# Titles and Abstracts

• Spiros Adams-Florou (University of Edinburgh)

Local to Global Results in Topology

Topology deals with theorems in which local conditions have global implications, and their converses. For example, the classic Vietoris theorem states that a map of reasonable spaces with acyclic point inverses induces isomorphisms in homology. The property of having acyclic point inverses is however not homotopy invariant. The modern controlled and bounded topology provides global categorical conditions on a map which ensure that it is at least homotopic to a map with acyclic point inverses, a kind of converse to the Vietoris theorem. A homeomorphism is a homotopy equivalence, but a homotopy equivalence of manifolds is not in general homotopic to a homeomorphism. I shall discuss how controlled and bounded algebra can be used to decide if a homotopy equivalence of manifolds is homotopic to a homeomorphism.

• Hugo Bacard (Université de Nice)

Segal Enriched Categories

A category C is said to be enriched over a category M if for any pair of objects (A,B), the “space of morphisms” C (A,B) is an object of M. For short we say that C is ‘M-category’. The category M needs to have a ‘product’ (e.g a monoidal category) to carry the operations giving the composition in C . Taking M to be (Set,×), (Ab,⊗Z), (Top,×),..., an M-category is, respectively, an ordinary category, a pre-additive category, a pre-topological category, etc. The composition in C induces a “multiplication” on C (A,A), so that C (A,A) becomes a monoid of M, where the unit is the identity map IA. Conversely any monoid of M can be taken to be the “endomorphismobject” C (A,A) of some M-category C . This allows us to identify monoids of M with M-categories with one object. In this talk I will give the definition of Segal enriched categories and will show that in the one-object case we recover the notion of “up-to-homotopy monoid” introduced by Leinster. The formalism adopted extends the classical theory of Segal categories and gives rise, in particular, to a linear version of Segal categories which did not exist so far.

• Olivia Bellier (Université de Nice)

Koszul Duality Theory

In 1970, Priddy introduced the Koszul duality theory, that gives good (projective for instance) resolutions, for quadratic associative algebras.
To encode the multilinear operations acting on a vector space, one defines the notion of operad. An associative algebra can be seen as an operad considering elements of the algebra as unary operations. Ginzburg-Kapranov and Getzler-Jones extended the Koszul duality theory from quadratic associative algebras to quadratic operads.
Furthermore, Salvatore-Wahl studied some operads equipped with a structure of H-module, H being a Hopf algebra, called H-operads. My work is to develop the Koszul duality theory for quadratic H-operads.

• Søren Boldsen (University of Bonn)

Homology stability of the mapping class group

The mapping class group of a surface is the group of connected components of the diffeomorphisms of the surface, and it plays a role in many different areas of mathematics. One way to study the mapping class group is via its homology, and it turns out to be stable: The homology in degree n of the mapping class group of a surface of genus g does not depend on g, if g is large enough compared to n. The goal is then to make the stability range as good as possible. In this talk I will explain these concepts and indicate the ideas that can be used to show homology stability.

• Aneta Cwalinska (N. Copenicus University)

On algebraic maps of products of spheres

Let $S^n(F)$ be the unit sphere over a field $F$ and write $T^n(F)$ for the corresponding $n$-torus. We examine polynomial and regular maps from products of spheres $S^{m_1}(F)\times\cdots\times S^{m_k}(F)$ into $T^n(F)$ and vice versa. In particular, for $F=R,C$ the fields of reals and complex numbers, we follows Loday's results to analyse their homotopy types.

• Rosona Eldred (University of Illinois at Urbana-Champaign)

A Brief Introduction to a Calculus of Functors

Homology theories are one of the first tools we use to study spaces. They are also one of the first tools used in the study of functors from Spaces to Spaces (or Spectra); homology theories are linear functors. As we use linear functions to locally approximate more complicated functions, we can use linear functors to 'locally' approximate arbitrary functors . In this talk, we will introduce the notion of linearity of a functor, the derivative of a functor, and, given time, the associated Taylor polynomials, drawing strong parallels with the calculus of real-valued functions.

• Alex Gonzalez (Universitat Autonoma de Barcelona)

Locally finite approximations for $p$-local compact groups

$p$-local compact groups were introduced in [BLO3] by C. Broto, R. Levi and B. Oliver as simplicial models for the $p$-completion of classifying spaces of compact Lie groups and $p$-compact groups, and still very few is known about their general properties, as their mod $p$ cohomology rings, properties about mapping spaces (which are already known for $p$-local finite groups), and more. It is known, however, that one can construct (very explicitely) infinitely many unstable Adams operations of increasing power on a given $p$-local compact group (this was studied in his thesis by F. Junod).
We face the question of whether, given such a family of unstable Adams operations on a fixed $p$-local compact group, we can obtain a family of $p$-local finite groups as the fixed-points of these actions, and such that the classifying space of the $p$-local compact group is homotopy equivalent to the $p$-completion of the homotopy colimit of the classifying spaces of the $p$-local finite groups.
We present then a construction of these fixed points and their properties. These fixed points are in fact triples that behave almost as $p$-local finite groups, although they depend on a rather technical condition in order to be actual $p$-local finite groups. We will see some examples in which we do obtain $p$-local finite groups, and we will also study some consequences of having such an approximation for $p$-local compact groups.

