
15 Jun 2016  17 Jun 2016
8:00 am  5:00 pm
Organisers: Philip Hackney (Universität Osnabrück) and Marcy Robertson (University of Melbourne)
Program
Time  Wednesday, 15 June 2016  Thursday, 16 June 2016  Friday, 17 June 2016 
08:00 – 09:00  Breakfast  Breakfast  Breakfast 
9:45 – 10:45  The quasicategory of homotopy coherent monads in a (∞, 2)category Dimitri Zaganidis 
What do homotopy algebras form? Chris Rogers 
Weakly globular structures in homotopy theory and higher category theory Simona Paoli 
10:45 – 11:30 
Morning Tea 
Morning Tea  Morning Tea 
11:30 – 12:30  Not even wrong! Nora Ganter 
Operadic convolution in probability theory Gabriel C. DrummondCole 
Etale homotopy theory for higher stacks David Carchedi 
12:45 – 14:00  Lunch  Lunch 
Lunch 
14:15 – 15:15  Sullivan diagrams and homological stability Daniela Egas Santander 
Chain models for moduli space operads Ben Ward 
On the Ktheory of monoid algebras Christian Haesemeyer 
15:15 – 15:30 
Afternoon Tea 
Afternoon Tea 
Afternoon Tea 
15:30 – 16:30  Moduli spaces of bialgebras, higher Hochschild cohomology and formality Sinan Yalin 
Hierarchical networks and coloured modular operads Sophia Raynor 
Operads and polynomial 2monads Mark Weber 
19:00  Dinner  Dinner 
Dinner 
Abstracts
Wednesday 15th June
Dimitri Zaganidis, EPFL
Title: The quasicategory of homotopy coherent monads in a (∞, 2)category.
Abstract: Homotopy coherent diagrams in a simplicial category K can be encoded as simplicial functors C → K, where C is a well chosen simplicial category. This idea goes back at least to Cordier and Porter (Math. Proc. Cambridge Philos. Soc. 1986) and originated in earlier work of Vogt (Math. Z. 134, 1973) on homotopy coherent diagrams. For instance, the homotopy coherent nerve is constructed in this way. In Riehl and Verity’s paper (Adv. Math. 2016), C is the universal 2category containing the object of study, either a monad or an adjunction. For instance they define homotopy coherent monads as simplicial functors Mnd → K, where K = qCat∞ , the category of quasicategories enriched over itself, and where Mnd is the universal 2category containing a monad.
In this talk, we define a cosimplicial object Mnd[] in 2categories which induces a nerve NMnd : sCat → sSet. When K is a 2category, NMnd(K) = NMnd(K), where Mnd(K) is the 1category of monads in K, as defined by Street in (JPAA 1972). We show that when K is enriched in quasicategories and sufficiently complete, NMnd (K) is a quasicategory whose objects are the homotopy coherent monads in K.
Nora Ganter, University of Melbourne
Title: Not even wrong!
Abstract: I will share some thoughts on the foundations of string theory. This is joint work in progress with M. Ando.
Daniela Egas Santander, Freie Universität Berlin
Title: Sullivan diagrams and homological stability
Abstract: In string topology one studies the algebraic structures of the chains of the free loop space of a manifold by defining operations on them. Recent results show that these operations are parametrized by certain graph complexes that compute the homology of compatifications of the Moduli space of Riemann surfaces. Finding nontrivial homology classes of these compactifications is related to finding nontrivial string operations. However, the homology of these complexes is largely unknown. In this talk I will describe one of these complexes: the chain complex of Sullivan diagrams and give some computational results and further work on this direction.
Sinan Yalin, University of Copenhagen
Title: Moduli spaces of bialgebras, higher Hochschild cohomology and formality
Abstract: Algebras over props provide a good formalism to parametrize various structures of bialgebras as well as their homotopy version, which naturally appears in problems related to topology, geometry and mathematical physics. A relevant idea to understand their deformation theory is to gather them in a moduli space of algebraic structures in the setting of ToenVezzosi’s derived algebraic geometry. Infinitesimal deformations of such structures are then controlled by tangent Lie (or Linfinity) algebras naturally providing the corresponding cohomology theories and obstruction theories.
In a work in collaboration with Gregory Ginot, we apply these results to several open problems relating the deformation theory of E₂algebras with the deformation theory of associative and coassociative bialgebras. In particular, relying on the higher Deligne conjecture (now a theorem), we solve two longstanding conjectures of GerstenhaberSchack and Kontsevich: the existence of an E₃structure refining the deformation complex of a dg bialgebra, and the E₃formality of this deformation complex in the case of a symmetric bialgebra. This E₃formality theorem provides, in turn, a new proof of EtingofKazdhan deformation quantization independent from the choice of a Drinfeld associator.
