It is difficult to determine when a graph G can be edge-covered by edge-disjoint copies of a fixed graph F. That is, when it has an F-decomposition. However, if G is large and has a high minimum degree then it has an F-decomposition, as long as some simple divisibility conditions hold. Recent research allows us to prove bounds on the necessary minimum degree by studying a relaxation of this problem, where a fractional decomposition is sought.
I will show how a relatively simple random process can give a good approximation to a fractional decomposition of a dense graph, and how it can then be made exact. This improves the best known bounds for this problem.
- Combinatorial Theory Seminar
In this talk, first we address the convergence issues of a standard finite volume element method (FVEM) applied to simple elliptic problems. Then, we discuss discontinuous finite volume element methods (DFVEM) for elliptic problems with emphasis on computational and theoretical advantages over the standard FVEM. Further, we present a natural extension of DFVEM employed for the elliptic problem to the Stokes problems. We also discuss suitability of these methods for the approximation of incompressible miscible displacement problems.
- Numerical Analysis Group Internal Seminar
Fake accounts detection and users’ polarization are two very well known topics concerning the social media sphere, that have been extensively discussed and analyzed, both in the academic literature and in everyday life. Social bots are autonomous accounts that are explicitly created to increase the number of followers of a target user, in order to inflate its visibility and consensus in a social media context. For this reason, a great variety of methods for their detection have been proposed and tested. Polarisation, also known as confirmation bias, is instead the common tendency to look for information that confirms one's preexisting beliefs, while ignoring opposite ones. Within this environment, groups of individuals characterized by the same system of beliefs are very likely to form. In the present talk we will first review part of the literature discussing both these topics. Then we will focus on a new dataset collecting tweets from the last Italian parliament elections in 2018 and some preliminary results will be discussed.
- Networks Seminar
After outlining the principles of Algebraic Quantum Field Theory (AQFT) I will describe the generalization of Hochschild cohomology that is relevant to describing deformations in AQFT. An interaction is described by a cohomology class.
- Quantum Field Theory Seminar
Answering a question of Milnor, Grigorchuk constructed in the early eighties the
first examples of groups of intermediate growth, that is, finitely generated
groups with growth strictly between polynomial and exponential.
In joint work with Laurent Bartholdi, we show that under a mild regularity assumption, any function greater than exp(n^a), where `a' is a solution of the equation
is a growth function of some group. These are the first examples of groups
of intermediate growth where the asymptotic of the growth function is known.
Among applications of our results is the fact that any group of locally subexponential growth
can be embedded as a subgroup of some group of intermediate growth (some of these latter groups cannot be subgroups in Grigorchuk groups).
In a recent work with Tianyi Zheng, we provide near optimal lower bounds
for Grigorchuk torsion groups, including the first Grigorchuk group. Our argument is by a construction of random walks with non-trivial Poisson boundary, defined by
a measure with power law decay.
- Topology Seminar
We consider a collisionless kinetic equation describing the probability density of particles moving in a one-dimensional domain subject to partly diffusive reflection at the boundary. It was shown in 2017 by Mokhtar-Kharroubi and Rudnicki that for large times such systems either converge to an invariant density or, if no invariant density exists, exhibit a so-called “sweeping phenomenon” in which the mass concentrates near small velocities. This dichotomy is obtained by means of subtle arguments relying on the theory of positive operator semigroups. In this talk I shall review some of these results before discussing how, under suitable assumptions both on the boundary operators (which in particular ensure that an invariant density exists) and on the initial density, one may even obtain estimates on the rate at which the system converges to its equilibrium. This is joint work with Mustapha Mokhtar-Kharroubi (Besançon).
- Partial Differential Equations Seminar
I will present a mathematical model for the genetic evolution of a population which is divided in discrete colonies along a linear habitat, and for which the population size of each colony is random and constant in time. I will show that, under reasonable assumptions on the distribution of the population sizes, over large spatial and temporal scales, this population can be described by the solution to a stochastic partial differential equation with constant coefficients. These coefficients describe the effective diffusion rate of genes within the population and its effective population density, which are both different from the mean population density and the mean diffusion rate of genes at the microscopic scale. To do this, I will present a duality technique and a new convergence result for coalescing random walks in a random environment.
- Stochastic Analysis Seminar
Recent tools make it possible to partition the space of rational Dehn
surgery slopes for a knot (or in some cases a link) in a 3-manifold into
domains over which the Heegaard Floer homology of the surgered manifolds
behaves continuously as a function of slope. I will describe some
techniques for determining the walls of discontinuity separating these
domains, along with efforts to interpret some aspects of this structure
in terms of the behaviour of co-oriented taut foliations. This talk
draws on a combination of independent work, previous joint work with
Jake Rasmussen, and work in progress with Rachel Roberts.
- Topology Seminar
In this paper, we investigate the limit properties of frequency of empirical averages when random variables are described by a set of probability measures and obtain a law of large numbers for upper-lower probabilities. Our result is an extension of the classical Kinchin's law of large numbers, but the proof is totally different.
keywords: Law of large numbers,capacity, non-additive probability, sub-linear expectation, indepence
paper by: Zengjing Chen School of Mathematics, Shandong University and Qingyang Liu Center for Economic Research, Shandong University
- Stochastic Analysis Seminar