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Stream: community: positions

Topic: Postdoc at São Paulo - Brazil

Ana Luiza Tenorio (May 23 2023 at 14:47):

"The Yablonowsky Postdoctoral Fellowship aims to attract young researchers in all areas of Mathematics, including Applied Mathematics, Statistics, and Computer Science who have concluded a doctorate in other institutions of excellence and recognized in the area, to develop research at IME-USP, for a period of up to two years.''
Deadline: Aug 14, 2023

I'm a PhD student at IME-USP under supervision of Hugo Luiz Mariano and he would gladly receive anyone interested in this position. Our group is currently working with notions of sheaves on quantales; non-intuitionistic versions of toposes; categorical approaches to smooth commutative algebra; multirings and the abstract theories of quadratic forms; and more. Your project do not have to be related with those topics.

Morgan Rogers (he/him) (May 24 2023 at 08:01):

Independently of the postdoc, I would be interested in hearing more about the topics you mention!!

Ana Luiza Tenorio (Jul 01 2023 at 16:28):

Hi, @Morgan Rogers (he/him)! Thanks for the interest and sorry for the delay.
I can talk a bit about sheaves on quantales since it is part of my Ph.D. thesis.
In our group we are only using semicartesian quantales ($u \otimes v \leq u,v, \forall u,v \in Q$). This is because if we have a presheaf $F: Q^{op} \to Set$ then we have maps $F(u) \to F(u \otimes v)$ and $F(v) \to F(u \otimes v)$.
So, we say that $F$ is a sheaf on $Q$ if for every cover $u = \bigvee_{i\in I} u_i$ we obtain that $F(u)$ is an equalizer of the parallel arrows $\prod_{i} F(u_i) \to \prod_{i,j} F(u_i \otimes u_j)$ that are almost the same as in the localic case, except that we replace $\wedge$ with the multiplication $\otimes$ of the quantale.

The underlying idea is straightforward: keep the same notion of covering but replace meet with multiplication whenever you want a distributive law. Then we obtain a generalization of the notion of sheaves on locales.
Some consequences: the category of sheaves on quantales $Sh(Q)$ looks like the category of sheaves on locales $Sh(L)$ (some categorical properties still hold) but $Sh(Q)$ is not a topos, because the poset of subobjects of the terminal sheaf in $Sh(Q)$ is isomorphic to the quantale $Q$, which does not have to be a locale/Heyting algebra. In particular, $Sh(Q)$ is not a Grothendieck topos and part of my thesis is about generalizing Grothendieck pretopologies in way that encompass $Sh(Q)$.
I also extended Čech cohomology to define Čech cohomology groups of a commutative ring (the ideals play the role of open subsets of a topological space). I do not have many examples/applications yet.

Two of my colleagues are currently exploring quantale-valued sets and our long-term project aims to develop a linear version of a topos that encompasses both the category of sheaves $Sh(Q)$ I am studying and their category of quantale-valued sets.

The other topics are not related to my work but I can provide a few links: https://www.lajm.ufscar.br/index.php/capa/article/view/5 and https://cgasa.sbu.ac.ir/article_101430_bb0f3ec01ae0ca89d6f8082aef1b5b76.pdf