TopMath Award 2020
Awards for D. Schmid and C. Stadlmayr
The TopMath Award is a prize for PhD Students within the TopMath Program, in recognition of exceptional achievements in research. It is awarded once a year. Upon recommendation from their mentor or the subject study advisor, the candidates are requested to apply for the award. Annually in November, the TopMath Board meets to discuss the applications and to nominate the awardee. The TopMath Award includes prize money of 500 Euros, and is usually presented within the graduation ceremony of the Department of Mathematics.
Winners of the TopMath Award 2020 are in equal measure Dominik Schmid (Chair for Probability) and Claudia Stadlmayr (Research Group Algebra).
Dominik Schmid held numerous talks at international math meetings in 2019/2020, travelled for research work to renowned universities, and wrote four scientific contributions based on international cooperations, for example with Evita Nestoridi (Princeton University) and Nicos Georgiou (University of Sussex). The TopMath Award is in recognition of his paper on the exclusion process with open boundaries, which he wrote together with Nina Gantert (TUM) and Evita Nestoridi.
For Claudia Stadlmayr 2019/20 was also a very successful year. Her previous research fruited in two contributions, one of which was written in cooperation with Gebhard Martin (University of Bonn) and the other as sole author. The high interest of the math community in her work led to numerous invitations for research visits and to give talks at international conferences. With the TopMath Award 2020, the Board acknowledges this remarkable level of achievement, which is exceptional at such an early stage in her career.
We congratulate both winners on their achievements and wish them every success in attaining their next mathematical goals!
Dominik Schmid: Mixing times for the exclusion process
Text: Dominik Schmid
In a joint work with Nina Gantert and Evita Nestoridi, we study the simple exclusion process with open boundaries. The simple exclusion process with open boundaries is an interacting particle system. It is used to describe the motion of particles in gases or the formation of traffic jams. This model has the following intuitive description: consider a segment of the integer lattice. We place particles on the lattice points such that each lattice point is occupied by at most one particle. The particles now try to move independently to a randomly chosen adjacent position. However, if a particle tries to move to an already occupied position, this move is suppressed. In addition, we allow the creation and annihilation of the particles at the endpoints of the segment.
Our goal is to study the convergence of the simple exclusion process with open boundaries to its equilibrium, starting from the worst-case initial configuration. We quantify this convergence using mixing times. We identify different regimes for the mixing times. These regimes depend on the transition rates of the particles on the lattice and the rates of creating and annihilating particles at the endpoints. In particular, we show in several cases that the cutoff phenomenon, a sharp transition in reaching the equilibrium, occurs.
Claudia Stadlmayr: Symmetries and singularities on del Pezzo surfaces
Text: Claudia Stadlmayr
So-called "del Pezzo surfaces" form a special class of smooth, projective surfaces, which, after having been studied first by Pasquale del Pezzo (1887), have become fundamental objects in algebraic geometry and basic building blocks in the classification of algebraic varieties over the past 100 years. By combining the classical study of symmetry groups of such surfaces with the modern point of view of automorphism schemes, Gebhard Martin and I were able to classify all weak del Pezzo surfaces with global vector fields over fields of arbitrary characteristic in a joint project.
See: "Weak del Pezzo surfaces with global vector fields" (with G. Martin)
Omitting the smoothness condition and instead allowing mild singularities, so called "rational double points", the concept of a del Pezzo surface can also be extended to singular projective surfaces. Such surface singularities can be divided into different types (according to their associated Dynkin diagram). In 1934, Patrick Du Val posed another classical question, namely which types of such singularities occur on del Pezzo surfaces. In my second project, I was able to settle this question for arbitrary characteristics, degrees and Picard ranks.
Quite encouragingly, both of these articles have piqued the interest of the mathematical community, which gave me the opportunity to present my second project online in the Moscow Iskovskikh Seminar and to present the results of my work on weak del Pezzo surfaces in the "Zoom Algebraic Geometry Marathon". The idea of this "marathon" was to celebrate each of the 24 hours of the "Day of Knowledge" with a research lecture on algebraic geometry by speakers located across all time zones. To represent our Central European time zone and to give my very first conference talk in this unique format as the youngest among the speakers, and also as the only speaker without a doctoral degree, was a great honor for me.
See also "Claudia Stadlmayr at the ZAG Marathon"
Finally, after the two research stays of two months each in Moscow at the Steklov Institute and in Salt Lake City at the University of Utah, which I had planned for the last year, could unfortunately not take place due to the pandemic, I am very grateful for the fruitful virtual collaboration and the opportunities to feel like becoming an appreciated member of the community of mathematicians working in algebraic geometry through my online lectures.
Acknowledgement of the prize winners and their achievements
"Dominik Schmid is interested in discrete probability theory, in particular interacting particle systems, mixing times for Markov chains and stochastic processes in random media. Several of his contributions have been internationally recognized and the topics have been taken up by well-known probabilists. His PhD thesis is based on five papers, the common theme being exclusion processes. In his new preprint
'Mixing times for the TASEP in the maximal current phase', not included in the PhD thesis, he solves a general conjecture for the particular case of Totally Asymmetric Exclusion Process (TASEP).
He has also contributed to different topics as random walks on trees and self-similar fragmentations.
Dominik Schmid has won a DAAD PRIME fellowship for the project 'Analytic and geometric properties of exclusion processes' and he will move to Princeton University in fall 2021." - Prof. Dr. Nina Gantert (TUM), Mentor of Dominik Schmid
"Already within her Bachelor thesis 'Deformations and Resolutions of Rational Double Points’, Claudia Stadlmayr showed an extraordinarily deep understanding of algebraic geometry. She has to date written two preprints presenting her own results: 'Weak del Pezzo surfaces with global vector fields' (50 pages, together with Gebhard Martin) and 'Which rational double points occur on del Pezzo surfaces?' (20 pages). Both preprints have already attracted the attention of the experts in the field and led to invitations to talks all over the world. Her work addresses central problems in the minimal model program (also known as Mori program) in dimension 2 and she completely solved them. In pure mathematics, this level of achievement at such an early career stage is highly remarkable and extraordinary." - Prof. Dr. Christian Liedtke (TUM), Mentor of Claudia Stadlmayr