Mateusz Michałek has build up an impressive Max-Planck history: during his postdoc years he initially worked at the institute in Bonn before he came to MiS in 2016 first as a postdoc and later as a visiting scholar. In March of 2017, he took the leap of becoming a research group leader of the group “Combinatorial Algebraic Geometry”. Together with his group he looked for applications of algebraic methods to topological problems, improved methodologies in fast matrix multiplication and investigated toric geometry and tensors, as well as the combinatorial aspects of polytopes. Besides his postdoc endeavors at the mathematical MPIs he also stayed at Barcelona University, FU Berlin, UC Berkeley, RIMS Kyoto and is an assistant professor with the Polish Academy of Sciences. Together with Bernd Sturmfels he authored the book "Invitation to Nonlinear Algebra", which aims to provide the theoretical basis for recent advances in computation and other applications. In October 2020 Mateusz became professor of Geometry (Real Geometry and Algebra) at the University of Konstanz.
MPI MiS: How did you get into math?
Mateusz Michałek: I guess I always knew my calling was to become a mathematician. In remember receiving only a ‘very good’ evaluation in math and ‘exceptional’ in Polish in first grade of elementary school. I couldn’t fathom why, as I was sure that I was way better in math than in Polish. It was only after some years that my parents told me that ‘exceptional’ was not granted in first grade, since there isn’t all that much math to speak of at that stage. It was thanks to my fourth-grade math teacher Mrs. Teresa Lekawska that I dived deeper into the realm of math, as she encouraged me to do extra exercises, take part in competitions and follow additional courses. Another tremendously positive influence was my uncle, Tomasz Michałek. He was a young computer arts student in Cracow when we started our weekly meetings, where he taught me some things that went way beyond my education at school and instead delved into his university education. This sounds extraordinary, however in Poland the university math courses start with more basic things than primary education. Formal introduction of set theory, functions, and induction, it surely takes some time until you learn that 1+1=2. At this point mathematical notation came more natural to me than using the 'standard' language. I was very much disappointed to discover years later that true mathematicians actually write articles in 'standard' language - not just mathematical symbols. My parents also told me later that they were often hesitant to enter the room when Tomasz and I were doing math, as we were actually fighting and shouting pretty regularly in the process. It was a lot of fun!
MPI MiS: What would you would like to accomplish?
Mateusz Michałek: As most mathematicians, I would like to solve some fundamental problem. The one I am currently working on is related to finding optimal ways of computing the determinant or multiplying matrices. This is very much related to the question that people were studying for millennia: how to solve a system of linear equations?
MPI MiS: Why did you choose to become a scientist at MiS?
Mateusz Michałek: It was a combination of quite a few factors. I already spent a year at the MPI in Bonn, so I knew that MPIs offer the highest possible research standards. At the same time my wife was offered a postdoc at the UFZ in Leipzig and Bernd Sturmfels, with whom I had worked before, became MiS director. I was offered an absolutely great position; probably one of the best in the world a young mathematician may hope for. Thus, the choice was very clear for me.
MPI MiS: What advice would you give young researchers starting out at the institute?
Mateusz Michałek: In many games, like chess or go, we start by learning the rules. Then we learn some tricks that allow us to outperform people who merely know the rules. The next, extremely powerful, step is usually to start planning a few moves ahead. While this is not a universal advice to solve mathematical problems, I think it’s a wise choice to somewhat plan out your mathematical career. You don’t have to follow these plans to a T, which is usually impossible anyways, but it is good to have them - even if it’s just to have the option to ignore them later.
MPI MiS: Which scientist would you love to work with?
Mateusz Michałek: To be honest, I simply contact anyone that I’d like to cooperate with. To the surprise of my fellow students I could get acquainted with senior professors at the university by simply knocking on their doors and had no problems contacting a Fields medalist even as a bachelor student. The main reason for the ease of these approaches was that I always felt very welcome: thus, maybe one more advice for young researchers is to not be afraid to discuss math with anyone you like!
How fast can matrices be multiplied? How can quantum states of many particles be represented efficiently? How can Markov processes be reconstructed, represented and applied to study the evolution of species?
All of these, at first glance unrelated questions are the motivation for some of Mateusz' research directions. They are all related to tensors, algebra and geometry. Classical linear algebra provides some of the most useful and most frequently applied tools, both in pure and applied mathematics. Novel and challenging problems, that come with advances in computational sciences, often arise in non-linear algebra. Just as matrices form the basic objects of linear algebra, tensors constitute the focal point of multilinear algebra. It turns out that their properties are much more intricate than those of matrices. This leads to fundamental open problems that often can be phrased in both a geometric and an algebraic language. As tensors are ubiquitous in many different branches of mathematics, so are the tools to study them, as they often require an interplay of different methods. Mateusz works on tensor problems by trying to solve them through the application of combinatorial, algebraic and geometric methods.
Honored with the Kazimierz Kuratowski Award in 2015 by the Institute of Mathematics of the Polish Academy of Sciences and the Polish Mathematical Society.
