Emmy Noether research group - Numerical and Probabilistic Nonlinear Algebra
Paul Breiding (Email)
+49 (0) 341 - 9959 - 770
+49 (0) 341 - 9959 - 658
The goal of this group is to develop and establish numerical and probabilistic methods for problems in nonlinear algebra. The focus of the group lies both on the theoretical foundations and on applications in the sciences. We aim to solve complicated problems in nonlinear algebra by applying mathematical theories and developing dedicated software.
A sample of points from the state space of the molecule cyclooctane (C8H16), projected on a three-dimensional space. Code and image courtesy of Paul Breiding and Sascha Timme.
An important task in the mathematics of data is to identify geometric structures underlying the data. When the geometry of data is defined by polynomials, methods from nonlinear algebra can exploit the algebraic structure to extract information. An example of this is molecular geometry: the state space of a molecule is defined by constraints on the distances between single atoms. These distances give rise to polynomial equalities and inequalities. Numerical and probabilistic methods in nonlinear algebra can be used to explore such state spaces and to compute quantities of interest. This is an example from the sciences . In this group we want to study another example: algebraic structures in computer vision.
Numerical and probabilistic methods in nonlinear algebra can also be applied in more theoretical contexts. An example of this is the article , where we contribute to the so-called Steiner's conic problem. For this, the application of numerical algorithms was essential for obtaining rigorous proofs. Furthermore, considering probability in algebra provides new insights about the geometric structure of problems. In this group we want to study the classical subject of Schubert Calculus from the probabilistic perspective.
I did my PhD (2017) at Technische Universität Berlin under the supervision of Peter Bürgisser. Before, I did my Masters (2013) at the University of Göttingen under the supervision of Preda Mihailescu.
For my Master thesis I studied number theory, but for my PhD I moved to numerical mathematics, probability and algebraic geometry. Since then I have been working on a variety of different topics including: sensitivity of tensor decompositions, eigenvalues of random tensors, solving systems of polynomial equations, sampling probability distributions, and topology of random algebraic varieties. In general, I enjoy combining methods and ideas from different fields in my research.
I did my PhD (2021) and Masters (2018) at the University of Washington advised by Jayadev Athreya.
I enjoy working with math where I can draw pictures. My research thus far has used dynamical, geometric, algebraic, and analytic techniques to study translation surfaces.Translation surfaces are a collection of polygons in the plane with parallel sides identified by translation to form a surface with a singular Euclidean structure. Understanding the geometry and behavior of flows on translations surfaces and their moduli spaces is a rich research area filled with mathematicians of all backgrounds. In particular I work on understanding counting and pair correlations for the set of closed geodesics on translation surfaces.
In my free time I love the outdoors and animals. So I can usually be found horseback riding, hiking with my partner Max, or hanging out with my two cats.
I completed my Master degree in Mathematical Engineering at Polytechnic University of Tirana in 2019. In my master thesis I studied numerical methods for solving nonlinear PDEs.
My research interests involve Applied Algebraic Geometry and Numerical Nonlinear Algebra. I will start my PhD on "Sensitivity in Computer Vision" at the MPI MiS in April 2021.