One Board Seminar

Talks

Arrow's Impossibility Theorem as a combinatorial game , Erlend D. Børve (University of Graz).

The colorblind Snow White and the p dwarfs., Barna Schefler (Eötvös Loránd University).

Who cares about sheaves?, Alexander Zahrer (University of Graz).

Tensor extriangulated categories , Raphael Bennett-Tennenhaus (University of Bielefeld).

Viewing matrices as graphs , Özgür Esentepe (University of Graz).

Inequalities and nonlinear flows, Gaspard Jankowiak (University of Graz).
Affine toric varieties, Nebojsa Pavic (University of Graz).
An algebraic variety from inverse problems for elastic materials, Daniel Windisch (University of Edinburgh).
On dual objects , Zahra Nazemian (University of Graz).

Reflexive modules over Arf local rings , Özgür Esentepe (University of Graz).

The first email

This is the email that started the seminar:

Hello everyone, I would like to start a seminar on Mondays called the One Board Seminar.

The seminar will take place in my office. I have one blackboard, 9 available seats and a Turkish teapot. We will make tea, eat snacks (oder mittagessen) and one person will speak on the board. There is no time limit and you can talk about anything you would like (it could be about your research, it could be something you proved, it could be something you could not prove, it could be something you thought you proved but there was a mistake and you can tell us about the mistake, it could be about some cool math fact you learned unrelated to your research, it could be a cool problem from one of the courses you are teaching, it could be about your grandmother's soup recipe). But there will be one constraint: you are not allowed to erase the board.

We will make tea and serve tea throughout the talk, you can bring snacks for yourself or for everyone, you can also bring your lunch if you wish. Currently I am thinking 12:30 on Monday but if this does not work for you, please let me know.

Please share with people who might be interested. Anyone who does not have a permanent position is welcome :) I am not sending this email to everyone but please share with your officemates etc.

I can start the first talk.

Abstracts

Speaker: Özgür Esentepe (University of Graz)
Date: November 11, 2024
Title: Reflexive modules over Arf local rings
Abstract: From linear algebra, we know that if V is a finite dimensional vector space over a field k, then the dual space V*=Hom_(V,k) is isomorphic to V. This would also tell us that the double dual V**=(V*)* is isomorphic to V. One particular isomorphism from V to V** is the natural map taking v to the linear map F k which sends a linear transformation f to f(v). This does not hold true when the ground ring is not a field. Given a finitely generated module M over a commutative Noetherian ring R, we may have that the natural map from M to M** is not an R-linear isomorphism. But sometimes it happens and we call such modules reflexive modules. They have nice properties and are important things to study for people from different areas. However, M being isomorphic to M** does not imply that M is isomorphic to M*. In this talk, I will tell you exactly when this happens for one dimensional rings.

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Speaker: Zahra Nazemian (University of Graz)
Date: November 18, 2024
Title: On dual objects
Abstract: Let $R$ be any ring and $M$ any left $R$-module. We call $M$ a dual module if $_R M \cong \operatorname{Hom}(N, R)$ for some right $R$-module $N$. Denoting $N^\star := \operatorname{Hom}(N, R)$, we show that if $M$ is dual, then the natural map $\sigma_M: M \to M^{\star \star}$ is injective if and only if it is a section map. We then discuss the conditions under which $\sigma_M$ is an isomorphism. Finally, we explore the possibility of generalizing this concept to certain classes of monoidal categories.

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Speaker: Daniel Windisch (University of Edinburgh)
Date: November 25, 2024
Title: An algebraic variety from inverse problems for elastic materials
Abstract: Recovering the constitution of an elastic material from partial measurements on the material while it is deforming is a classical inverse problem. I will introduce an algebraic variety coming up in this area, the so-called slowness hypersurface. We will discuss how this variety is studied in the literature and how its properties are used for the original problem.

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Speaker: Nebojsa Pavic (University of Graz)
Date: December 9, 2024
Title: Affine toric varieties
Abstract: This talk will be a short introduction to affine toric varieties. We are going to define affine toric varieties as affine varieties containing a torus with as a (Zariski) dense open subset, such that its action on itself extends to the whole affine variety. We then give an alternative definition in terms of strongly convex polyhedral rational cones.

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Speaker: Gaspard Jankowiak (University of Graz)
Date: January 13, 2025
Title: Inequalities and nonlinear flows
Abstract: I will introduce the so-called fast diffusion equation, and discuss its links to two families of functional inequalities, which are dual in some sense. The focus will be on optimal constants.

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Speaker: Özgür Esentepe (University of Graz)
Date: January 27, 2025
Title: Viewing matrices as graphs
Abstract: Some time ago, I read in Tai-Danae Bradley's blog that one can see matrices as graphs. I thought it was interesting and I gave it as a project to some linear algebra students during the pandemic. It seems that I almost forgot how it works so I am going to refresh my memory by talking with you about her ideas. The goal is to be able to show how matrix multiplication works.

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Speaker: Raphael Bennett-Tennenhaus (University of Bielefeld)
Date: March 11, 2025
Title: Tensor extriangulated categories
Abstract: The unbounded derived category of complexes of all modules over a non-artinian ring, equipped with the pure exact triangles, is an example of a tensor extriangulated category that is neither exact nor triangulated. I will explain what this means and why this is true. The talk is based on a paper on the arxiv 2502.18257.

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Speaker: Alexander Zahrer (University of Graz)
Date: March 17, 2025
Title: Who cares about sheaves?
Abstract: I have an absurdly general question for you... intended, melodramatic pause... What is a space? You are free to scream, at the top of your lungs, terms like "topological spaces", "euclidean spaces", "manifolds", "algebraic varieties", "schemes", "measure spaces", etc. and that would be somewhat satisfying answers. I propose the following more loose answer: A space is something that is coherently glued together from some given building bricks in such a way so as to satisfy a local-to-global property (e.g.a manifold is patched together from euclidean patches, just like any scheme may be constructed by glueing together certain affine schemes). We will try to motivate this line of thought, which will lead us to the concept of a presheaf and then in turn to the notion of a sheaf. The goal of this talk is to convince you that the answer to the above is, in fact, sheaves (and more generally stacks).

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Speaker: Barna Schaffler (Eötvös Loránd University)
Date: March 31, 2025
Title: The colorblind Snow White and the p dwarfs.
Abstract: Consider some disks divided into p circular sectors such that each sector is coloured either red or green. The p dwarfs have all the different disks (up to cyclic rotation). Unfortunately, Snow White is colour blind and she can not differentiate between the colours red and green, so all the disks are the same for her. The dwarfs play a game to help Snow White to identify the disks.

Speaker: Erlend D. Børve (University of Graz)
Date: April 07, 2025
Title: Arrow's Impossibility Theorem as a combinatorial game.
Abstract: Arrow's celebrated Impossibility Theorem (1951) is central to Social Choice Theory. It proves that no ranking-based voting system may satisfy three reasonable criteria simultaneously. Subsequent work of Chichilinsky (1979), Baryshnikov (1993), and Tanaka (2007) re-asserted Arrow's insights in topological terms, ultimately proving that Arrow's Impossibility Theorem is equivalent to Brouwer's Fixed Point Theorem. In this one-board talk, we derive Arrow's Impossibility Theorem from Sperner's Lemma, one of the many results known to be equivalent to Brouwer's Fixed Point Theorem. The speaker will pursue his dictatorial ambitions by playing a combinatorial game with the audience. Under his rule, a recent note of Miku [arXiv:2212.12251] will be disambiguated.