- M. Michałek and Y. Shitov, Quantum Version of Wielandt’s Inequality Revisited, IEEE Transactions on Information Theory, vol. 65, no. 8, pp. 5239-5242, Aug. 2019.
- J. Buczyński, T. Januszkiewicz, J. Jelisiejew, M. Michałek, Constructions of \(k\)-regular maps using finite local schemes, J. Eur. Math. Soc. 21 (2019), 1775-1808.
- J. Landsberg, M. Michałek, On the geometry of border rank algorithms for matrix multiplication and other tensors with symmetry, SIAM Journal on Applied Algebra and Geometry 1 no. 1 (2017), 2–19.
- M. Michałek, B. Sturmfels, C. Uhler, P. Zwiernik, Exponential varieties, Proceedings of London Mathematical Society 112 no. 1 (2016), 27-56.
- M. Lason, M. Michałek, On the toric ideal of a matroid, Advances in Mathematics 259 (2014), 1–12.
We would to like to congratulate our former scientists Martina Hofmanová and Julian Fischer who rank among Europe's best young researchers. For their excellent work they have each been awarded an ERC Starting Grant.
Martina Hofmanová currently holds a professorship for mathematics at the University of Bielefeld and is a member of the Collaborative Research Center 1283 "Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications". She received her Master's degree in mathematics at Charles University in Prague, the largest university in the Czech Republic, and her doctorate at the École Normale Supérieure de Cachan, Atenne de Bretagne in France. At our institute she worked as a postdoc in the group of Prof. Felix Otto.
Her ERC project „Mathematical analysis of fluid flows: the challenge of randomness” is based on the conviction that a probabilistic description is indispensable in modeling of fluid flows to capture the chaotic behavior of deterministic systems after blow-up, and to describe model uncertainties due to high sensitivity to input data or parameter reduction. For a set of selected models, she investigates with her research group different aspects of the underlying deterministic and stochastic PDE dynamics.
Picture: Sarah Jonek / University of Bielefeld
Randomness is also in the focus of Julian Fischer's ERC project, which aims to achieve a deeper mathematical understanding of the role of randomness in multi-scale problems in physics and mechanics. With his group he wants to investigate largely unexplored aspects of random and multiscale PDEs. His project will focus on the impact of random microstructure on nonlinear materials, homogenization theories and numerical methods in the absence of clear scale separation, and the stabilizing effect of randomness on interface evolution equations.
In 2017, at the age of 27 Julian Fischer became an Assistant Professor at IST Austria after completing his PhD at the University of Erlangen-Nuremberg and subsequently holding two postdoctoral positions at the University of Zurich, Switzerland, and at our institute in the research group of Prof. Felix Otto.
Picture: Paul Pölleritzer / IST Austria
Tinggui Zhang and Damián Blasi, both former Ph.D. students of Jürgen Jost, have been awarded the "South China Sea Masters " youth project funding and the Robert J. Glushko dissertation prizes in cognitive science, respectively.
Tinggui Zhang studied quantum information and computation as a Ph.D. student under the supervision of Xianqing Li-Jost and Jürgen Jost from August 2010 until July 2014, when he obtained his doctorate. Afterwards he returned to China and worked in Hainan Normal University. The "South China Sea Masters" youth project focuses on the selection and training of a group of young talents with strong scientific research abilities, innovative spirits and great potential. Tinggui Zhang is the only candidate from mathematics in this project. He will be awarded 300,000 RMB (~39,000 €) and the title of top talent in Hainan Province.
Damián Blasi originally studied physics in Argentina and after coming to Leipzig was simultaneously affiliated to both our institute and the department of linguistics at the MPI for evolutionary anthropology from July 2012 until August 2015. Damián received a Ph.D. in computer sciences for his dissertation on "Linguistic Diversity Through Data". This outstanding thesis has earned him the $10,000 (~9,200 €) award by the Cognitive Science Society and the Glushko-Samuelson Foundation, which is an annual honor for five young scientists that conduct ground-breaking research in the cognitive sciences. Consistent with their vision that "that understanding how minds work will require the synthesis of many different empirical methods" a dissertation has to transcend any one of the individual fields comprising cognitive science in order to be considered for the honor. After his postdoc at the university of Zürich, Switzerland, he is now the current Maury Green Fellow at the Radcliffe Institute for Advanced Studies at Harvard University, as well as being affiliated to the Department of linguistic and cultural evolution at the MPI for the science of human history in Jena.
Angkana Rüland has been a Max-Planck research group leader of the group “Rigidity and Flexibility in PDEs” at the MPI MiS since October 2017. In joint work with her group she has investigated inverse problems, nonlocal equations, problems from the calculus of variations and free boundary value problems. Before starting at the institute, she studied Mathematics in Bonn and Leipzig and had been a postdoctoral research associate at the University of Oxford. In April 2020 she will become a professor of applied mathematics at the Ruprecht-Karls-University in Heidelberg.
MPI MiS: Why does mathematics fascinate you?
