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Abstract. We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N> 0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KR-homology of knots with 9 crossings or fewer. 1.

Abstract. We introduce the Farrell-Jones Conjecture with coefficients in an additive category with G-action. This is a variant of the Farrell-Jones Conjecture about the algebraic K- or L-Theory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties. Since known proofs using controlled algebra carry over to the set-up with coefficients we obtain new results about the original Farrell-Jones Conjecture. The conjecture with coefficients implies the fibered version of the Farrell-Jones Conjecture. 1.

Many questions concerning a zero-dimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the A-module structure of the dual space � A. An important feature of our algorithms is that we do not require � A to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 n D 5/2) and O(n2 n D 5/2) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.

Abstract. Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on computations inside and outside of the system (twolevel approach). 1 Formal Computational Proofs 1.1 Machines and the Quest for Correctness It is generally considered that modern mathematical logic was born towards the end of 19 th century, with the work of logicians like Frege, Peano, Russell or Zermelo, which lead to the precise definition of the notion of logical deduction and to formalisms like arithmetic, set theory or early type theory. From then on, a mathematical proof could be understood as a mathematical object itself, whose correction obeys some well-defined syntactical rules. In most formalisms, a formal proof is viewed as some tree-structure; in natural deduction for instance, given to formal proofs σA and σB respectively of propositions A and B, these can be combined in order to build a proof of A ∧ B: σA σB ⊢ A ⊢ B ⊢ A ∧ B To sum things up, the logical point of view is that a mathematical statement holds in a given formalism if there exists a formal proof of this statement which follows the syntactical rules of the formalism. A traditional mathematical text can then be understood as an informal description of the formal proof. Things changed in the 1960-ties, when N.G. de Bruijn’s team started to use computers to actually build formal proofs and verify their correctness. Using the fact that data-structures like formal proofs are very naturally represented in a computer’s memory, they delegated the proof-verification work to the machine; their software Automath is considered as the first proof-system and is the common

Let K be a function field, let ϕ ∈ K(T) be a rational map of degree d ≥ 2 defined over K, and suppose that ϕ is not isotrivial. In this paper, we show that a point P ∈ P 1 ( ¯ K) has ϕ-canonical height zero if and only if P is preperiodic for ϕ. This answers affirmatively a question of Szpiro and Tucker, and generalizes a recent result of Benedetto from polynomials to rational functions. We actually prove the following stronger result, which is a variant of the Northcott finiteness principle: there exists ε> 0 such that the set of points P ∈ P 1 (K) with ϕ-canonical height at most ε is finite. Our proof is essentially analytic, making use of potential theory on Berkovich spaces to study the dynamical Green’s functions gϕ,v(x, y) attached to ϕ at each place v of K. For example, we show that every conjugate of ϕ has bad reduction at v if and only if gϕ,v(x, x)> 0 for all x ∈ P 1 Berk,v, where P1 Berk,v denotes the Berkovich projective line over the completion of ¯ Kv. In an appendix, we use a similar method to give a new proof of the Mordell-Weil theorem for elliptic curves over K.

Holonomic functions (respectively sequences) satisfy linear ordinary differential equations (respectively recurrences) with polynomial coefficients. This class can be generalized to functions of several continuous or discrete variables, thus encompassing most special functions that occur in applications, for instance in mathematical physics. In particular, all hypergeometric functions are holonomic. This work makes several contributions to the theory of holonomic functions and sequences. In the first part, new methods are introduced to show that a given function or sequence is not holonomic. First, number-theoretic methods are applied, and connections to the theory of transcendental numbers are pointed out. A new application of the saddle point method from asymptotic analysis to a concrete function is given, which proves its non-holonomicity. The second part addresses questions of positivity of holonomic (and more general) sequences. First, two new methods for proving positivity of sequences algorithmically

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Abstract. This is an overview article on finite-dimensional algebras and quivers, written for the Encyclopedia of Mathematical Physics. We cover path algebras, Ringel-Hall algebras and the quiver varieties of Lusztig and Nakajima. 1.

ABSTRACT. This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p: ( ˜Q, ˜q) → (Q,q). This in turn leads us to define the algebraic fundamental group π1(Q,q): = Aut(p) = {g ∈ Adj(Q) ′ | q g = q}, where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected coverings of (Q,q) and subgroups of π1(Q,q). Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire’s algebraic covering theory of perfect groups. A detailed investigation also reveals some crucial differences, which we illustrate by numerous examples. As an application we obtain a simple formula for the second (co)homology group of a quandle Q. It has long been known that H1(Q) ∼ = H 1 (Q) ∼ = Z[π0(Q)], and we construct natural isomorphisms H2(Q) ∼ = π1(Q,q)ab and H 2 (Q,A) ∼ = Ext(Q,A) ∼ = Hom(π1(Q,q),A), reminiscent of the classical Hurewicz isomorphisms in degree 1. This means that whenever π1(Q,q) is known, (co)homology calculations in degree 2 become very easy.