Coefficients for the Farrell-Jones Conjecture

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.     

Remarks on Harada’s conjecture

An open conjecture by Harada from 1981 gives an easy characterization of the p-blocks of a finite group in terms of the ordinary character table. Kiyota and Okuyama have shown that the conjecture holds for p-solvable groups. In the present work we extend this result using a criterion on the decomposition matrix. In this way we prove Harada’s Conjecture for several new families of defect groups and for all blocks of sporadic simple groups. In the second part of the paper we present a dual approach to Harada’s Conjecture.     

The Atiyah Conjecture and $$L^2$$-cohomology computations

If G is a discrete group with bounded torsion, the Strong Atiyah Conjecture predicts that the \(L^2\)-Betti numbers of any G-space are rational, with denominators determined by the orders of the torsion subgroups. We prove the conjecture for some new examples of groups, including the class of virtually sparse special groups. We then show that the conjecture can be used to compute the \(L^2\)-cohomology of certain Coxeter groups that have highly symmetric nerves.     

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel n-Unipotent Conjecture and the Vanishing n  -Massey Conjecture for n≥3$n\ge 3$ were formulated. These conjectures evolved in the last forty years as a byproduct of the application of topological methods to Galois cohomology. We show that both of these conjectures are true for odd rigid fields. This is the first case of a significant family of fields where both of the conjectures are verified besides fields whose Galois groups of p-maximal extensions are free pro-p-groups. We also prove the Kernel Unipotent Conjecture for Demushkin groups of rank 2, and establish various filtration results for free pro-p-groups, provide examples of pro-p-groups which do not have the kernel n-unipotent property, compare various Zassenhaus filtrations with the descending p-central series and establish new type of automatic Galois realization.     

Isomorphism Conjecture for homotopy K-theory and groups acting on trees

We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the Farrell-Jones Conjecture in algebraic K-theory.     

A solution to de Groot’s absolute cone conjecture

A compactum X is an ‘absolute cone’ if, for each of its points x, the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n-dimensional absolute cone is an n-cell. In this paper, we give a complete solution to that conjecture. In particular, we show that the conjecture is true for n≤3 and false for n≥5. For n=4, the absolute cone conjecture is true if and only if the 3-dimensional Poincaré Conjecture is true.     

Estimates on conjectures of Minkowski and woods II

Let ? ⁿ be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < Rwhich contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in ? ⁿ of radius \(\sqrt n /2\) contains a point of ∧. This is known to be true for n ≤ 8. Recently we gave estimates on a more general conjecture of Woods for n ≥ 9. This lead to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms. Here we shall refine our method to obtain improved estimates for Woods Conjecture. These give improved estimates of Minkowski’s conjecture for 9 ≤ n ≤ 31.     

A maximum degree theorem for diameter-2-critical graphs

A graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ?n ²/4? and that the extremal graphs are the complete bipartite graphs K _?n/2?,?n/2?. Fan [Discrete Math. 67 (1987), 235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98] proved the conjecture for n > n ₀ where n ₀ is a tower of 2’s of height about 10¹⁴. The conjecture has yet to be proven for other values of n. Let Δ denote the maximum degree of G. We prove the following maximum degree theorems for diameter-2-critical graphs. If Δ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and Δ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.     

An upper bound for permanents of nonnegative matrices

A recent conjecture of Caputo, Carlen, Lieb, and Loss, and, independently, of the author, states that the maximum of the permanent of a matrix whose rows are unit vectors in lp is attained either for the identity matrix I or for a constant multiple of the all-1 matrix J.The conjecture is known to be true for p=1 (I) and for p?2 (J).We prove the conjecture for a subinterval of (1,2), and show the conjectured upper bound to be true within a subexponential factor (in the dimension) for all 1<p<2. In fact, for p bounded away from 1, the conjectured upper bound is true within a constant factor.     

Partial augmentations power property: A Zassenhaus Conjecture related problem

Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group G is conjugate in the rational group algebra of G to an element in ±G. We review the known weaker versions of this conjecture and introduce a new condition, on the partial augmentations of the powers of a unit of finite order in $\mathbb{Z}G$, which is weaker than the Zassenhaus Conjecture but stronger than its other weaker versions.We prove that this condition is satisfied for units mapping to the identity modulo a nilpotent normal subgroup of G. Moreover, we show that if the condition holds then the HeLP Method adopts a more friendly form and use this to prove the Zassenhaus Conjecture for a special class of groups.     

The Anderson-Badawi conjecture for commutative algebras over infinite fields

In [1], Anderson and Badawi conjecture that every n-absorbing ideal of a commutative ring is strongly n-absorbing. In this article we prove their conjecture in certain cases (in particular this is the case for commutative algebras over an infinite field). We also show that an affirmative answer to another conjecture in [1] implies the Anderson-Badawi Conjecture.     