Angkana Rüland: I was always fond of mathematics in school, but could also have imagined studying other subjects at university. When encountering math at the university level, I was fascinated by the clarity of mathematical arguments and the idea of being completely rigorous about a statement. Furthermore, having always been interested in the natural sciences, mathematics was an ideal way to combine precise, rigorous mathematical analysis with working on relevant applications in the natural sciences. The interaction between exciting, new math, which is already signiﬁcant from an inner-mathematical point of view and in this sense purely curiosity-driven, and its application to problems from the natural sciences is an important source of motivation for me. I love the fact that once a problem is “understood” many more fascinating new problems related to it emerge.
MPI MiS: What motivates you in your research?
Angkana Rüland: The curiosity of understanding something new is one of my main sources of motivation. Often, when trying to understand and to “solve” a problem, you run into a multitude of challenges and many ideas do not work the way you had hoped. Of course, this can sometimes be very frustrating. However, learning from these experiences, exploring how far you can push an idea and ﬁnding ways out of seemingly dead ends, makes it extremely rewarding when all ingredients suddenly ﬁt together and give a “clear picture” of the situation. Of course my research ﬁeld is another major motivation for me. I love to work at the interface between mathematics and the sciences and to discuss problems, not only with other mathematicians, but also with engineers and physicists.
Unique continuation and inverse problems
Inverse problems are ubiquitous in nature: Animals like bats navigate with sonar, medical applications like X-ray tomography allow for non-invasive measurements and diagnoses and spectroscopy methods allow to indirectly determine the composition of chemical substances. In all of these examples one is interested in reconstructing information in various settings from physics, engineering or medicine for which only indirect, non-invasive measurements are available. The theory of inverse problems for which unique continuation results yield important tools allows one to deduce such results in a mathematically precise way.
Free boundary value problems and their regularity properties
Free boundary value problems arise in many processes in the sciences and our daily lives. A prototypical example for instance includes the melting of ice in water, the so-called Stefan Problem. While there are evolution equations for the ice and the water, the interface of the ice and the water is not explicitly given in the description of this problem. It is part of the problem to determine, to predict and to analyze its evolution. This makes these problems very «nonlinear» and poses interesting mathematical challenges in particular in terms of the regularity theory of the solutions and interfaces.
Problems from the calculus of variations in particular the dichotomy between rigidity and ﬂexibility in the modelling of shape-memory alloys
Shape-memory alloys are materials with a thermodynamically very interesting behaviour making them promising materials for a number of industrial applications. Just as in the melting of ice to water, these materials also undergo a phase transformation, however from one solid state with very high symmetry at high temperatures to another solid state with less symmetry at low temperatures. This loss of symmetry implies that the materials have different variants of their low temperature state and are thus very ﬂexible. When heating up the material again, it loses this ﬂexibility and has to recover its original shape – it has «a memory». Mathematically, the presence of multiple variants of the low energy states provides these materials with a rich energy landscape leading to a variety of different microstructures. It has been one of the objectives of the research group «Rigidity and Flexibility in PDEs» to study their rigidity and ﬂexibility properties.
- Angkana Rüland, Mikko Salo, The fractional Calderón problem: low regularity and stability, Nonlinear analysis / A, 193 (2020), 111529.
- Angkana Rüland, Jamie M. Taylor, and Christian Zillinger, Convex integration arisingin the modelling of shape-memory alloys : some remarks on rigidity, ﬂexibility and some numerical implementations, Journal of nonlinear science, 29 (2019) 5, p. 2137-2184.
- Herbert Koch, Angkana Rüland, Wenhui Shi, The Variable Coeffcient Thin Obstacle Problem: Carleman Inequalities, Advances in Mathematics, Vol 301 (2016), pp.820-866.
Future proof — Tim Laux appointed Bonn Junior Fellow (07.02.2020)
We would like to congratulate our former IMPRS and PhD student Tim Laux on his appointment as Bonn Junior Fellow at the Hausdorff Center for Mathematics (HCM) in January of this year. He continues to be interested in problems of geometric analysis with his current research group on variational methods and mathematical aspects of materials science. At the end of July Tim will be hosting his first workshop at the HCM on the topic of geometric and applied analysis (registration is open until May 31, respectively March 31, 2020, for funding opportunities).
Tim wrote his dissertation under the supervision of Felix Otto on the topic of "Convergence of phase-field models and thresholding schemes via the gradient-flow structure of multi-phase mean-curvature flow". In 2018 he was awarded the prestigious Otto Hahn medal for his outstanding performance as a junior scientist. After his post-doc at the institute, he became a Morrey Visiting Assistant Professor at the University of California, Berkeley.
- Future proof — Mateusz Michałek (19.10.2020)
- Future proof — ERC Starting Grants for Martina Hofmanová and Julian Fischer (07.09.2020)
- Future proof — South China Sea Masters and Glushko dissertation prize awarded to former MiS Ph.D. students (23.04.2020)
- Future proof — Angkana Rüland (18.03.2020)
- Future proof — Tim Laux appointed Bonn Junior Fellow (07.02.2020)