Jacob Lurie

Moduli Problems for Ring Spectra

In algebraic geometry, it is common to study a geometric object X (such as a scheme) by means of the functor R 7! Hom(SpecR,X) represented by X. In this paper, we consider functors which are defined on larger classes of rings (such as the class of ring spectra which arise in algebraic topology), and sketch some applications to deformation theory.

On simply laced Lie algebras and their minuscule representations

Abstract. In this paper we adapt a known construction for the simply laced, semisimple Lie algebras (over Z), and thereby obtain a very simple construction for all minuscule representations of those Lie algebras (again over Z). We apply these results to give explicit formulas for tensors invariant under the exceptional algebras

Anti-admissible sets

E_6

The Effective Content of Surreal Algebra

and

Higher Topos Theory

E_7

On the classification of topological field theories

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Derived Algebraic Geometry III: Commutative Algebra

Aczels theory of hypersets provides an interesting alternative to the standard view of sets as inductively constructed, well-founded objects, thus providing a convienent formalism in which to consider non-well-founded versions of classically well-founded constructions, such as the “circular logic” of [3], This theory and ZFC are mutually interpretable; in particular, any model of ZFC has a canonical “extension” to a non-well-founded universe. The construction of this model does not immediately generalize to weaker set theories such as the theory of admissible sets. In this paper, we formulate a version of Aczels antifoundation axiom suitable for the theory of admissible sets. We investigate the properties of models of the axiom system KPU − , that is, KPU with foundation replaced by an appropriate strengthening of the extensionality axiom. Finally, we forge connections between “non-wellfounded sets over the admissible set A ” and the fragment L A of the modal language L ∞ .

Topological Quantum Field Theories from Compact Lie Groups

This paper defines and explores the properties of several effectivizations of the structure of surreal numbers. The construction of one of previously investigated systems, the metadyadics, is shown to be effectively equivalent to the construction of the surreals in . This equivalence is used to answer several open questions concerning the metadyadics. Results obtained seem to indicate that the metadyadics best capture the notion of a recursive surreal number.