Efthymios Sofos
Max Planck Institute, Bonn
Vivatsgasse 7, 53111, Bonn, Germany

Email: sofos@mpim-bonn.mpg.de


Conferences organisation With Alex Bartel we organise a conference on the Cohen–Lenstra heuristics at the Max Planck Institute for Mathematics, Bonn, during 3 to 7 June 2019.
Its website is here.

Curriculum Vitae in pdf

Research Interests

My research interests lie in arithmetic geometry, analytic and algebraic number theory, the fibration method, the Hardy-Littlewood circle method, the distribution of class groups of number fields, probability theory, random walks, prime number theory and sieve theory.

I finished my Ph.D. in Bristol in 2015. 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.

During 2016 and 2017 I was in Leiden, where I 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.

Name Affiliation Country
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 Max Planck Institute, Bonn Germany
Damaris Schindler Utrecht University Netherlands
Erik Visse-Martindale University of Leiden Netherlands
Yuchao Wang Shanghai University China

Master and Bachelor Students
Name Place Country
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

Michel Zoeteman has converted his thesis to a paper, see [arXiv:1810.04942].
During September 2018 I supervised two undegraduate interns (Carlos Esparza and Lukas Gehring) in the Max Planck Institute. Their work resulted in a paper, see [arXiv:1810.08679].

In the second semester of 2016-17 in Leiden University I taught a Masters course in Topics in Analytic Number Theory.
In the first semester of 2016-17 I taught with Jan-Hendrik Evertse in Leiden University a bachelor course in Analytic Number Theory.
In the second semester of 2015-16 I taught a Bachelor course in Statistics in the Leiden University College (which collaborates with the Leiden University in teaching).
In September 2018 I am teaching a short course on the prime number theorem in Max Planck. The lectures are scheduled for the interns, which are bachelor students that are doing a one month internship in the Max Planck Institute. Here are the notes.

Seminars organised
I am co-organising the PLeaSANT seminar in Max Planck Institute, Bonn.
I was the organiser of the Kloosterman seminar in the Math Department, Leiden.