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Abstracts for Seminar on Higher Structures

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What is higher Lie theory?

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Speaker: 
Dmitry Roytenberg
Affiliation: 
Utrecht/Amsterdam
Date: 
Fri, 08/01/2016 - 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Higher Structures

Lie theory" refers to a functorial correspondence between
finite-dimensional Lie algebras and 1-connected Lie groups, elucidated
by Sophus Lie and summarized in his three theorems, Lie I, II and III.
The first hint that there might be more to the story came from van Est
who noticed (and proved) that integrating higher Lie algebra cocycles
required higher connectivity assumptions on the Lie group. Furthermore,
attempts to build a Lie theory for Lie algebroids (and
infinite-dimensional Lie algebras) generally run into obstructions of a

Generalized cohomology of M2/M5-branes

Posted in
Speaker: 
Urs Schreiber
Affiliation: 
CAS Prague/MPIM
Date: 
Fri, 15/01/2016 - 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Higher Structures

While it is well-known that the charges of F1/Dp-branes in type II string theory need to be refined from de Rham cohomology to certain twisted generalized differential cohomology theories, it is an open problem to determine the generalized cohomology theory for M2-brane/M5-branes in 11 dimensions. I discuss how a careful re-analysis of the old brane scan (arXiv:1308.5264 , arXiv:1506.07557, joint with Fiorenza and Sati) shows that rationally and unstably, the M2/M5 brane charge is in degree-4 cohomotopy.

$L_\infty$ algebras governing simultaneous deformations in algebra and geometry

Posted in
Speaker: 
Yaël Frégier
Affiliation: 
U d'Artois/MPIM
Date: 
Fri, 29/01/2016 - 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Higher Structures

We will explain how simultaneous deformations problems in algebra and geometry are naturally governed by non quadratic $L_\infty$ algebras and how such algebras can be constructed by super geometry or operad theory, depending on the applications.

 This will be illustrated with examples. We will consider simultaneous deformations of pairs such as couples of algebras/morphisms, coisotropic/Poisson, Dirac/Courant, generalized complex/Courant (if time permits).

Open-closed homotopy algebras

Posted in
Speaker: 
Martin Doubek
Affiliation: 
Charles Univ. Prague/MPIM
Date: 
Fri, 19/02/2016 - 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Higher Structures

Topological types of string worldsheets can be organised into a modular operad QP. For closed strings, algebras over this operad are essentially topological quantum field theories formalized by monoidal functors from cobordisms into vector spaces or by commutative Frobenius algebras. There is an analogous picture for open and open/closed strings. In the closed case, QP can be freely generated from certain cyclic suboperad P via modular envelope: in fact P=Com. In the open case, analogous result holds for P=Ass. In open-closed case, this problem is subtler.

Rational homotopy of the little cubes operads and graph complexes

Posted in
Speaker: 
Benoit Fresse
Affiliation: 
Université de Lille/MPIM
Date: 
Fri, 26/02/2016 - 14:00 - 15:00
Location: 
MPIM Seminar Room
Parent event: 
Seminar on Higher Structures

I will report on a joint work with Victor Turchin and Thomas Willwacher about the rational homotopy of the little cubes operads (equivalently, the little discs operads).

Ginzburg--Kapranov criterion for Koszulness of operads, and its limitations

Posted in
Speaker: 
Vladimir Dotsenko
Affiliation: 
Trinity College Dublin
Date: 
Fri, 04/03/2016 - 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Higher Structures

In their seminal paper on operadic Koszul duality, Ginzburg
and Kapranov established a remarkable functional equation that holds
whenever an operad is Koszul. I shall discuss some examples
demonstrating that non-Koszul operads do not have to violate that
criterion, in either of two possible ("weak" or "strong") senses.

What good is a semi-model structure on algebras over an operad?

Posted in
Speaker: 
David White
Affiliation: 
Denison University, Granville
Date: 
Fri, 11/03/2016 - 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Higher Structures

In general, it is difficult to transfer a model structure from a monoidal
model category M to the category of algebras over a (colored) operad P in M.
Often, one only ends up with a semi-model structure (i.e. where half the
factorization and lifting axioms only hold for maps with cofibrant domain),
and even this traditionally requires P to be Sigma-cofibrant. I?ll explain
what standard techniques still work in the context of semi-model categories,
so that users of model categories can get by if they only have a semi-model

An Algebraic Combinatorial Approach to Opetopic Structure

Posted in
Speaker: 
Marcelo Fiore
Affiliation: 
U of Cambridge, UK
Date: 
Wed, 23/03/2016 - 10:30 - 12:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Higher Structures

The starting point of this talk will be an algebraic connection to be  presented briefly between
the theory of abstract syntax of [1,2] and the approach to opetopic sets of [4].  This realization
conceptually allows us to transport viewpoints between these mathematical theories and I
will explore it here in the direction of higher-dimensional algebra leading to opetopic
categorical structures.  The technical work will involve setting up a microcosm principle
for near-semirings and subsequently exploiting it in the cartesian closed bicategory of

 

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