One Board Seminar

Talks

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.