Andrew Baker - Recent work on invariants for derived commutative rings

The term 'derived commutative ring' refers to some notion of 'commutative ring + homotopy theory'. The main examples to think about are simplicial commutative rings, E-infinity algebras and commutative S-algebras (aka E-infinity ring spectra).

I'll discuss work done during the past decade or so on constructing analogues of classical invariants such as Picard and Brauer groups for the topological examples provided by commutative S-algebras, however, these should be regarded as test cases for other settings.

Michael Batanin - Polynomial monads and homotopy theory of operads and algebras

Polynomial monads appear in mathematics in many different contexts and under various names: containers, rigid operads, $\Sigma$-free coloured operads etc. Examples of polynomial monads include monads for monoids, nonsymmetric and symmetric operads, cyclic and modular operads, $n$-operads, dioperads, properads and generalised (wheeled) PROP's, their higher-order and coloured extensions etc.

In my talk, I will explain from a $2$-categorical perspective why polynomial monads and their algebras have special categorical properties suitable for explicit combinatorial calculations. As an application, I will explain how this $2$-categorical approach gives combinatorial criteria for the existence and properness of the Quillen model structure on operads and algebras over operads generalising and extending many earlier results.

Gwyn Bellamy - Counting resolutions of symplectic quotient singularities

If V is a symplectic vector space and G a finite subgroup of Sp(V), then the quotient singularity V/G is a very interesting object to study, both from the geometric and representation-theoretic point of view. One of the motivational problems in trying to understand the singularities of V/G is that of deciding whether V/G admits a symplectic resolution or not. More generally, one can ask how many symplectic resolutions it admits. The goal of this talk is to explain how one can count the number of symplectic resolutions of V/G. I'll present an explicit formula for this number in terms of the dimension of a certain Orlik-Solomon algebra. The key to deriving this formula is to relate the resolutions of V/G to the Calogero-Moser deformations, where one can use the representation theory of symplectic reflection algebras.

Chris Braun - Derived noncommutative localisation

The localisation of commutative rings and modules is a basic construction in commutative algebra. It is straightforward, well-behaved and well-understood. On the other hand, noncommutative localisation is much more involved and less well-behaved, but also more interesting. It comes up in various contexts, for example, the derived category can be viewed as the localisation with respect quasi-isomorphisms. By working with a derived version of noncommutative localisation and employing methods from homotopical algebra we can obtain a better-behaved theory of noncommutative localisation.

In this theory, the localisation of a dg algebra, or more generally a dg category, can be seen to be, in a certain precise sense, equivalent to the Bousfield localisation of its category of dg modules. This general abstract result has a wide range of concrete applications including, amongst others, a general version of the group completion theorem and a derived version of the Riemann-Hilbert correspondence.

Kei Yuen Chan - Homological algebras for graded Hecke algebras

Graded Hecke algebras were introduced by Lusztig for studying the representation theory of p-adic groups and Iwahori-Hecke algebras. In this talk, I will discuss some recent study on the homological properties of modules for graded Hecke algebras. In particular, I will present a duality on the Ext-groups, which is analogue to the Poincaré duality.

Denis-Charles Cisinski - The universal property of mixed motives

Carmelo di Natate - Grassmannians in Derived Algebraic Geometry

The goal of this talk is to explain how to extend Grassmannians to the world of derived stacks, i.e. how to construct a satisfying derived enhancement of Grassmannian varieties. I will begin by discussing some background about derived geometric stacks and in particular, I will focus on Artin-Lurie-Pridham representability theorem, which provides us with a "computational" criterion to check whether a simplicial presheaf of derived algebras is a derived geometric stack. Then I will use such a result in order to study derived moduli of perfect complexes and filtered perfect complexes over a base scheme; finally, the derived Grassmannian will arise as some suitable homotopy limit of such stacks. Time permitting I will end by sketching how to use this derived version of the Grassmannian to obtain a derived version of Griffiths' period mapping.

