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Abstracts and Talk Materials

In response to a question of Milnor, a collaboration of Mike Hill, Mike Hopkins, Mark Mahowald, and myself have been attempting to determine in which dimensions exotic spheres exist. We will review
the work of Kervaire-Milnor, which reduces the problem to questions in the stable homotopy groups of spheres. Thanks to the solution of the Kervaire invariant one problem, the problem is reduced to producing certain infinite families in the stable stems. I will discuss what we know, and don't know about the stable stems these days, and discuss some joint work with Mark Mahowald, producing a plethora of 192-periodic families (the Hurewicz image of TMF) which give partial answers to the original question.

The talk is an introduction to problems and results in Dynamics centering around contributions to the field by Milnor.

January 30, 2012

In 1956 John Milnor startled the mathematical community by
constructing smooth 7-dimensional manifolds that are homeomorphic but
not diffeomorphic to the standard seven-sphere. His discovery opened a
new branch of research in topology and won him the Fields Medal in
1962. In this animation we present Milnor's construction and give a
way of visualizing these manifolds.

Read More...

In his 1968 book on singularities of complex hypersurfaces, Milnor asked a question about the "unknotting number" of knots that arise as the links of singular points of complex plane curves. The question was eventually answered in the affirmative, using gauge theory, by Kronheimer and Mrowka in 1992. A proof requiring only combinatorial techniques was found much later, by Rasmussen, using Khovanov homology. In this talk we will explore a surprising relationship between these two proofs: an interplay between gauge theory and Khovanov homology.

I will discuss the homological structure of the classifying space of 2d-dimensional (d-1)-connected manifolds. This generalizes the case of surfaces where the stable homology was predicted by the Mumford conjecture and the actual homology by the Harer stability theorem. The methods involve rational homotopy theory and surgery theory.

In this talk I will recall and discuss the conjecture of John Milnor on the homology of Lie groups made discrete, as well as its algebraic analogue, the Friedlander conjecture.

In a work still partially in progress, we give a proof of that conjecture for algebraic groups G over algebraically closed fields. I will sketch some ideas behind this proof, in particular the role of A1-homotopy theory (already used by V. Voevodsky to prove other conjectures of John Milnor concerning mod 2 Galois cohomology and quadratic forms) and the role of a new object attached to G, its simplicial building. We will emphasize the case G = SL_2, SL_3,...

In a work still partially in progress, we give a proof of that conjecture for algebraic groups G over algebraically closed fields. I will sketch some ideas behind this proof, in particular the role of A1-homotopy theory (already used by V. Voevodsky to prove other conjectures of John Milnor concerning mod 2 Galois cohomology and quadratic forms) and the role of a new object attached to G, its simplicial building. We will emphasize the case G = SL_2, SL_3,...

Heegaard Floer homology is an invariant for knots and three-manifolds
defined using methods from Lagrangian Floer homology and symplectic
geometry. I will describe a concrete model for computing this
invariant for knots in S^{3}. Time permitting, I will turn to some
further developments in the theory. The material I will describe is
joint with collaborators, including Robert Lipshitz, Ciprian
Manolescu, Dylan Thurston, and Zoltán Szabó.

The Milnor conjectures relating the graded Witt ring and the mod-2 Galois cohomology ring of a field have been a driving force in the algebraic theory of quadratic forms. The degree 2 norm
residue isomorphism, due to Merkurjev, established the first case of the Milnor conjectures and ushered in a new era in the theory of quadratic forms. We shall explain some of the consequences of the
Milnor conjectures, with particular reference to invariants of fields associated to quadratic forms.

The 1963 work of Kervaire-Milnor on the classification of exotic spheres in terms of the stable homotopy groups of spheres left one unanswered question in dimensions congruent to two modulo four. Subsequent work by Brown-Peterson and Browder gave the answer in all dimensions that are not two less than a power of two. We solve the problem in all remaining cases except 126. Our answer is the opposite of the one that many sought in the 1970s.

For the last 50 years or longer, it has been a basic question in topology to study differential structures of a given manifold. In the early 80's, Donaldson used anti-self solutions of the Yang-Mills equation to study differentiable 4-manifolds and construct a very useful class of differentiable invariants for 4-manifolds. In particular, he was able to show that a topological 4-manifold may have different differentiable structures. In this talk, I will discuss geometric methods in studying 4-manifolds and in particular, some recent progress.

Cobordisms have played an important role in the classification of
manifolds ever since the 1950s. In a different way, they are fundamental to
the mathematical formulation for topological quantum field theory. We will
explain how recent results shed new light on both theories.