A New Matrix-Tree Theorem

The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have exactly three vertices) the spanning trees are generated by the Pfaffian of a suitably defined matrix. This result can be interpreted topologically as an expression for the lowest order term of the Alexander-Conway polynomial of an algebraically split link. We also prove some algebraic properties of our Pfaffian-tree polynomial.Comment: minor changes, 29 pages, version accepted for publication in Int. Math. Res. Notice

Matrix factorizations and singularity categories for stacks

We study matrix factorizations of a section W of a line bundle on an algebraic stack. We relate the corresponding derived category (the category of D-branes of type B in the Landau-Ginzburg model with potential W) with the singularity category of the zero locus of W generalizing a theorem of Orlov. We use this result to construct push-forward functors for matrix factorizations with relatively proper support.Comment: 29 page

On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields

Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation \gal(\bar{K}/K) \to \gl_2(\ff_\ell). A famous theorem of Serre states that as long as $E$ has no Complex Multiplication (CM), the map \gal(\bar{K}/K) \to \gl_2(\ff_\ell) is surjective for all but finitely many $\ell$. We say that a prime number $\ell$ is exceptional (relative to the pair $(E,K)$) if this map is not surjective. Here we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of $E$. We show in particular that conditionally on the Generalized Riemann Hypothesis (GRH), the largest exceptional prime of an elliptic curve $E$ without CM is no larger than a constant (depending on $K$) times $\log N_E$, where $N_E$ is the absolute value of the norm of the conductor. This answers affirmatively a question of Serre

Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.Comment: v1: 45 pages, v2: added the generalized HRR theorem (Cardy condition) and new examples with the boundary-bulk maps, v3: 54 pages, to appear in Duke Math.