# Lectures

Speaker Title & Abstract
Paul Balmer

### The Eilenberg-Moore adjunction in triangulated categories

Nature provides an abundance of triangulated categories but it is usually quite difficult to construct new triangulated categories out of old ones, without using models. The most notorious example of such an abstract construction is localization (following Verdier or Bousfield) and this has numerous applications already. However, in equivariant examples, it is quite difficult to pass from the category associated to a group G to the category associated to a subgroup and localization is of no help there. We shall see that the classical construction of Eilenberg-Moore actually solves that problem in the so-called "separable" case, which covers not only Bousfield localization but also restriction to subgroups and much more. We shall then turn to the question of describing the tt-geometry of such Eilenberg-Moore extensions, in the presence of a tensor structure.

Wednesday, July 24, 2013 - from 11:00 to 12:00

John Greenlees

### Torus equivariant cohomology theories from algebraic geometry

There is a complete algebraic model A(G) for rational G-equivariant cohomology theories for a torus G. The category A(G) has a formal structure very similar to that of a category of sheaves over an algebraic variety, and this means that suitable algebraic geometric data lets one write down an object of A(G) representing a cohomology theory E^*_G(.). The most familiar example comes from a complex elliptic curve, giving rise to equivariant elliptic cohomology, but related constructions give other interesting examples. (Joint with Brooke Shipley and Pokman Cheung).

Thursday, July 25, 2013 - from 11:00 to 12:00

Martin Hyland

### 45 years on: Some Abstract Mathematics of Eilenberg

Samuel Eilenberg had a distinctive taste for abstract mathematics which extended beyond his famous contributions to algebraic topology and homological algebra. In this talk I shall discuss how Eilenberg's taste manifested itself in some of his other work. I shall consider in particular the book Recursiveness written with Calvin Elgot and the paper Automata in General Algebras joint with Jesse Wright. In the first of these the theory of recursive functions is developed using subcategories of the category of relations. The basic setting is of the free monoid of words on a finite alphabet. The second presents an approach to automata with a very contemporary flavour. I shall sketch the main mathematical ideas and attempt to place them in the context of more recent developments.

Thursday, July 25, 2013 - from 09:30 to 10:30

Andre Joyal

### Simplicial sets in perspective

The notion of simplicial set was introduced by Eilenberg and Zilber in 1950, originally for applications to algebraic topology. It was a giant step forward, since the notion is currently used in many branches of mathematics, from homotopical algebra to algebraic geometry, higher category theory and constructive logic. For instance, it was used by Daniel Kan for the foundation of combinatorial homotopy theory, by Daniel Quillen for homotopical algebra, by Bertrand Toën and Gabriele Vezzosi for homotopical algebraic geometry, by Jacob Lurie for higher topos theory and by Vladimir Voevodsky for the semantics of Martin-Löf type theory. The foundation of higher category theory is currently based on simplicial structures of different forms: simplicial categories, Segal categories, Rezk categories and quasi-categories. The latter is just a simplicial set satisfying the Boardman condition. After briefly recalling the history of simplicial sets, I will describe their applications to higher category theory and then mention some recent generalizations.

Monday, July 22, 2013 - from 09:30 to 10:30

William Lawvere

### Conceptions of Space and the Role of Axiomatization

The axioms in the Foundations of Algebraic Topology were not an idealist starting point but rather an extraction of structure inherent in previous practice, creatively made explicit to serve as a guide to enlightened further practice. This work with Steenrod was an important instance of Sammy’s progressive use of axiomatization as a tool for ongoing research. It deeply impressed the students at Columbia University in the work on the foundations of universal algebra, of set theory and logic. It strongly influenced the conceptual underpinning of the topos foundations of algebraic geometry and of smooth geometry, namely the determination of a space by its incidence relations between figures (which were whimsically called ‘points’ by some) as clearly expressed in the first paragraph of the 1950 paper of Eilenberg & Zilber that introduced Simplicial Sets. Sammy’s philosophy of mathematics is reflected in his successful collaborations, with Steenrod, with Zilber, with Mac Lane, with Cartan. This philosophy and these collaborations were part of the attraction that led to the remarkable concentration of students and young researchers at Columbia in 1960. Also in 1960, in Cartan’s Paris seminar where simplicial sets had been further studied, Grothendieck elaborated a categorical approach to Complex Analytic Geometry. The simplicial and analytic categories were two of the first toposes studied explicitly but, as categories of all spaces of a given kind, they are different from the localic sort associated to a given topological space. In our struggle today to clarify the nature of the several classes of toposes and their relationships, Sammy’s axiomatic method still provides fundamental guidelines. In particular, his hero Hurewicz had done crucial work in the 1940’s on map spaces, that Sammy axiomatized in his 1965 collaboration with Kelly (using the crucial tool made explicit by his student Kan in 1958). The results of that collaboration continue to play a foundational role in the study of toposes and their applications to functional analysis.

Wednesday, July 24, 2013 - from 09:30 to 10:30

Haynes Miller

### Infinite loop spaces and the Steenrod algebra

There is a rather obvious spectral sequence converging to the cohomology of the zero-space of a connective spectrum, whose $E_2$ term is determined homologically from the spectrum cohomology as a module over the Steenrod algebra. I will describe joint work with Rune Haugseng in which we explore the computation of this $E_2$ term. We use simplicial methods with which Eilenberg and Mac Lane would have been comfortable. I will also report on current work which seeks to extend this to finite loop spaces, providing a context for my work with John Harper, from the 1980's, on looping spaces whose cohomology is of the form $U(M)$.

Tuesday, July 23, 2013 - from 09:30 to 10:30

Ieke Moerdijk

### Dendroidal sets and stable homotopy theory

The category of dendroidal sets models the homotopy theory of operads and of "operads-up-to-homotopy". Segal's category of Gamma-spaces models the homotopy theory of connective spectra. In this lecture, I will recall the basic definitions concerning these theories, and explain how they are related.

Tuesday, July 23, 2013 - from 11:00 to 12:00

Bertrand Toën

### Derived geometry and quantization

In this lecture I will report on recent progress on interactions between on the one hand derived algebraic geometry and on the other hand the quantum world (quantum group, deformation quantization ...). For this, I will start by presenting a theorem, stating that the derived moduli spaces of G-bundles on oriented manifolds have canonical quantization. An important part of the lecture will be devoted to explain this result in more details: to make precise the various notions and explain its relations with previously known results and concepts (quantum groups, quantum invariants of manifolds, hypersurface singularities ...). Finally, I will say few words about its proof and its possible future applications.

Monday, July 22, 2013 - from 11:00 to 12:00

Shmuel Weinberger

### Functoriality in surgery theory: geometric remarks and implications

It is a remarkable fact that the surgery theoretic classification of manifolds - in the topological setting - has some functoriality properties: One can push forward a manifold homotopy equivalent to M to one homotopy equivalent to N under a continuous map from M to N, under appropriate dimension and orientation conditions. We shall discuss this and equivariant and stratified variants, and show how things like the existence of nonresolvable homology manifolds, counterexamples to the equivariant Borel conjecture, embedding theorems and calculations of certain types of equivariant structure sets follow quite painlessly from such statements. I will also attempt to give some indication of the sources of these functorialities. (Much of this talk will be based on joint work with Sylvain Cappell and Min Yan.)

Friday, July 26, 2013 - from 11:00 to 12:00

Mariusz Wodzicki

### TBA

Friday, July 26, 2013 - from 09:30 to 10:30