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Archive : Discrete Mathematics

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Irene Villa, University of Trento, ItalyIrene Villa, University of Trento, ItalyZoom:<br> <br> Title: The isotopic shift for APN and planar functions <br> <br> Abstract: In the study of functions over finite fields with good (cryptographic) properties, much interest is put in the analysis, construction and classification of APN and planar maps. In this talk we give an overview of APN functions, planar functions and their connection. Then we consider a recently introduced method for constructing these maps, the isotopic shift construction. The isotopic shift idea was originated in the setting of quadratic planar functions and then it was applied to APN functions. We discuss some properties of the isotopic shift and present some results obtained using this construction method.4/7/2021 5:20:00 PM4/7/2021 6:20:00 PMFalse
Bhargav Narayanan, Rutgers UniversityBhargav Narayanan, Rutgers UniversityZoomTitle: Friendly bisections of random graphs Abstract: Can we partition the vertices of a random graph G(n,1/2) into two parts of equal size in which almost all the vertices have more neighbours on their own side than across? Furedi conjectured in 1988 that we can, and in this talk, I will finally show that this is indeed the case. Joint work with Ferber, Kwan, Sah and Sawhney3/31/2021 5:20:00 PM3/31/2021 6:20:00 PMFalse
Nikolay Kaleyski, University of Bergen, NorwayNikolay Kaleyski, University of Bergen, NorwayZoom<br> Title: The minimum distance between APN functions <br> Abstract: Almost perfect nonlinear (APN) functions are defined to be the ones that offer the best possible resistance to differential cryptanalysis. Despite the primary motivation for the study coming from the design of cryptographic ciphers, APN functions have a much wider significance and correspond to optimal objects in various other areas of study, including coding theory, combinatorics, projective geometry, etc. Finding new constructions and characterizing the properties of such functions is, therefore, an important direction of research, but it is also very difficult since the cryptographic optimality of APN functions translates to an absence of apparent structure or patterns in them. In this talk, we discuss the problem of finding a lower bound on the Hamming distance between APN functions. This has a direct connection to other important open problems in the area (most notably, the maximum value of the algebraic degree of an APN function), and to certainmethods of constructing new APN functions from known ones. We recall anefficiently computable lower bound on this distance introduced previously in our work, and discuss its theoretical and practical implications. We give a survey of our computational data and overview some promising directions for future work on the topic.3/24/2021 5:20:00 PM3/24/2021 6:20:00 PMFalse
Cynthia Vinzant, North Carolina State and IASCynthia Vinzant, North Carolina State and IASZoom<br> Title: Log-concavity, matroids, and expanders <br> Abstract: Matroids are combinatorial objects that model various types of independence. They appear several fields mathematics, including graph theory, combinatorial optimization, and algebraic geometry. In this talk, I will introduce the theory of matroids along with the closely related class of polynomials called strongly log-concave polynomials. Strong log-concavity is a functional property of a real multivariate polynomial that translates to useful conditions on its coefficients. Closed related classes are real stable and Lorentzian polynomials. Discrete probability distributions defined by these coefficients inherit several of these nice properties. I will discuss the beautiful real and combinatorial geometry underlying these polynomials and describe applications to random walks on the faces of simplicial complexes. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. 3/17/2021 5:20:00 PM3/17/2021 6:20:00 PMFalse

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