Successful Autumn

Dr. Tim Seynnaeve

Defense: October 28, 2020
Thesis: “Algebraic geometry for tensor networks, matrix multiplication, and flag matroids”
Advisor: Dr. Mateusz Michałek

The first part of Tim’s thesis is focused on tensors from an algebro-geometric point of view. He studies the geometry matrix product states, certain tensor arising in quantum information theory. Furthermore, by using methods from representation theory he contributes towards determining the complexity of matrix multiplication. In the second part of his thesis Tim investigates matroids, which are central to modern combinatorics. He reviews a beautiful connection between matroids (combinatorial objects) and Grassmannians (geometric objects), and generalizes it to flag matroids and flag varieties.

Tim is now working at the Mathematical Institute of the University of Bern, Switzerland.

Dr. Paul Görlach

Defense: November 10, 2020
Thesis: “Projective geometry, toric algebra and tropical computations”
Advisors: Prof. Bernd Sturmfels and Dr. Mateusz Michałek

The thesis is centered around the interplay of projective geometry with combinatorial structures from toric and tropical algebra. It studies low-dimensional injections of projective varieties, corresponding to a dimensionality reduction technique in polynomial system solving. Moreover, it advances computational methods in tropical geometry and uses them to study degenerations of algebraic families to toric settings.

Paul is now working at the Chemnitz University of Technology, Germany.

Dr. Konstantinos Zemas

Defense: November 11, 2020
Thesis: “Geometric rigidity estimates for isometric and conformal maps from \(\mathbb{S}^{n-1}\) to \(\mathbb{R}^n\)”
Advisor: Prof. Stephan Luckhaus

The goal of this thesis is the study of qualitative as well as quantitative stability aspects of isometric and conformal maps from \(\mathbb{S}^{n-1}\) to \(\mathbb{R}^n\). Motivated by the classical rigidity theorem of Liouville on the one hand, and the abundance of such maps in this “codimension-one” setting on the other, stability of rigid motions (respectively Möbius transformations) among maps from \(\mathbb{S}^{n-1}\) to \(\mathbb{R}^n\) is obtained in terms of appropriately defined deficits. Unlike results similar in flavor for maps defined on domains of \(\mathbb{R}^n\) and mapping into \(\mathbb{R}^n\), not only an isometric (respectively conformal) deficit is necessary in this more flexible setting, but also a deficit measuring how much the maps distort \(\mathbb{S}^{n-1}\). The latter is phrased in terms of an isoperimetric type of deficit associated to the maps in consideration.

Konstantinos is now working at the Department of Mathematics of Westfälische-Wilhelms-Universität Münster, Germany.

Dr. Orlando Marigliano

Defense: November 25, 2020
Thesis: “The Algebraic Statistics of Sampling, Likelihood, and Regression”
Advisors: Prof. Bernd Sturmfels and Dr. Christiane Görgen

Orlando's thesis is about statistical models and algebraic varieties. Algebraic Statistics unites these two concepts, turning algebraic structure into statistical insight. The thesis features three types of models that have such an algebraic structure: Linear Gaussian covariance models, Discrete models with rational maximum likelihood estimator, and algebraic manifolds. Orlando closes his thesis with a chapter on mathematics in the sciences featuring case studies from soil ecology and nonparametric statistics.

Orlando is now working at the Math Data Lab at KTH Royal Institute of Technology Stockholm, Sweden.