Vladimir Dotsenko - Many facets of Givental group action

I shall talk about how homotopical algebra can help to interpret the Givental action on (genus 0) cohomological field theories in several different ways. The main ingredients that allow for those interpretations are homotopy theory for Batalin--Vilkovisky algebras and the formalism of gauge symmetries in Lie-infinity algebras.

Interpretation of the Givental action via gauge symmetries permits, first, to extend this action to cohomological field theories up to homotopy, and second, to prove the claim of Kontsevich on how the Givental action can be described via homotopically trivial circle actions. This is based on joint work with Sergey Shadrin and Bruno Vallette.

Nicola Gambino - Univalent Foundations of Mathematics and homotopical algebra

The Univalent Foundations of Mathematics programme, formulated by Vladimir Voevodsky around 2009, seeks to develop an alternative approach to the foundations of mathematics. A distinguishing feature of this approach is that it combines ideas from mathematical logic and homotopy theory. I will give a general introduction to the Univalent Foundations programme and explain its connections with the theory of Quillen model categories.

Victor Ginzburg - Counting indecomposables over a finite field

Mark Gross - Canonical bases for cluster algebras

I will talk about recent joint work with Hacking, Keel and Kontsevich. We apply techniques developed in mirror symmetry (scattering diagrams and tropical geometry) to cluster algebras. We give a very general method for constructing elements of upper cluster algbras which, in pleasant circumstances, give canonical bases. In general, the method yields a proof of the positivity of the Laurent phenomenon for all skew-symmetrizable cluster algebras.

Vladimir Hinich - Enriched infinity-categories

We propose a notion of infinity-category enriched over arbitrary (=not necessarily cartesian) monoidal infinity-category with colimits. Our definition is basically equivalent to the one using simplicial objects with Segal conditions in the cartesian case; also left-tensored infinity-categories give rise to enrichment in our sense.

Dominic Joyce - New Donaldson-Thomas style counting invariants for Calabi-Yau 4-folds

Pantev, Toen, Vezzosi and Vaquie (arXiv:1111.3209) introduced the notion of "k-shifted symplectic derived schemes and stacks" in Derived Algebraic Geometry. They showed that moduli stacks of coherent sheaves and complexes on a Calabi-Yau m-fold Y are (2-m)-shifted symplectic. So, in particular, Calabi-Yau 3-fold moduli stacks are -1-shifted symplectic, and Calabi-Yau 4-fold moduli stacks are -2-shifted symplectic. In previous work with Ben-Bassat, Brav, Bussi, Dupont, Meinhardt, and Szendroi we studied -1-shifted (3-Calabi-Yau) geometry and generalizations of Donaldson-Thomas theory. Today we move on to the -2-shifted case.

Using a "shifted symplectic Darboux Theorem" by Brav, Bussi and Joyce, we prove that a -2-shifted symplectic derived scheme X over C can be given the structure of a "derived smooth manifold" (d-manifold, or M-Kuranishi space) X*, uniquely up to bordisms of X* fixing the underlying topological space.

If X is proper and has an "orientation" (similar to Kontsevich-Soibelman orientation data in the 3-Calabi-Yau case), then X* is a compact, oriented derived manifold, and so has a virtual class (e.g. in bordism), which is an integer if vdim X = 0.This should give virtual classes for proper Calabi-Yau 4-fold moduli schemes, and lead to new Donaldson-Thomas style invariants "counting" (semi)stable coherent sheaves on a Calabi-Yau 4-fold Y, which will be unchanged under continuous deformations of Y.

This is joint work with Dennis Borisov. It is related to recent work of Cao and Leung (arXiv:1407.7659).

Alastair King - Cluster categories and dimer models

I will explain how the combinatorics of Grassmannian cluster algebras can be refined to give a direct categorical link with dimer models on a disc.

