Finiteness conjecture and subdivision

The finiteness conjecture by J.C. Lagarias and Y. Wang states that the joint spectral radius of a finite set of square matrices is attained on some finite product of such matrices. This conjecture is known to be false in general. Nevertheless, we show that this conjecture is true for a big class of finite sets of square matrices used for the smoothness analysis of scalar univariate subdivision schemes with finite masks.     

A conjecture on bipartite graphical regular representations

In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR.First, we find a natural obstruction that prevents a finite group from admitting a bipartite GRR. Then we give a complete classification of the finite groups satisfying this natural obstruction and hence not admitting a bipartite GRR. Based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite GRR.Next, we prove the existence of bipartite DRRs for most of the finite groups not admitting a bipartite GRR found in this paper. Actually, we prove a much stronger result: we give an asymptotic enumeration of the bipartite DRRs over these groups. Again, based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite DRR.     

The Tate conjecture for K3 surfaces over finite fields

Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.     

About the Connes embedding conjecture

In his celebrated paper in 1976, A. Connes casually remarked that any finite von Neumann algebra ought to be embedded into an ultraproduct of matrix algebras, which is now known as the Connes embedding conjecture or problem. This conjecture became one of the central open problems in the field of operator algebras since E. Kirchberg’s seminal work in 1993 that proves it is equivalent to a variety of other seemingly totally unrelated but important conjectures in the field. Since then, many more equivalents of the conjecture have been found, also in some other branches of mathematics such as noncommutative real algebraic geometry and quantum information theory. In this note, we present a survey of this conjecture with a focus on the algebraic aspects of it.     

The graph formulation of the union-closed sets conjecture

The union-closed sets conjecture asserts that in a finite non-trivial union-closed family of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The conjecture is trivially true for non-bipartite graphs and we show that it holds also for the classes of chordal bipartite graphs, subcubic bipartite graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs. We derive that the union-closed sets conjecture holds for all union-closed families being the union-closure of sets of size at most three.     

On the Gray index conjecture for phantom maps

We study the Gray index, a numerical invariant for phantom maps. It has been conjectured that the only phantom map between finite-type spaces with infinite Gray index is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid if the target spaces of the phantom maps are restricted to being simply connected finite complexes.As a result of the counter example, we can show that SNT^∞(X) can be non-trivial for some space X of finite type.     

A note on the Nakai conjecture

The conjectures of Zariski-Lipman and of Nakai are still open in general in the class of rings essentially of finite type over a field of characteristic zero. However, they have long been known to be true in dimension one. Here we give counterexamples to both conjectures in the class of one-dimensional pseudo-geometric local domains that contain a field of characteristic zero. Likewise, in connection with a recent result of Traves on the Nakai conjecture, we also show that their hypothesis of finite generation of the integral closure cannot be removed even in the class of local domains containing a field of characteristic zero.

     

On a long-standing conjecture by Pólya–Szeg? and related topics

The electrostatic capacity of a convex body is usually not simple to compute. We discuss two possible approximations of it. The first one is related to a long-standing conjecture by Pólya–Szegö. It states that, among all convex bodies, the “worst shape” for the approximation exists and is the planar disk. We prove the first part of this conjecture, and we establish some related results which give further evidence for the validity of the second part. We also suggest some complementary conjectures and open problems. The second approximation we study is based on the use of web functions.     

On a long-standing conjecture by Pólya–Szegö and related topics

The electrostatic capacity of a convex body is usually not simple to compute. We discuss two possible approximations of it. The first one is related to a long-standing conjecture by Pólya–Szegö. It states that, among all convex bodies, the “worst shape” for the approximation exists and is the planar disk. We prove the first part of this conjecture, and we establish some related results which give further evidence for the validity of the second part. We also suggest some complementary conjectures and open problems. The second approximation we study is based on the use of web functions.Received: September 29, 2003     

The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case

Publications mathématiques de l'IHÉS - In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups $U_{n}\times U_{n+1}$ in all the...     

A reduction theorem for the McKay conjecture

The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL₂(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2 ^e or q=3 ^e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J ₁, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.     

