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SUMMARY:Dmitriy Voloshyn (IBS Center for Geometry and Physics)
DTSTART:20240503T013000Z
DTEND:20240503T030000Z
DTSTAMP:20260423T010005Z
UID:gapkias/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/gapkias/11/"
 >Cluster algebras and Poisson geometry</a>\nby Dmitriy Voloshyn (IBS Cente
 r for Geometry and Physics) as part of Geometry\, Algebra and Physics at K
 IAS\n\nLecture held in Room 1423\, Korea Institute for Advanced Study.\n\n
 Abstract\nCluster algebras are commutative rings with distinguished sets o
 f generators characterized by a remarkable combinatorial structure. Discov
 ered by S. Fomin and A. Zelevinsky in the early 2000s\, these algebraic st
 ructures have found applications across diverse mathematical fields\, incl
 uding integrable systems\, total positivity\, Teichmüller theory\, Poisso
 n geometry\, knot theory and mathematical physics.\n  Fomin and Zelevinsky
  conjectured that numerous varieties in Lie theory are equipped with a clu
 ster structure. Early examples include double Bruhat cells\, Grassmannians
  and simple complex algebraic groups. M. Gekhtman\, M. Shapiro and A. Vain
 shtein observed that cluster algebras in these examples are compatible wit
 h certain Poisson brackets. Specifically\, for any given cluster $𝑥_1\,
 𝑥_2\,...\,𝑥_n$\, there exist constants 𝟂ij such that $\\{𝑥_i\,
 𝑥_j\\} = \\omega_{ij} 𝑥_i 𝑥_j$. This observation led to a program
  aiming to construct cluster algebras by addressing the inverse problem: g
 iven a Poisson bracket in the coordinate ring of an algebraic variety and 
 a collection of regular functions $(𝑥_1\,𝑥_2\,...\,𝑥_n)$ satisfyi
 ng $\\{𝑥_i\,𝑥_j\\} = \\omega_{ij} 𝑥_i 𝑥_j$\, does there exist 
 a compatible cluster algebra? The research initiative led to the formulati
 on of the GSV conjecture: for a given simple complex algebraic group and a
  Poisson bracket from the Belavin-Drinfeld class\, there exists a compatib
 le cluster structure.\n  The plan for the talk is as follows. First\, we w
 ill discuss an example of a cluster structure on ${\\rm GL}_3(\\mathbb{C})
 $. Then we will explore the connection between cluster algebras and Poisso
 n geometry\, as well as discuss how to construct a cluster structure compa
 tible with a Poisson bracket. After that\, we will discuss the recent resu
 lts on the three main families of objects: simple connected simple complex
  algebraic groups\, their Drinfeld doubles and their Poisson duals.\n
LOCATION:https://researchseminars.org/talk/gapkias/11/
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