Algebraic Statistics and the Study of Multistate Models: Theory and Applications, April 1-4, 2025, Aalto and U. Helsinki
This Zoom seminar is also watchable in M2! Tensors are higher-dimensional generalisations of matrices, and likewise the main notion of complexity on matrices - their rank - may be extended to tensors. Unlike in the matrix case however, there is no single canonical notion of rank for tensors, and the most suitable notion often depends on the application that one has in mind. The most frequently used notion so far has been the tensor rank (hence its name), but several other notions and their applications have blossomed in recent years, such as the slice rank, partition rank, analytic rank, subrank, and geometric rank. Unlike their counterparts for the rank of matrices, many of the basic properties of the ranks of tensors are still not well understood. After reviewing the definitions of several of these rank notions, I will present a number of results of a type that arises in many cases when one attempts to generalise a basic property of the rank of matrices to these ranks of tensors: the naive extension of the original property fails, but it admits a rectification which is simultaneously not too complicated to state and in a spirit that is very close to that of the original property from the matrix case. Link to Zoom: https://aalto.zoom.us/j/66860024175
We classify arrangements of three ellipsoids in space up to rigid isotopy classes, focusing on nondegenerate configurations that avoid singular intersections. Our approach begins with a combinatorial description of differentiable closed curves on the projective plane that intersect a given arrangement of lines transversally. This framework allows us to label classes of spectral curves associated with ellipsoid configurations, which are real plane quartic curves. We determine necessary and sufficient conditions for these classes to be inhabited through arguments coming from linear algebra, algebraic geometry, combinatorics, and by computations in Mathematica and Macaulay2.
Group Testing is an area in information and communication sciences that is as well-established as Coding Theory and Cryptography. The author of this talk stumbled over this amazingly interesting topic during the recent COVID-19 pandemic and came to the moderately surprising observation that (non-adaptive) group testing in both the noiseless and the noisy (=error-correcting) case, may be considered as coding theory over the Boolean semi-field (1+1=1). Following this path, he discovered new and re-discovered known results of the theory that now allow for a presentation in a new skin. This talk will delve into the topic and show how Noiseless and Noisy Group Testing can be connected to Partially Ordered Sets, Residuation, Partial Linear Spaces, Configurations, Barbilian Spaces, and Block Designs, which gives raise to further applications of Finite Geometry and Order Theory.
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