University of Chicago
Department of Mathematics

We address the question of how elegantly to combine a number of different structures on a category, such as finite product structure, monoidal structure, and colimiting structure. Extending work of Marmolejo and Lack, we develop the definition of a pseudo-distributive law between pseudo-monads, and we show how the definition and the main theorems about it may be used to model several such structures simultaneously. Specifically, we address the relationship between pseudo-distributive laws and the lifting of one pseudo-monad to the 2-category of algebras and to the Kleisli bicategory of another. This generalises the main result of the theory of ordinary distributive laws between monads; I will begin the talk with an overview of this theory before presenting its generalisation.

The combinatorics of 1-fold loop spaces is well understood in terms of geometry of Stasheff's polytopes and is extremely useful. The combinatorics of n-fold loop spaces for n>1 is much more mysterious. In my lecture I will show how higher categorical ideas lead to a new recognition principle for n-fold loop spaces which reveals their hiden combinatorics. If time allows me I will tell about higher dimensional analogues of Stasheff's polytopes and corresponding notions of E_n-algebras (including n= \infty).

I will give a simple and very general definition of `homotopy-algebra' for an operad. This extends Segal's Gamma-space idea (homotopy topological commutative monoids), but works in arbitrary monoidal categories, not just those in which the product is cartesian. For instance, applied to the operad whose algebras are monoids, it gives a notion of homotopy differential graded algebra. Along the way, we will find a new perspective on what a simplicial object is.

In this talk I will give an overview of the spin foam approach to quantum gravity, developed by Baez, Barrett, Crane, Reisenberger, Smolin and others. States for quantum geometry are given by labelled simplicial complexes, and transition amplitudes are defined using fun diagrammatic methods in SU(2) representation theory. After giving the background, I will discuss the computational difficulties that come up when trying to do calculations in this theory and will describe new algorithms due to Greg Egan, John Baez and myself which have allowed us to make the first computations. No previous exposure to the subject will be assumed.

(joint work with Albert Ruiz) The concept of $p$-local finite group arise in the work of Broto-Levi-Oliver as a generalization of the classical concept of finite group. Therefore, the classification of $p$-local finite groups has interest, not only by itself but, as an opportunity to enlighten one of the highest mathematical achievements in the last decades: The Classification of Finite Simple Groups. In this work we classify all $p$-local finite group over the $p$-groups of type $p^{1+2}_+$. In this classification three new exotic $7$-local finite groups arise.

If $G$ is a free product of finite groups, let $\Sigma Aut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {\em Symmetric Automorphisms of Free Products}, Mem. Amer. Math. Soc. 122 (1996), no. 582] and Guti\'{e}rrez-Krsti\'{c} [M. Guti\'{e}rrez and S. Krsti\'{c}, {\em Normal forms for the group of basis-conjugating automorphisms of a free group}, International Journal of Algebra and Computation 8 (1998) 631-669] derived (also see Bogley- Krsti\'{c} [W. Bogley and S. Krsti\'{c}, {\em String groups and other subgroups of $Aut(F_n)$}, preprint] space of pointed trees is an $\underline{E} \Sigma Aut_1(G)$-space for these groups. (Joint with H. Glover and Y. Chen.)

In this talk, we use algebraic topology to develop invariants useful in describing shapes and locating and classifying features (i.e. singular points of various types). The invariants we propose can be computed on "point cloud data", i.e. finite but large sets of points sampled from an underlying topological space in R^n, and we will discuss how topological invariants are obtained from point cloud data. We will also discuss an algorithm for locating singular points of particular types.

In this talk we discuss an approach to the descent problem for K-theory of fields in which we construct an analogue to the assembly map for group rings which applies to the K-theory of fields instead. In this case, the domain of this assembly map is a space attached to the complex representation theory of the absolute Galois group. We will conjecture that this map is an equivalence, and discuss some consequences.

In joint work with Bjorn Ian Dundas and Paul Arne Ostvaer, I constructed a model for the motivic stable homotopy category which is equipped with a nice smash product. I will outline how Voevodsky's spectrum representing motivic cohomology extends to a commutative monoid in this model.