• Jesper Grodal (University of Copenhagen)

Lectures: Homotopical Group Theory

I'll survey some recent advances in the homotopy theory of classifying spaces, and homotopical group theory. The first talk will focus on the classification of p–compact groups in terms of root data over the p–adic integers, and the second will discuss some of its consequences e.g., for finite loop spaces and polynomial cohomology rings. The talks are based on my recent survey "The classification p-compact groups and homotopical group theory" https://www.math.ku.dk/~jg/papers/icm.pdf

The symmetric squaring construction in bordism

Looking at the cartesian product of a topological space with itself, a natural map to be considered on that object is the involution that interchanges the coordinates, i.e. that maps (x,y) to (y,x).
The so-called "symmetric squaring construction" in Cech homology with Z/2-coefficients was introduced in [arXiv:0709.1774 ] as a map from the k-th Cech homology group of a space X to the 2k-th Cech homology group of X \times X divided by the above mentioned involution. It turns out to be a crucial construction in the proof of a Borsuk-Ulam-type theorem.
In [arXiv:1002.1449 ] a generalization of this construction to Cech homology with integer coefficients is given for even dimensions k, which is proven to equally satisfy the useful properties of the original construction.
The symmetric squaring construction can be generalized further to give a map in bordism, which will be the main topic of this talk. More precisely, it will be shown that there is a well-defined map from the k-th singular bordism group of X to the 2k-th bordism group of X\times X divided by the involution as above. Moreover, this squaring works as well for oriented bordism groups in even dimensions and really is a generalization of the Cech homology case since it is compatible with the passage from bordism to homology via the orientation homomorphism.

• Ib Madsen (University Of Copenhagen)

Lectures: On the Classification of Manifolds

The two lectures will outline results from the classification theory of manifolds. The topics to be touched upon are:
1.The Pontryagin-Thom cobordism theory.
2.The moduli space of manifolds of a given homotopy type.(Surgery)
3.Concordance theory and diffoemorphism groups of high dimensional manifolds.
4.The cobordism category of surfaces and the generalized Mumford conjecture.
5.Topological B-fields.
6.The future?

• Arjun Malhotra (University Of Sheffield)

Positive scalar curvature and the Gromov-lawson-Rosenberg conjecture

The Gromov-Lawson-Rosenberg conjecture for a group G says that a compact spin manifold with fundamental group G admits a metric of positive scalar curvature if and only if a certain topological obstruction vanishes. The plan is to discuss the conjecture, and sketch how to prove it for some finite groups.

• Stephen Miller (University of Manchester)

Quasitoric Manifolds

I will introduce quasitoric manifolds and explain their role in complex cobordism. My own work includes the construction of a universal space which can be used to simplify cobordism calculations.

• Joan Millès (Université de Nice)

How to study unital associative algebra ?

The Hochschild cohomology theory and the notion of homotopy algebras concern augmented unital associative algebras. However, unital associative algebras are not necessarily augmented. A simple example is given by the real algebra of complex number.
To find a "good" cohomology theory or a "good" homotopy theory associated to unital associative algebras, we see algebras as representation of an object, called operad, and we extend the Koszul duality theory to the operad encoding unital associative algebras.
In this talk, we describe algebras as representations and we introduce the homotopy theory on topological and algebraic examples. We finish by some applications of the Koszul duality theory.
(Some parts of this talk comes from a joint work with Joseph Hirsh.)

• John Olsen (University of Rochester)

Three Dimensional Manifolds All of Whose Geodesics Are Closed

We will discuss some results on the Morse Theory of the energy function on the free loop space of S^3 for metrics all of whose geodesics are closed. We will show what implications these results have for the Berger conjecture in dimension three. If time permits, we will sketch the proof of the main result, which states that the energy function is perfect with respect to the S^1-equivariant cohomology with rational coefficients.

• Mehmetcik Pamuk (Middle East Technical University)

Four Manifolds wih Free Fundamental Group

In this talk we are going to give the s-cobordism classification of 4-manifolds with free fundamental group by studying the group of homotopy classes of homotopy self-equivalences of such 4-manifolds.

• Semra Pamuk (Middle East Technical University)

Equivariant CW-Complexes and The Orbit Category

This talk is about the paper which is a joint work with I. Hambleton and E. Yalcin. A good algebraic setting for studying actions of a group with isotropy given in a given family of subgroups is provided by the modules over the orbit category. In this talk, I will briefly talk about the orbit category and then I will give the example of the group S_5 with isotropy in the family of cyclic subgroups.

• Mark Powell (University of Edinburgh)

Knot Concordance

I will recall the notion of knot concordance and talk about my project to use the symmetric chain complex of the universal cover of the knot complement to obtain obstructions to null-concordance.

