Algebraic Statistics and the Study of Multistate Models: Theory and Applications, April 1-4, 2025, Aalto and U. Helsinki
Order polytopes and chain polytopes, associated with partially ordered sets, were introduced by Stanley in 1986. In 2016, Hibi and Li proposed the following conjecture concerning the number of faces of these polytopes in each dimension: (a) For any dimension i (≥1), the number of i-dimensional faces of an order polytope does not exceed that of the corresponding chain polytope. (b) If the numbers of i-dimensional faces of both polytopes coincide for some i (≥1), then the two polytopes are unimodularly equivalent. In this talk, we will provide an overview of the current progress on this conjecture and discuss results obtained specifically for faces of simplices.
I will report on two recent papers establishing some geometric and categorical properties of character sheaves. In one, joint with Gonin and Ionov, we give a new proof and an extension to non-unipotent setting of the t-exactness of the composition of Radon and Harish-Chandra transforms for character sheaves. In another, joint with Bezrukavnikov, Ionov and Varshavsky, we show that this can be used to compute the Drinfeld center of the abelian Hecke category attached to the same reductive group.
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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