Joachim Kock - Rudiments of Homotopy Combinatorics

Where classical combinatorics deals with finite sets of structures, homotopy combinatorics deals with finite homotopy types of structures. The basic notion is that of homotopical species. I will explain two theorems: the first (joint work with David Gepner) is a 'Joyal theorem' for homotopical species, characterising their associated analytic functors in terms of exactness conditions. The second (joint work with Imma Galvez and Andy Tonks), is a 'Schmitt theorem', to the effect that restriction species give rise to incidence coalgebras, via the notion of decomposition space.

Motivation comes on one hand from program semantics (generic datatypes), and on the other hand from quantum field theory (Feynman graphs). In both cases, the need for a homotopical setting comes from the presence of symmetries. (1-groupoids would actually be enough to deal with these examples, but it is practical to develop the theory in the setting of infinity-groupoids.)

Muriel Livernet - Massey operadic products and non-formality of operads

In this talk, I will prove that the operad Swiss-cheese is not formal. In order to do this, I will introduce the Swiss-cheese operad, and compare it to the little cube operad. I will introduce Massey operadic products and compute them for the Swiss-cheese operad.

Wendy Lowen - The curvature problem for formal and infinitesimal deformations

We interpret all Maurer-Cartan elements in the formal Hochschild complex of a small dg category which is cohomologically bounded above in terms of torsion Morita deformations. This solves the "curvature problem", i.e. the phenomenon that such Maurer-Cartan elements naturally parameterize curved A_infinity deformations. We also discuss the more subtle situation for infinitesimal deformations. This is joint work with Michel Van den Bergh.

Ieke Moerdijk - The Homotopical Algebra of Operads and Trees

The theory of operads arose in the 1970s with the aim of describing the algebraic structure of loop spaces in topology and has developed into a tool to study algebraic structures in many other parts of mathematics. It has been known for several decades now that different occurrences of operads can be described and compared by methods of Quillen's homotopical algebra, providing a natural way to study algebraic structures in which the equations hold only "up to homotopy". I will describe how this development has recently been enriched with an entirely new way of looking at operads, as realisations of combinatorial structures built up from trees, which is in many ways inspired by and analogous to the way in which one studies topological spaces as realisations of combinatorial simplicial complexes.

Peter Neumann - Aspects of Algebra in Britain 1865–2015

I have been invited to contribute to the LMS sesquicentenary celebrations by speaking on the development of algebra in Britain, 'with particular reference to notable mathematicians who have also had close ties with the LMS or where the LMS has supported and facilitated their work'. I shall endeavour to take us at a gallop through 150 years of mathematical developments at a rate of three years per minute.

Jon Pridham - K-theory and real Deligne cohomology

Beilinson's conjecture predicts that the K-theory of a smooth projective variety is closely related to its real Deligne cohomology. I will indicate how the real Deligne cohomology of a complex manifold arises locally as a new kind of K-theory, given by a topological vector space completion of the analytic Lie groupoid of holomorphic vector bundles.

Vic Snaith - Monomial resolutions of admissible representations

The purpose of this talk is to describe a functorial embedding of the category of admissible k[G]-representations of a locally profinite topological group G into the derived category of the additive category of the admissible k[G]-monomial module category.

Admissible smooth representations of locally profinite Lie groups are the central characters of the Langlands Programme (LP). As such they related in a very important manner to modern number theory and arithmetic-algebraic geometry. The LP attempts to generalise abelian class field theory to several non-abelian setting settings.

Abelian class field is about one-dimensional representations. The k[G]-monomial module category is constructed entirely from one-dimensional representations. My idea, therefore, is to embed LP into a context made from abelian data.

I shall concentrate on the homological algebra side of the construction, which I believe to be new even for finite groups.

Bruno Valette - Pre-Lie deformation theory

In this talk, I will develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, I will provide a homotopical description of the associated Deligne groupoid. This will permit me to give a clear proof of the ubiquitous Homotopy Transfer Theorem by means of gauge action: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its underlying homology if and only if it gauges trivial, which is the ultimate generalization of the d-dbar lemma. (Joint work with Vladimir Dotsenko and Sergey Shadrin. Reference: arXiv:1502.03280.)