Consider the curves of genus g>0 and(principally polarized) abelian varieties of dimension g>0, all defined overa field of positive characteristic p>0. Denote with M_g and A_g respectivelytheir moduli spaces - the spaces that parametrize isomorphism classes of these.There are some invariants used to better understand M_g and A_g, for example, p-rank,Ekedahl-Oort type, and Newton polygon. The latter gives us a way to define themain objects of this project, the supersingular curves/abelian varieties. For g= 4, 5, and p = 2, we discuss some properties of the supersingular locus S_g inA_g, such as irreducibility. Further, we describe a piece of geometry relatedto the intersection of S_g with the image of M_g in A_g. Lastly, inspired by apaper that (algorithmically) determined all the non-isomorphic curves of genusg = 4 defined over a field with two elements, we discuss a similar problem forg = 5.
The first part of the talk should be accessibleto anyone interested in mathematics, and it will contain some basic examplesfrom algebraic geometry. Later, we will explain the notion of being supersingularand motivate what ideas led us to consider some of the problems, which we willdiscuss in the last part of the talk.
Written as part of aninternship at Air Traffic Control the Netherlands (LVNL) under the supervisionof Marcel Opbroek (LVNL) with Rob Bisseling as project supervisor and PaulZegeling as second reader.
MS Teams code: 5lazxzb (for UU members) or link (forguests): https://teams.microsoft.com/l/meetup-join/19%3a7K6DsIK5wjDbJCiJk4JvJwiAzAdw2_6j93pjKG_sumI1%40thread.tacv2/1623226838812?context=%7b%22Tid%22%3a%22d72758a0-a446-4e0f-a0aa-4bf95a4a10e7%22%2c%22Oid%22%3a%221db3264e-330c-4c2f-bb88-ef423780f0b8%22%7d
Abstract: Increasing landing capacity is important to reduce delaysand therefore costs, fuel burn and emissions. Decreasing separation distancesbetween aircraft is a way to increase the landing capacity. A safe distancebetween aircraft depends, among other things, on the level of turbulence anaircraft experiences. Turbulence can be caused by wind or weather conditions,but also by wake vortices generated by nearby aircraft. Therefore, aircraftcannot fly too close behind each other and separation minima have beenestablished by ICAO. To increase landing capacity, LVNL will implementRECAT-TBS at Amsterdam Airport Schiphol by the end of this year to safelydecrease these separation minima.The goal of this thesis is to implement analgorithm that can detect and measure wake turbulence in the approach areausing Mode S radar data. This can give insight in a possible increase in thenumber of wake turbulence encounters or might yield a warning system whendecreasing the separation distances in final approach by the end of this year.The algorithm that will be analysed is an algorithm proposed in a paper byXavier Olive and Junzi Sun. They implemented and analysed the algorithm foren-route air traffic to detect turbulence. The question is whether thisalgorithm can be used or adapted to be applicable for approaching air trafficand if it is possible to distinguish wake turbulence encounters.
Zoom details can be found in the website of the seminar: https://www.few.vu.nl/~trt800/ddtg.html
Writtenunder the supervision of Dr Wioletta Ruszel with Dr Cristian Spitoni as secondreader
Abastract: Continuousdiffusions (without jumps) are a ubiquitous model in especially financialmathematics, among other disciplines. While adding jumps to such models complicatestheir study, their addition is invaluable for applications. For example, inmarket analysis, jumps play the rôle of market fluctuations, which aretypically the events one is most interested in.
In my talk, Iwill expound on how to estimate the parameters of a general jump-diffusionmodel. These parameters quantify important information, like jump sizes andfrequency, for instance. A numerical case study and insight into prooftechniques will be provided.