Fulkersonʼs Conjecture and Loupekine snarks

Fulkersonʼs Conjecture says that every bridgeless cubic graph has six perfect matchings such that each edge belongs to exactly two of them. In 1976, F. Loupekine created a method for constructing new snarks from already known ones. We consider an infinite family of snarks built with Loupekineʼs method, and verify Fulkersonʼs Conjecture for this family.     

Liouville numbers and Schanuel’s Conjecture

In this paper, using an argument of P. Erd?s, K. Alniaçik, and É. Saias, we extend earlier results on Liouville numbers, due to P. Erd?s, G.J. Rieger, W. Schwarz, K. Alniaçik, É. Saias, E.B. Burger. We also produce new results of algebraic independence related with Liouville numbers and Schanuel’s Conjecture, in the framework of ${G_\delta}$ -subsets.     

Hardy–Littlewoodʼs inequalities in the case of a capacity

Hardy–Littlewood?s inequalities, well known in the case of a probability measure, are extended to the case of a monotone (but not necessarily additive) set function, called a capacity. The upper inequality is established in the case of a capacity assumed to be continuous and submodular, the lower — under assumptions of continuity and supermodularity.     

Chen’s Conjecture and Its Generalization

Let l1, l2, ..., lg be even integers and x be a sufficiently large number. In this paper, the authors prove that the number of positive odd integers k ≤ x such that （k ＋l1）^2, （k ＋l2）^2, ..., （k ＋lg）^2 can not be expressed as 2^n＋p^α is at least c（g）x, where p is an odd prime and the constant c（g） depends only on g.     

Chang’s Conjecture and weak square

We investigate how weak square principles are denied by Chang’s Conjecture and its generalizations. Among other things we prove that Chang’s Conjecture does not imply the failure of ${\square_{\omega_1, 2}}$ , i.e. Chang’s Conjecture is consistent with ${\square_{\omega_1, 2}}$ .     

Zhang’s Conjecture and the Effective Bogomolov Conjecture over function fields

We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang’s Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of good reduction, for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang’s Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.     

Nakajimaʼs remark on Hennʼs proof

We fill up a gap in Hennʼs proof concerning large automorphism groups of function fields of degree 1 over an algebraically closed field of positive characteristic.     

Perverse equivalences and Brouéʼs conjecture

We give a new approach to the construction of derived equivalences between blocks of finite groups, based on perverse equivalences, in the setting of Broué?s abelian defect group conjecture. We provide in particular local and global perversity data describing the principal blocks and the derived equivalences for a number of finite simple groups with Sylow subgroups elementary abelian of order 9. We also examine extensions to automorphism groups in a general setting.     

Baumgartnerʼs conjecture and bounded forcing axioms

We study the spectrum of forcing notions between the iterations of σ-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of α-proper forcings for indecomposable countable ordinals α, the Axiom A forcings and forcings completely embeddable into an iteration of a σ-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartner?s Axiom A. This resolves a conjecture of Baumgartner from the 1980s.     

Hedetniemi’s Conjecture and Dense Boolean Lattices

The category D{\mathcal{D}} of finite directed graphs is Cartesian closed, hence it has a product and exponential objects. For a fixed K, let K^DK^{\mathcal{D}} be the class of all directed graphs of the form K ^G , preordered by the existence of homomorphisms, and factored by homomorphic equivalence. It has long been known that K^DK^{\mathcal{D}} is always a Boolean lattice. In this paper we prove that for any complete graph K _n with n ≥ 3, K_n^DK_n^{\mathcal{D}} is dense, hence up to isomorphism it is the unique countable dense Boolean lattice. In graph theory, the structure of K_n^DK_n^{\mathcal{D}} is connected to the conjecture of Hedetniemi on the chromatic number of a categorical product of graphs.     

Effective Constructions in Plethysms and Weintraub’s Conjecture

We give a short proof of Weintraub’s conjecture (Weintraub J Algebra 129:103–114, 1990), first proved in Bürgisser et al. (J Algebra 328:322–329, 2011), by constructing explicit highest weight vectors in the plethysms S ^p (?∧?^2q W).     

Order of Selbergʼs and Ruelleʼs zeta functions for compact even-dimensional locally symmetric spaces

We prove that Selberg?s and Ruelle?s zeta functions considered by U. Bunke and M. Olbrich can be represented as quotients of two entire functions of order not larger than the dimension of the underlaying compact, even-dimensional, locally symmetric space.     

On Alperin’s Conjecture and Certain Subgroup Complexes

We prove a new formula about local control of the number of p-regular conjugacyclasses of a finite group. We then relate the results to Alperins weight conjecture to obtain newresults describing the number of simple modules for a finite group in terms of weights of solvablesubgroups. Finally, we use the results to obtain new formulations of Alperins weight conjecture,and to obtain restrictions on the structure of a minimal counterexample.