Remarks on the Milnor conjecture over schemes

The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging from sums of squares to the structure of absolute Galois groups. Here, we survey some recent work on generalizations of the Milnor conjecture to the context of schemes (mostly smooth varieties over fields of characteristic not 2). Surprisingly, a version of the Milnor conjecture fails to hold for certain smooth complete p-adic curves with no rational theta characteristic (this is the work of Parimala, Scharlau, and Sridharan). We explain how these examples fit into the larger context of an unramified Milnor question, offer a new approach to the question, and discuss new results in the case of curves over local fields and surfaces over finite fields.Comment: 23 page

A-branes and Noncommutative Geometry

We argue that for a certain class of symplectic manifolds the category of A-branes (which includes the Fukaya category as a full subcategory) is equivalent to a noncommutative deformation of the category of B-branes (which is equivalent to the derived category of coherent sheaves) on the same manifold. This equivalence is different from Mirror Symmetry and arises from the Seiberg-Witten transform which relates gauge theories on commutative and noncommutative spaces. More generally, we argue that for certain generalized complex manifolds the category of generalized complex branes is equivalent to a noncommutative deformation of the derived category of coherent sheaves on the same manifold. We perform a simple test of our proposal in the case when the manifold in question is a symplectic torus.Comment: 15 pages, late

Weakly infinite dimensional subsets of R^N

The Continuum Hypothesis implies an Erd\"os-Sierpi\'nski like duality between the ideal of first category subsets of \reals^{\naturals}, and the ideal of countable dimensional subsets of \reals^{\naturals}. The algebraic sum of a Hurewicz subset - a dimension theoretic analogue of Sierpinski sets and Lusin sets - of \reals^{\naturals} with any compactly countable dimensional subset of \reals^{\naturals} has first category.Comment: 16 pages, corrected Statements of Theorem 6 and Lemma 8, Inserted Problem 1, Inserted remarks by R. Pol, solving Problem 3, in "Added in Proof" sectio

Some remarks on orbit sets of unimodular rows

We give a cohomological interpretation of orbit sets of unimodular rows of length d+1 over smooth algebras of Krull dimension d.Comment: 18 page

More about vanishing cycles and mutation

The paper continues the discussion of symplectic aspects of Picard-Lefschetz theory begun in "Vanishing cycles and mutation" (this archive). There we explained how to associate to a suitable fibration over a two-dimensional disc a triangulated category, the "derived directed Fukaya category" which describes the structure of the vanishing cycles. The present second part serves two purposes. Firstly, it contains various kinds of algebro-geometric examples, including the "mirror manifold" of the projective plane. Secondly there is a (largely conjectural) discussion of more advanced topics, such as (i) Hochschild cohomology, (ii) relations between Picard-Lefschetz theory and Morse theory, (iii) a proposed "dimensional reduction" algorithm for doing certain Floer cohomology computations.Comment: 33 pages, LaTeX2e, 9 eps figure

Colours, Corners And Complexity

"There is a philosophical question as to what one really sees". Wittgenstein's remark raises all sorts of questions: Does one see tables and chairs, people jumping up and down, their jumps, their sadness ? Does one see colours and forms, coloured forms, dynamic and static, that are above or to the left of other coloured forms ? If the latter, are these things one sees private entities or public entities as are, presumably, tables and chairs ? If both answers are legitimate (sometimes, or whenever we see ?) what are the relations between the people we see and the coloured forms that we also see ? In other words, is what is presented to me in my visual field private, public or partly private and partly public

An Introduction To The Web-Based Formalism

This paper summarizes our rather lengthy paper, "Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions," and is meant to be an informal, yet detailed, introduction and summary of that larger work.Comment: 50 pages, 40 figure

Hermitian-holomorphic (2)-Gerbes and tame symbols

We observe that the line bundle associated to the tame symbol of two invertible holomorphic functions also carries a fairly canonical hermitian metric, hence it represents a class in a Hermitian holomorphic Deligne cohomology group. We put forward an alternative definition of hermitian holomorphic structure on a gerbe which is closer to the familiar one for line bundles and does not rely on an explicit ``reduction of the structure group.'' Analogously to the case of holomorphic line bundles, a uniqueness property for the connective structure compatible with the hermitian-holomorphic structure on a gerbe is also proven. Similar results are proved for 2-gerbes as well. We then show the hermitian structures so defined propagate to a class of higher tame symbols previously considered by Brylinski and McLaughlin, which are thus found to carry corresponding hermitian-holomorphic structures. Therefore we obtain an alternative characterization for certain higher Hermitian holomorphic Deligne cohomology groups.Comment: Sections on comparisons for hermitian connective structures added at referee's request. Some new results on compatibility between hermitian and analytic connective structure