Algebraic topology uses techniques of algebra to describe and solve problems in geometry and topology. From its inception with Poincare's work on the fundamental group and homology, the field has exploited natural ways to associate numbers, groups, rings, and modules to various spaces. As the field evolved, two dominant themes emerged: "what are the invariants we can construct, and how do we compute them?" and "what is the general framework in which we can do algebraic topology?".
As people grew to better understand the invariants and computations techniques, they saw that they enjoy extra structure and sit in natural families similar to those seen in algebraic geometry. They suffer from the drawback that cohomology theories are not quite as rigid as rings. Here various cooperations with those studying the framework of homotopy theory has allowed ways to make precise the connections with algebraic geometry. This has culminated in the Hopkins-Miller theory of topological modular forms, which records information about elliptic curves and integral modular forms. Lurie's derived algebraic geometry naturally associates ring spectra to deeply signicant objects in algebraic geometry and number theory. This has lead to striking cross-overs in which algebraic topologists and number theorists focus on the same objects: abelian varieties and their moduli.
Algebraic topology on the whole has enjoyed several exciting advances of late, and all of them arise from blending the computational and foundational techniques. These hybridized results harken back to Poincare: algebraic topology should illuminate the geometry, and the interactions of the schools allows a brighter picture. The solution of Hill-Hopkins-Ravenel to the Kervaire Invariant One problem, where a panoply of computations techniques blended with very elementary geometry to solve this 40 year old problem. Lurie's proof of the Cobordism Hypothesis, synthesizing decades of work on topological quantum field theories and intuition about the geometry of manifolds.
The MSRI program will build on this cooperative narrative. A primary goal of the MSRI program is to draw together algebraic topologists of all stripes, reintroducing each to the tools of the others and providing a synergistic research forum. Algebraic topologists, both those focused on the families of invariants and those focused on the framework, will have the opportunity to explore the descriptive language employed by the other. This leads to ferreting out underlying commonalities and grappling with the deeper structures inherent to the subject.
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|January 23, 2014 - January 24, 2014||Connections for Women: Algebraic Topology|
|January 27, 2014 - January 31, 2014||Introductory Workshop: Algebraic Topology|
|April 07, 2014 - April 11, 2014||Reimagining the Foundations of Algebraic Topology|