About lievenlb

Also blogs at NeverEndingBooks.

antwerp algebra seminar november-december 2012

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

Antwerp Algebra Seminar – october 2012

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

Friday october 5th at 16h in room G.005 (Middelheim campus), lecture by
Boris Shoikhet (UA) : “Deligne conjecture for higher-monoidal categories”

Friday october 12th at 16h in room G.005 (Middelheim campus), lecture by
Liyu Liu (UA) : “Twisted Calabi-Yau property of skew-polynomial extensions”

Friday october 19th at 16h in the hall of building G (Middelheim campus),
drink on the occasion of the retirement of Fred Van Oystaeyen and Guy Van Steen

Friday october 26th, one-day workshop in honor of Fred Van Oystaeyen’s 65th birthday
organizers : S. Caenepeel (VUB) and Y. Zhang (UHasselt), details follow

Friday november 2nd : university closed, no seminar.

With kind regards,

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

Prep-notes dump

Here are the scans of my rough prep-notes for some of the later seminar-talks. These notes still contain mistakes, most of them were corrected during the talks. So, please, read these notes with both mercy are caution!

Hurwitz formula imples ABC : The proof of Smirnov’s argument, but modified so that one doesn’t require an $\epsilon$-term. This is known to be impossible in the number-theory case, but a possible explanation might be that not all of the Smirnov-maps $q~:~\mathsf{Spec}(\mathbb{Z}) \rightarrow \mathbb{P}^1_{\mathbb{F}_1}$ are actually covers.

Frobenius lifts and representation rings : Faithfully flat descent allows us to view torsion-free $\mathbb{Z}$-rings with a family of commuting Frobenius lifts (aka $\lambda$-rings) as algebras over the field with one element $\mathbb{F}_1$. We give several examples including the two structures on $\mathbb{Z}[x]$ and Adams operations as Frobenius lifts on representation rings $R(G)$ of finite groups. We give an example that this extra structure may separate groups having the same character table. In general this is not the case, the magic Google search term is ‘Brauer pairs’.

Big Witt vectors and Burnside rings : Because the big Witt vectors functor $W(-)$ is adjoint to the tensor-functor $- \otimes_{\mathbb{F}_1} \mathbb{Z}$ we can view the geometrical object associated to $W(A)$ as the $\mathbb{F}_1$-scheme determined by the arithmetical scheme with coordinate ring $A$. We describe the construction of $\Lambda(A)$ and describe the relation between $W(\mathbb{Z})$ and the (completion of the) Burnside ring of the infinite cyclic group.

Density theorems and the Galois-site of $\mathbb{F}_1$ : We recall standard density theorems (Frobenius, Chebotarev) in number theory and use them in combination with the Kronecker-Weber theorem to prove the result due to James Borger and Bart de Smit on the etale site of $\mathsf{Spec}(\mathbb{F}_1)$.

New geometry coming from $\mathbb{F}_1$ : This is a more speculative talk trying to determine what new features come up when we view an arithmetic scheme over $\mathbb{F}_1$. It touches on the geometric meaning of dual-coalgebras, the Habiro-structure sheaf and Habiro-topology associated to $\mathbb{P}^1_{\mathbb{Z}}$ and tries to extend these notions to more general settings. These scans are unintentionally made mysterious by the fact that the bottom part is blacked out (due to the fact they got really wet and dried horribly). In case you want more info, contact me.