Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO. Naturally, two interesting questions arise:

1. What is the number \lambda of topological types of locally finite trees?

2. What are the possible sizes of an equivalence class of locally finite trees?

For (1), clearly, \omega_0 \leq \lambda \leq c and Matthiesen refined it to \omega_1 \leq \lambda \leq c. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address both questions by showing - entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:

- \lambda = \omega_1, and

- the size of an equivalence class can only be either 1 or c.

# Past Forthcoming Seminars

In joint work with Gareths Boxall and Jones we prove a poly-logarithmic bound for the number of rational points on the graph of functions on the disc that exhibit a certain decay. I will present an application of this counting theorem to the arithmetic of dynamical systems. It concerns fields generated by the solutions of equations of the form $P^{\circ n}(z) = P^{\circ n}(y)$ for a polynomial $P$ of degree $D \geq 2$ where $y$ is a fixed algebraic number. The general goal is to show that the degree of such fields grows like a power of $D^n$.

We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over Z. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a finite field, and refines earlier work by N.O. Nygaard and J.-D. Yu. Two important ingredients in the proof are integral p-adic Hodge theory, and a description of CM points on Shimura stacks in terms of associated Galois representations. References: arXiv:1711.09225, arXiv:1707.01236.

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.

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.

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.

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.

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

2^(3-3/x)+ 2^(2-2/x)+2^(1-1/x)=2,

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.