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Abstracts for Learning seminar on deformation theory

Alternatively have a look at the program.

Deformations of complex structures and associative algebras

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Speaker: 
Sylvain Lavau
Datum: 
Don, 2018-10-11 10:00 - 12:00
Location: 
MPIM Seminar Room

Deformation problems

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Speaker: 
Alex Betts
Datum: 
Don, 2018-10-18 10:00 - 12:00
Location: 
MPIM Seminar Room

In this talk we will continue the introduction to deformation problems by taking a different point of view on them:

We will discuss local artinian rings, and how some natural deformation problems can be posed as functors from local artinian rings to sets, or better, to simplicial sets. We will also see what kind of properties such functors satisfy. The talk will include a reminder on the needed definitions and not assume previous knowledge of simplicial sets.

Deformation functors - a modern approach

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Speaker: 
João Nuno Mestre
Datum: 
Don, 2018-10-25 10:30 - 12:30
Location: 
MPIM Seminar Room

We will continue the study of deformation problems by getting to a modern defintion of a deformation functor. These are functors from local Artinian k-algebras to simplicial sets. We will see how the Maurer-Cartan equation provides a such a deformation functor out of a dgLa.

The talk includes an introduction to the necessary tools on simplicial sets.

The Chevalley-Eilenberg complex of a dgLa

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Speaker: 
Joost Nuiten
Datum: 
Don, 2018-11-08 10:30 - 12:30
Location: 
MPIM Seminar Room

We describe how to associate to any differential graded Lie algebra g a differential graded coalgebra C(g), whose underlying complex is the Chevalley-Eilenberg complex. We then use this construction to give a new description of the Maurer-Cartan elements of g.

The model category dgLa

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Speaker: 
Lory Kadiyan
Datum: 
Don, 2018-11-15 10:30 - 12:30
Location: 
MPIM Seminar Room

In this talk we will see what is a model category structure, describe a model structure on chain complexes, and how to transfer it to a model category structure on dgLa's via a free-forgetful adjunction.

Relating dgLa and cdga: $C^*$ and its adjoint

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Speaker: 
João Nuno Mestre
Datum: 
Don, 2018-11-22 10:30 - 12:30
Location: 
MPIM Seminar Room

In this talk we will see that the functor $C^*$ from dgLa to cdga admits a right adjoint up to homotopy $D$. In order to understand $D$, we will use the free-forgetful adjunction between dgLa and chain complexes: We will see that up to homotopy, $C^*$ composed with the free functor $F$ has a particularly simple form, which in the following week will let us write a simple expression for $D$ (up to homotopy, when computed on certain nice enough algebras).

Relating dgLa and cdga: $C^*$ and its adjoint. Part 2

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Speaker: 
Sylvain Lavau
Datum: 
Don, 2018-11-29 10:30 - 12:30
Location: 
MPIM Seminar Room

In the previous talk we studied the existence of a right adjoint $D$ to the cohomological Chevalley-Eilenberg  functor $C^*$ between the $\infty$-categories $dgLa[W^{-1}]$ and $(cdga^{aug}[W^{-1}])^{op}$. We also discussed that $C^*$ applied to free dgLa's is (homotopically) very simple: It is equivalent to applying the functor that sends a cochain complex $V$ to the augmented cdga $k\oplus V^*[-1]$, equipped with the product determined by $(1,a)(1,b)=(1,a+b)$.

The main theorem

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Speaker: 
Christian Blohmann
Datum: 
Don, 2018-12-06 10:30 - 12:30
Location: 
MPIM Seminar Room

Proof of the main theorem

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Speaker: 
David Carchedi
Datum: 
Don, 2018-12-13 10:30 - 12:30
Location: 
MPIM Seminar Room

Deformations as Maurer-Cartan elements

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Speaker: 
Joost Nuiten
Datum: 
Don, 2018-12-20 10:30 - 12:30
Location: 
MPIM Seminar Room
In the previous talks we have discussed the equivalence between formal moduli problems and dg-Lie algebras. In this talk, we will see how this equivalence relates to the more classical description of deformation problems via solutions to the Maurer-Cartan equation. If time permits, we will also discuss the formal moduli problem describing deformations of algebras.
 
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