Thursday 16th June
Chris Rogers, University of Louisiana
Title: What do homotopy algebras form?
Abstract: In this talk, I will describe joint work with V. Dolgushev in which we use ideas from deformation theory to construct an enriched category whose objects are homotopy algebras of a fixed type, e.g., L∞, A∞, or Ger∞ algebras. The enrichment is over a certain symmetric monoidal category of L∞algebras. Roughly, this is a “nonabelian” analogue of the fact that chain complexes are enriched over themselves. From this L∞enriched category, we obtain a simplicial category by using a nonabelian analogue of the DoldKan functor. We show that the mapping spaces in this simplicial category are, in fact, Kan complexes, and that this construction produces an explicit model for the ∞category of homotopy algebras.
Gabriel C. DrummondCole, IBS Center for Geometry and Physics
Title: Operadic convolution in probability theory
Abstract: The algebra of random variables associated to a probability space is a toy model for the observables of an algebraic quantum field theory. In this context, operadic language provides a concise encapsulation of the relationship between the moments and the cumulants of the random variables. I hope to talk about two aspects of this relationship related to noncommutative probability theory as an advertisement of potentially fertile ground for exploration by experts in this language. First, over a commutative ground ring, the framework leads to an operad in some ways reminiscent of the cactus operad. Second, over a noncommutative groundring, the combinatorics of the relationship between moments and cumulants uses composition along trees and is even more intimately tied to operadic language.
Ben Ward, Simons Center for Geometry and Physics
Title: Chain models for moduli space operads.
Abstract: First, we will review several operads built from moduli spaces of punctured spheres and their compactifications. In particular we will recall that algebraic structures arising in homological mirror symmetry and string topology may be represented by such operads. I will then discuss the problem of lifting these structures to the chain level and describe some recent progress.
Sophia Raynor, University of Aberdeen
Title: Hierarchical networks and coloured modular operads
Abstract: In applications across science and technology, there is an increasing interest in hierarchical structures of complex networks. My work aims to develop a general conceptual framework to capture the structure and function of complex networks at multiple spatiotemporal scales in one single model.
In this talk, I will first explain how coloured modular operads, and, in particular, a sheaftheoretic approach similar to that of Joyal and Kock (2009), provide a suitable formalism for this problem. I will conclude with a short impression of how these ideas are being received by applied scientists, and some of the collaborations on ‘real world’ problems that are currently in the pipeline.
Friday 17th June
Christian Haesemeyer, University of Melbourne
Title: On the Ktheory of monoid algebras
Abstract: I will discuss work in progress with G. Cortiñas, M. Walker and C. Weibel regarding the algebraic Ktheory of commutative monoid algebras. The ingredients I will focus on in this talk are a computation in (topological) cyclic homology, and some results in the algebraic geometry of monoids.
David Carchedi, George Mason University
Title: Etale Homotopy Theory for Higher Stacks
Abstract: Etale homotopy theory, as originally introduced by Artin and Mazur in the late 60s, is a way of associating to a suitably nice scheme a proobject in the homotopy category of spaces, and can be used as a tool to extract topological invariants of the scheme in question. It is a celebrated theorem of theirs that, after profinite completion, the etale homotopy type of an algebraic variety of finite type over the complex numbers agrees with the homotopy type of its underlying topological space equipped with the complex analytic topology. We will present work of ours which offers a refinement of this construction which produces a proobject in the infinitycategory of spaces (rather than its homotopy category) and applies to a much broader class of objects, including all algebraic stacks. We will also present a generalization of the previously mentioned theorem of ArtinMazur, which holds in much greater generality than the original result.
Simona Paoli, University of Leicester
Title: Weakly globular structures in homotopy theory and higher category theory.
Abstract: ntypes are spaces whose homotopy groups vanish in dimension higher than n. They are the building blocks of spaces thanks to the Postnikov decomposition. The search for combinatorial structures to model algebraically ntypes leads to higher categorical structures. In this talk we discuss this problem using a novel approach, based on iterated internal categories and the notion of weak globularity. We discuss the resulting structure, called weakly globular nfold categories, and its relevance to homotopy theory and to higher category theory.
Mark Weber, Maquarie University
Title: Operads and polynomial 2monads
Abstract: Polynomial 2monads provide a framework for the discussion of operadic structures, and have been used by BataninBerger in the study of when transferred model structures on categories of operads exist. In this talk the theory of polynomial functors will be recalled, it will be explained how operads can be seen as polynomial monads in a few ways, and how a lot of operadic theory can be recovered from more general ideas in 2dimensional monad theory.
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Image: Wikimedia Commons
ASSOCIATED EVENTS
Program: Higher Structures in Geometry and Physics 6 June 2016 – 17 June 2016
Workshop: Higher Structures 6 June 2016 – 10 June 2016