You are cordially invited to attend the following activities of the Antwerp Algebra Seminar.

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Friday November 9th at 16h in room G0.05 (Middeheim campus) lecture by

Boris Shoikhet (UAntwerpen) : “Introduction to operads”

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Friday November 16th at 16h in room G0.05 (Middelheim campus) lecture by

Marcel Van de Vel (VUAmsterdam & UAntwerpen) : “Applicable Math from Scratch: n-permutron”

abstract : Most random number generators (RNGs) are vulnerable in a cryptographic environment. One of the simplest effective proposals to remedy such defects was made by Donald Knuth in his “Art of Computer Programming”. Our research originated from an attempt to eliminate a potential weakness of this proposal. An $n$-permutron is an $n \times n$ matrix of “digits” $0,1, ..,n-1$, each occurring $n$ times. (Latin squares are a special case of this.) It is operated by “indirection”, requiring $n$-digit input and producing $n$-digit output. The main result is that (for suitable dimensions $n$, including 16 and 32) any input sequence can be turned into any output sequence provided two shuffles are performed before each indirection.

The major steps of the proof have a combinatorial and geometric flavor: optimising a set of $n$ positions in an $n$-permutron, a “spaghetti crumbling effect” on $n$-sets, and finding a “metric partition” in a regular $n$-gon. Some arguments require computer assistance, resulting into tables with clear conclusions or drawings of large metric partitions.

If time permits, we will give some details on how a hexadecimal ($n=16$) and a duotrigesimal ($n=32$) permutron can be implemented. The physical model of an $n$-permutron is a torus with, for each of its two main directions, a system of $n$ rotating rings. Thinking of digits as colors, we obtain a relative of Rubik’s cube with a programmable solution of the generic “restauration problem”.

Our presentation is largely self-contained and can be followed by a general audience.

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Friday November 23rd at 16h in room G0.05 (Middelheim campus) lecture by

Mélanie Raczek (UCLouvain) : “Okubo algebras in characteristic 3 and valuations”

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Friday December 7th at 16h in room G0.05 (Middelheim campus) lecture by

Frédéric Bourgeois (ULBruxelles) : “Bilinearized Legendrian contact homology and the augmentation category”

abstract : We define an $A_\infty$-category associated to a Legendrian submanifold. The objects are augmentations of a differential graded algebra constructed using holomorphic curves. The homology are the morphism spaces form new invariants of Legendrian submanifolds, called bilinearized Legendrian contact homology. This refines the usual linearized Legendrian contact homology defined using the above DGA and an augmentation. This is joint work with Baptiste Chantraine.

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Friday December 14th at 16h in room G0.05 (Middelheim campus) lecture by

Sofie Beke (UGent) : “The Tits index of an algebra with involution under specialisation”

abstract : Let O be a valuation ring of a field F with residue field kappa. We consider a class of separable O-algebras with involution (A,sigma), for which (A_F,sigma_F) and (A_kappa,sigma_kappa) are algebras with involution of either the first or second kind. We are interested in how the isotropy of (A_F,sigma_F) is related to the isotropy of (A_kappa,sigma_kappa). We will indicate how one can use value functions to show that an isotropic right ideal of a certain type of (A_F,sigma_F) yields an isotropic right ideal of the same type of (A_kappa,sigma_kappa). Using the language of the Tits index, this can be phrased as follows: the Tits index of (A_F,sigma_F) is contained in the Tits index of (A_kappa,sigma_kappa). This generalises the behaviour of the Witt index of a quadratic form with good reduction with respect to a place.

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With kind regards,

Wendy Lowen, Karim Becher, Boris Shoikhet and Lieven Le Bruyn