Max Planck Institute, Bonn
Vivatsgasse 7, 53111, Bonn, Germany
My research interests lie in arithmetic geometry, analytic and algebraic number theory, prime number theory, sieve theory, the Hardy-Littlewood circle method, the distribution of class groups of number fields, probability theory and random walks.
In my early research, I worked on the Batyrev-Manin conjecture on counting rational points on smooth del Pezzo surfaces. I proved the lower bounds predicted by the conjecture for degree 1,2 and 3, as well as, upper and lower bounds agreeing with the Manin conjecture in general cases with degree 4. Very little was previously known for degrees 1 and 2 and the conjecture is only known for a single case in degree 4. Furthermore, I worked on Sarnak's saturation problem on the density of semi-prime solutions of Diophantine equations. In particular, my work covers low dimensional cases (e.g. smooth cubic surfaces), where standard methods do not apply, in addition to large dimensional complete intersections via the circle method.
While in Leiden I later became interested in the Cohen-Lenstra heuristics on the statistical behaviour of class groups of number fields. I worked on extending the heuristics to the setting of ray class groups and estimating the moments in the case of 4-ranks.
Most recently, I have been working on the behaviour of rational points in families of varieties. Serre's problem regards the distribution of the varieties in the family with a rational point. This problem is in character rather different from the Batyrev-Manin conjecture but it includes the B-M conjecture as a very special case. While there are examples where upper bounds are known, there are very few instances where the conjectured lower bounds (or asymptotics) can be proved. I have proved lower bounds and asymptotics for the case of conic bundles over the projective space and higher-dimensional bases respectively. The video of my talk in BIRS on the recent developments in the area.
Another way to understand the behaviour of rational points in families of varieties is to study the primes p for which the typical variety has no p-adic solution. In recent work I have shown that their distribution follows an arithmetic analogue of the Central Limit Theorem. I have also shown that by associating to each variety a path in the plane one can import results (such as the Feynman-Kac formula) from stochastic processes into arithmetic geometry. This is a new connection and it gives previously unknown distribution laws for p-adic solubility. There seem to be more structures that can be imported from probability theory into arithmetic geometry via this connection.
Briefly, my interests are in arithmetic geometry, the Cohen-Lenstra heuristics and probability theory.
|Stefanos Aivazidis||Queen Mary University of London||UK|
|Tim Browning||Institute of Science and Technology||Austria|
|Kevin Destagnol||Max Planck Institute, Bonn||Germany|
|Christopher Frei||University of Manchester||UK|
|Peter Koymans||University of Leiden||Netherlands|
|Daniel Loughran||University of Manchester||UK|
|Carlo Pagano||University of Leiden||Netherlands|
|Damaris Schindler||Utrecht University||Netherlands|
|Erik Visse-Martindale||University of Leiden||Netherlands|
|Yuchao Wang||Shanghai University||China|
|Michel Zoeteman||University of Leiden||Netherlands|
|Ingela Mennema||University of Leiden||Netherlands|
|Ho Yin Leung||University of Bonn||Germany|
|Lukas Klebonas||University of Bonn||Germany|
|Valerio Cini||University of Bonn||Germany|