A conjecture on B-groups

In this note, I propose the following conjecture: a finite group $G$ is nilpotent if and only if its largest quotient $B$ -group $\beta (G)$ is nilpotent. I give a proof of this conjecture under the additional assumption that $G$ be solvable. I also show that this conjecture is equivalent to the following: the kernel of restrictions to nilpotent subgroups is a biset-subfunctor of the Burnside functor.     

Frames and the Feichtinger conjecture

We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linearly independent sequences. We further show that the two conjectures are implied by the paving conjecture. Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences.

     

On Frobenius' conjecture

By using the classification theorem of finite simple groups, we have shown that “IfG is a finite group,H is a coprime operator group ofG, C _G(H)≤S(G), thenG is solvable.” As a direct corollary, we have completely proved the long-standing conjecture on fixed-point-free automorphism group. The author is grateful to Professor Chen Zhongmu for his supervision.     

Generalizations of the image conjecture and the Mathieu conjecture

We first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of Rⁿ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion.     

Semiorders and the 1/3–2/3 conjecture

A well-known conjecture of Fredman is that, for every finite partially ordered set (X, <) which is not a chain, there is a pair of elements x, y such that P(x, the proportion of linear extensions of (X, <) with x below y, lies between 1/3 and 2/3. In this paper, we prove the conjecture in the special case when (X, <) is a semiorder. A property we call 2-separation appears to be crucial, and we classify all locally finite 2-separated posets of bounded width.     

From the Kneser–Poulsen conjecture to ball-polyhedra

A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser [M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass, Arch. Math. 6 (1955) 382–390] and Poulsen [E.T. Poulsen, Problem 10, Math. Scand. 2 (1954) 346]. According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that the distance between the centers of every pair of balls is decreased, then the volume of the union (resp., intersection) of the balls is decreased (resp., increased). In the first half of this paper we survey the state of the art of the Kneser–Poulsen conjecture in Euclidean, spherical as well as hyperbolic spaces with the emphases being on the Euclidean case. Based on that it seems very natural and important to study the geometry of intersections of finitely many congruent balls from the viewpoint of discrete geometry in Euclidean space. We call these sets ball-polyhedra. In the second half of this paper we survey a selection of fundamental results known on ball-polyhedra. Besides the obvious survey character of this paper we want to emphasize our definite intention to raise quite a number of open problems to motivate further research.     

A conjecture on arithmetic fundamental groups

The conjecture is the following: Over an algebraic variety over a finite field, the geometric monodromy group of every smooth is finite. We indicate how to prove this for rank 2, using results of Drinfeld. We also show that the conjecture implies that certain deformation rings of Galois representations are complete intersection rings. This material is based upon work supported by the National Science Foundation under Grant No. 9970049.     

Fibred coarse embeddings,a-T-menability and the coarse analogue of the Novikov conjecture

The connection between the coarse geometry of metric spaces and analytic properties of topological groupoids is well known. One of the main results of Skandalis, Tu and Yu is that a space admits a coarse embedding into Hilbert space if and only if a certain associated topological groupoid is a-T-menable. This groupoid characterisation then reduces the proof that the coarse Baum–Connes conjecture holds for a coarsely embeddable space to known results for a-T-menable groupoids. The property of admitting a fibred coarse embedding into Hilbert space was introduced by Chen, Wang and Yu to provide a property that is sufficient for the maximal analogue to the coarse Baum–Connes conjecture and in this paper we connect this property to the traditional coarse Baum–Connes conjecture using a restriction of the coarse groupoid and homological algebra. Additionally we use this results to give a characterisation of the a-T-menability for residually finite discrete groups.     

Recasting the Elliott conjecture

Let A be a simple, unital, finite, and exact C^*-algebra which absorbs the Jiang–Su algebra tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases— -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of -stable C^*-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that -theoretic invariants will classify separable and nuclear C^*-algebras—with the recent appearance of counterexamples to its strongest concrete form. Research supported by the DGI MEC-FEDER through Project MTM2005-00934, and the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. A. S. Toms was also supported in part by an NSERC Discovery Grant.