This article treats the compactification of the space of higher spin curves, i.e. pairs (X,L) with L an r root of the canonical bundle of X. More precisely, for positive integers r and g, with g > 2, r dividing 2g − 2, and for a flat family of smooth curves f : X → T, an r-spin structure on X is a line bundle L such that L ∼= ωX /T . And an r-spin curve over T is a flat family of smooth curves ...

Let Y C Pm be a subvariety of codimension d defined by an ideal / in charp > 0 with //'(Y, 0 (-1)) = 0. If t is an integer greater than log (d) and Hi( Y, f/l"+i) = 0 for n » 0 and i =1,2, then Pic(Y) is an extension of a finite p-primary group of exponent at most pt by Z[ 0 (1 )1 and Br'(Y)(p) is a group of exponent at most pl'. If Y is also smooth and defined over a finite field with dim Y < ...

Within a constructive homological algebra approach, we study the factorization and decomposition problems for general linear functional systems and, in particular, for multidimensional linear systems appearing in control theory. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, time-delay operators), we first concentrate o...

We study deformations of associative submanifolds Y 3 ⊂ M 7 of a G 2 manifold M 7. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative submanifolds of M. The local equations at each associative Y are restrictions of a global eq...

In the previous lecture, we outlined some approaches to describing the cohomology of the classifying space of G-bundles M on a Riemann surface X. For example, we asserted that the cochain complex C * (M; Q) is quasi-isomorphic to a continuous tensor product ⊗ x∈X C * (BG x ; Q). Here it is vital that we work at the level of cochains, rather than cohomology: there is no corresponding procedure t...

Introduction. In [1], the Koszul complex was used to study the relationship between codimension and multiplicity. It also helped us investigate Macaulay modules and rings, and provided a context in which to prove the Cohen-Macaulay Theorem concerning the unmixedness of complete intersections. Now there is a generalization of the Cohen-Macaulay Theorem (known, we believe, as the generalized Cohe...

This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential operators on a compact manifold, namely those operators P with the property that dim ker P, dim coker P < ∞. In general these integers are very difficult to compute without some very precise information abou...

In this Note, using an idea due to Thomason [8], we define a “homology theory” on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find Balmers’s higher Witt groups. For more general rings, this homology is isomorphic to the KT-theory of Hornbostel [3], inspired by the work of Williams [9]. ...

We study deformations of associative submanifolds of a G 2 manifold. We show that deformation spaces can be perturbed to be smooth and finite dimensional , and they can be made compact by constraining them with an additional equation and reducing it to Seiberg-Witten theory. This allows us to associate in-variants of certain associative submanifolds. More generally we apply this process to cert...

We examine a bilinear form associated with a real arrangement of hyperplanes introduced in [Schechtman and Varchenko 1991]. Our main objective is to show that the linear algebraic properties of this bilinear form are related to the combinatorics and topology of the hyperplane arrangement. We will survey results and state a number of open problems which relate the determinant, cokernel structure...