The AGC seminar at Aalto aims to provide talks relating broadly to algebra, geometry, combinatorics and their interplay. The seminar is usually held live at the department in Aalto Undergraduate Centre on Mondays starting at 14:15.
The seminar is organised by the research groups of Oscar Kivinen and Kaie Kubjas. If you're interested in giving a talk send an email to the current managers
emilia[dot]takanen[at]aalto[dot]fi
nataliia[dot]kushnerchuk[at]aalto[dot]fi
You can find our old webpage on Google Sites.
This is an expository talk about polytopes. I will introduce what polytopes are and how to construct them. I will talk about faces, face lattices, f-vectors and duality of polytopes and what is meant by the "combinatorics of polytopes". I will talk about some classical results and conjectures on polytopes and how to turn combinatorial objects into polytopes. Everyone is welcome.
The first part of the talk is a high level "What is?" talk introducing homological algebra with a view towards algebraic geometry, a setting where there is a contravariant algebra of functions accompanying the topology. There will however also be examples from more familiar topologies to introduce simplicial sets. We will finish with an explanation of basic symplectic geometry from this viewpoint. The second part of the talk is an introduction to my line of research for the purposes of a midterm review. The previous part will be used to explain the intuition behind derived algebraic geometry and how symplectic geometry naturally lifts to this setting as shifted symplectic geometry. The (-1)-shifted symplectic setting has a nice local model and a natural connection to Milnor fibres and vanishing cycles. The motivic incarnations of these objects give well behaving invariants that have in some cases been shown to be functorial w.r.t. correspondences. Finally we show how character varieties of 3-manifolds have a natural (-1)-shifted symplectic structure and how this could be leveraged to obtain motivic invariants of 3-manifolds.
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