• Matan Prezma (Hebrew University of Jerusalem)

Homotopical normal maps

These are top. group maps G--->H s.t. the homotopy quotient is h.e. to a top. group map.
Thm: Normal maps are preserved by functors which preserve h.e. and finite products up to homotopy.

• Jenny Santoso (University of Stuttgart)

Local and Global Convexity of Maps

In 1982, Atiyah, and independently, Guillemin and Sternberg, proved for a torus action on a compact symplectic manifold that the image of the momentum map is a convex polytope. After a series of improvements (Kirwan, Condevaux, Dazord, Molino, Hilgert, Neeb, Plank, Birtea, Ortega, Ratiu, and many others), it turned out that a wide class of convexity results can be based on a local-to global principle, given in terms of almost pure topology, which generalizes Klee's theorem for locally convex spaces as well as geodesic convexity theorems for Riemannian manifolds. Using a new factorization of the momentum map, convexity of its image is proved without local fiber connectedness, and for almost arbitrary spaces of definition. Geodesics are obtained by straightening rather than shortening of arcs, which leads to a unified treatment and extension of previous convexity results where the target space is either a dual Banach space or a proper length metric space. The talk presents a new local-to global principle for convexity of continuous maps and shows an efficient way how convexity can be married with topology.

• Nora Seeliger (Universite de Paris 13)

Assigning a classifying space to a given fusion system up to F-isomorphism

Fusion Systems and Linking Systems are finite categories which model the conjugacy properties of a Sylow p-subgroup in a finite group. The main open problem in the theory of p-local finite groups, from the toplogical point of view, is to assign a classifying space to a given fusion system. In this talk we solve the problem up to F-isomorphism in the sense of Quillen.

• Marc Stephan (ETH Zurich)

Elmendorf's Theorem for Cofibrantly Generated Model Categories

Elmendorf's Theorem in equivariant homotopy theory states that for any topological group G, the model category of G-spaces is Quillen equivalent to the model category of continuous diagrams of spaces indexed by the opposite of the orbit category of G. For discrete G, Bert Guillou explored equivariant homotopy theory for any cofibrantly generated model category C and proved an analogue of Elmendorf's Theorem assuming that C has "cellular" fixed point functors. We will generalize Guillou's approach and study equivariant homotopy theory for topological, cofibrantly generated model categories. Elemendorf's Theorem will be recovered for G a compact Lie group.

• Vesna Stojanoska (Northwestern University)

Group Cohomology and Duality for Topological Modular Forms

We consider the spectrum of topological modular forms $TMF$ as the homotopy fixed points of $TMF(2)$, topological modular forms with level $2$-structure, under a natural action of the (finite) group $GL_2(\Z/2)\cong S_3$. Grothendieck-Serre duality makes $TMF(2)$ self dual; combining this with the self-duality of the Tate cohomology of $S_3$, we obtain Brown-Comenetz duality for $TMF$.

• Martin Stolz (University of Bergen)

Semi-Free Orthogonal Ring Spectra and the Convenient Model Structure

We use semi-free orthogonal spectra to give a model structure on the category of commutative orthogonal ring spectra which is convenient in the sense that cofibrant objects are already cofibrant as underlying orthogonal spectra. Special care has to be given to smallness and flatness conditions, that distinguish the topological case of orthogonal spectra from the simplicial world of symmetric spectra for which an analogue had been previously studied by Shipley.

• Alvise Trevisan (Vrije Universiteit Amsterdam)

The homology of abelian covers of the real Davis-Januszkiewicz space

Let X_R be the real part of a smooth projective toric variety X. The real variety X_R is equipped with an action of an elementary abelian 2-group with homotopy orbit space DJ_R , a real Davis-Januszkiewicz space. Then X_R is homotopy equivalent to an abelian regular cover of DJ_R with finitely many sheets. A lot is known about X_R , including its additive and multiplicative mod-2 cohomol- ogy. Nonetheless, virtually nothing was known about its integral (co)homology: this is one of the motivating ideas behind the study of covers of DJ_R . In this talk I explain how to compute the cohomology (and the cup products therein) with field coefficients of an abelian cover of DJ_R . As a corollary we compute the cohomology of X_R (and therefore its integral Betti numbers), but we also recover results of Goresky-MacPherson, deLongueville and Baskakov-Buchstaber-Panov on the cohomology of complements of subspace arrangements and of moment-angle complexes. This is joint work with Alexander I. Suciu.

• Inna Zakharevich (MIT)

Higher Scissors Congruence

The question of scissors congruence goes back to the Greeks (who knew that if two plane polygons had the same area, then you could cut one up and rearrange it into the other); the corresponding problem for three dimenasions was a Hilbert problem resolved (negatively) by Dehn. Further work in the 60s and 70s generalized this to other geometries by constructing groups which encode scissors congruence data. In this talk we give an alternate construction of these groups as the 0-th homotopy groups of K-theory spectra, and demonstrate how ring structures arising in the older work are the shadow of E_infty ring structures on these spectra.