmOTdh%40thread.v2/0?context=%7b%22Tid%22%3a%22d72758a0-a446-4e0f-a0aa-4bf95a4a10e7%22%2c%22Oid%22%3a%22fb36163f-a37b-418c-a69b-b6b41a1b6733%22%2c%22IsBroadcastMeeting%22%3atrue%7d