On the GLY Conjecture of upper estimate of positive integral points in real right-angled simplices

The GLY (Granville-Lin-Yau) Conjecture is a generalization of Lin, Xu and Yau's results. An important application of GLY is its use in characterizing an affine hypersurface in Cn as a cone over a nonsingular projective variety. In addition, the Rough Upper Estimate Conjecture in GLY, recently proved by Yau and Zhang, implies the Durfee Conjecture in singularity theory. This paper develops a unified approach to prove the Sharp Upper Estimate Conjecture for general n. Using this unified approach, we prove that the Sharp Upper Estimate Conjecture is true for n=4,5,6. After giving a counter-example to show that the Sharp Upper Estimate Conjecture is not true for n=7, we propose a Modified GLY Conjecture. For each fixed n, our unified approach can be used to prove this Modified GLY Conjecture.     

Partitioning edge-coloured complete graphs into monochromatic cycles

A conjecture of Erdös, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for $r=2$. In this note we show that in fact this conjecture is false for all $r⩾3$. We also discuss some weakenings of this conjecture which may still be true.     

A refinement of Alperin’s Conjecture for blocks of the endomorphism algebra of the Sylow permutation module

We present a refinement of Alperin’s Conjecture involving the blocks of the endomorphism algebra of the permutation module formed by the cosets of a p-subgroup. We prove the conjecture in two special cases where every weight module has a simple socle.     

Twisted Group Algebras, Normal Subgroups and Derived Equivalences

We study Rickard equivalences between p-blocks of twisted group algebras and their local structure, in connection with Dade's conjectures. We prove that an extended form of Broué's conjecture implies Dade's Inductive Conjecture in the Abelian defect group case; this is a consequence of the fact that Rickard equivalences induced by complexes of graded bimodules preserve the relevant Clifford theoretical invariants. As an application, we show that these conjectures hold for p-extensions of blocks with cyclic defect groups.     

Uniquely colorable perfect graphs

This paper defines the concept of sequential coloring. If G or its complement is one of four major types of perfect graphs, G is shown to be uniquely k-colorable it and only if it is sequentially k-colorable. It is conjectured that this equivalence is true for all perfect graphs. A potential role for sequential coloring in verifying the Strong Perfect Graph Conjecture is discussed. This conjecture is shown to be true for strongly sequentially colorable graphs.     

Icosahedral galois extensions and elliptic curves

Summary This paper is devoted to the last unsolved case of the Artin Conjecture in two dimensions. Given an irreducible 2-dimensional complex representation of the absolute Galois group of a number fieldF, the Artin Conjecture states that the associatedL-series is entire. The conjecture has been proved for all cases except the icosahedral one. In this paper we construct icosahedral representations of the absolute Galois group of ℚ(√5) by means of 5-torsion points of an elliptic curve defined over ℚ. We compute the L-series explicitely as an Euler product, giving algorithms for determining the factors at the difficult primes. We also prove a formula for the conductor of the elliptic representation. A feasible way of proving the Artin Conjecture in a given case is to construct a modular form whose L-series matches the one obtained from the representation. In this paper we obtain the following result: letρ be an elliptic Galois representation over ℚ(√5) of the type above, and letL(s, ρ) be the corresponding L-series. If there exists a Hilbert modular formf of weight one such thatL(s, f) ≡L(s, ρ) modulo a certain ideal above (√5), then the Artin conjecture is true forρ. This article was processed by the author using the LATEX style filecljour1m from Springer-Verlag.     

On the deque conjecture for the splay algorithm

Splay is a simple, efficient algorithm for searching binary search trees, devised by Sleator and Tarjan, that reorganizes the tree after each search by means of rotations. An open conjecture of Sleator and Tarjan states that Splay is, in essence, the fastest algorithm for processing any sequence of search operations on a binary search tree, using only rotations to reorganize the tree. Tarjan proved a basic special case of this conjecture, called theScanning Theorem, and conjectured a more general special case, called theDeque Conjecture. The Deque Conjecture states that Splay requires linear time to process sequences of deque operations on a binary tree. We prove the following results:

1. Almost tight lower and upper bounds on the maximum numbers of occurrences of various types of right rotations in a sequence of right rotations performed on a binary tree. In particular, the lower bound for right 2-turns refutes Sleator's Right Turn Conjecture.
2. A linear times inverse Ackerman upper bound for the Deque Conjecture. This bound is derived using the above upper bounds.
3. Two new proofs of the Scanning Theorem, one, a simple potential-based proof that solves Tarjan's problem of finding a potential-based proof for the theorem, the other, an inductive proof that generalizes the theorem.

     

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.