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### Session S35 - Moduli Spaces in Algebraic Geometry and Applications

Thursday, July 15, 15:20 ~ 15:40 UTC-3

## Moduli spaces of quiver representations and an open conjecture

### Kaveh Mousavand

#### Queen's University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakdd8e5d99deb8e2fae7978a2b77540a13').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addydd8e5d99deb8e2fae7978a2b77540a13 = 'm&#111;&#117;s&#97;v&#97;nd.k&#97;v&#101;h' + '&#64;'; addydd8e5d99deb8e2fae7978a2b77540a13 = addydd8e5d99deb8e2fae7978a2b77540a13 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_textdd8e5d99deb8e2fae7978a2b77540a13 = 'm&#111;&#117;s&#97;v&#97;nd.k&#97;v&#101;h' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloakdd8e5d99deb8e2fae7978a2b77540a13').innerHTML += '<a ' + path + '\'' + prefix + ':' + addydd8e5d99deb8e2fae7978a2b77540a13 + '\'>'+addy_textdd8e5d99deb8e2fae7978a2b77540a13+'<\/a>';

In 2014, Adachi-Iyama-Reiten introduced $\tau$-tilting theory of finite dimensional algebras as a modern generalization of tilting theory. The subject was primarily motivated by the notion of mutation in cluster algebras. It soon received a lot of attention and became an active area of research. Around the same time, Chindris-Kinser-Weyman studied the moduli spaces of quiver representations, in particular the behaviour of Schur representations of finite dimensional algebras. In this talk, we relate these two subjects. More specifically, we propose a conjectural geometric counterpart for the algebraic notion of $\tau$-tilting finiteness. For the sake of tractability, we avoid the technicality of $\tau$-tilting theory and state our conjecture as a modern analogue of the celebrated Brauer-Thrall conjectures. We verify the conjecture for some families of algebras and outline some strategies to treat the general case via a reductive argument.

Joint work with Charles Paquette (Royal Military College, Canada).