Arithmetic Sets in Groups

We define a notion of an arithmetic set in an arbitrary countable group and study properties of these sets in the cases of Abelian groups and non-abelian free groups

The Chow ring of the stack of cyclic covers of the projective line

In this paper we compute the integral Chow ring of the stack of smooth uniform cyclic covers of the projective line and we give explicit generators.Comment: 20 pages; final version, to appear in Ann. Inst. Fourie

On the Chow ring of the stack of rational nodal curves

From the introduction: In this thesis we start investigating the intersection theory of the Artin stack M0 of nodal curves of genus 0, following a suggestion of Rahul Pandharipande. Intersection theory on moduli spaces of stable curves has a not so long, but very intense history. It started at the beginning of the \u201980\u2019s with Mumford\u2019s paper [Mum2], where he laid the foundations and carried out the first calculations. Many people have contributed to the theory after this (such as Witten and Kontsevich), building an imposing structure. The foundations of intersection theory on Deligne-Mumford stacks have been developed by Gillet and Vistoli. The first step towards an intersection theory on general Artin stacks (like those that arise from looking at unstable curves) was the equivariant intersection theory that Edidin and Graham developed, following an idea of Totaro. Their theory associates a commutative graded Chow ring A (M) with every smooth quotient stack M of finite type over a field. Unfortunately many stacks of geometric interest are not known to be quotient stacks (the general question of when a stack is a quotient stack is not well understood). Later, A. Kresch developed an intersection theory for general Artin stacks; in particular, he associates a Chow ring A (M) with every smooth Artin stack M locally of finite type over a field, provided some technical conditions hold, which are satisfied in particular for stacks of pointed nodal curves of fixed genus. Since there is not yet a theory of Chow rings of such stacks that extends the theory of stacks of stable curves, there does not seem to be much else to do than look at specific examples: and the first example is the stack M0 of nodal connected curves of genus 0. However, even this case turns out to be extremely complicated. In this thesis we compute the rational Chow ring of the open substack M 3 0 consisting of nodal curves of genus 0 with at most 3 nodes: it is a Q-algebra with 10 generators and 11 relations. The techniques that we use, and the problems that we encounter, are discussed below

Preliminary studies for a future LHCb upgraded vertex-locator using 4D track reconstruction

This thesis reports a preliminary study aimed at finding realistic solutions for the future Vertex Locator of the LHCb experiment in the harsh scenario imposed by the High Luminosity phase of the Large Hadron Collider (LHC). One of these possible solutions to cope with a high-multiplicity-track environment is the introduction of the time information associated with hits on pixels. The addition of the fourth coordinate, the time, will improve the efficiency of track and primary vertices reconstruction, reducing the rate of the fast tracker. The impact of the time information is evaluated in the present work through a simulation. The fast simulation developed is based on the Upgrade-I detector geometry. It has been validated with a full, official, LHCb simulation. Differently from the latter, fast simulation has the advantages to be quicker and versatile, to cope with higher rates of Upgrade-II conditions. This work here presented is done in collaboration with the INFN project TimeSpot, which is developing a promising silicon sensor with a time resolution of O(30ps). Because of the high data-rate and the demand for real-time devices, FPGA solution is a possible candidate for clustering and real-time track reconstruction at read-out level, due to its high predisposition to perform easy and repetitive processes in a highly parallelized form. Since FPGAs receives in input binary data, a raw bank data-format is implemented in the fast simulation. Each raw bank encodes active pixels information of a detector sensor. Once this information is processed, FPGAs will reconstruct tracks from the hits in a parallel way. In conclusions, the thesis work here presented shows, trough simulation studies, that the time information in a pixel sensor, with a resolution of about 30 ps, will be necessary to achieve similar performance like the VeloPixel will do in the coming years