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We study some versions of the statement of Hadwiger's conjecture for finite as well as infinite graphs.

We prove the Baum--Connes conjecture with arbitrary coefficients for some classes of groups: (1) Linear algebraic groups over a non-archimedean local field. (2) Linear algebraic groups over the adeles of a global field k, provided that at…

The aim of this work is to prove a conjecture related to the Combinatorial Invariance Conjecture of Kazhdan-Lusztig polynomials, in the parabolic setting, for lower intervals in every arbitrary Coxeter group. This result improves and…

We prove a conjecture of Przytycki which asserts that the $n$-quandle of a link $L$ in the 3-sphere is finite if and only if the fundamental group of the $n$-fold cyclic branched cover of the 3-sphere, branched over $L$, is finite.

In this work we provide an upper bound for the multiplicity of a one-dimensional Cohen-Macaulay ring (under certain conditions), describe the rings attaining the equality for this bound, and outline a connection with Wilf's conjecture for…

We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…

We prove a theorem on how a conclusion on homological dimension of a family of coherent sheaves over a scheme can be done from homological dimension of the restriction of this family to the reduction of the base.

We state and prove a realization of King's Conjecture for a category glued from the derived categories of all of the toric varieties arising from a given Cox ring. Our perspective extends ideas of Beilinson and Bondal to all semiprojective…

In 1989, D. Happel pointed out for a possible connection between the global dimension of a finite-dimensional algebra and its Hochschild cohomology: is it true that the vanishing of Hochschild cohomology higher groups is sufficient to…

Let F be a global field, A its ring of adeles, G a reductive group over F. We prove the Baum-Connes conjecture for the adelic group G(A).

We prove the topological analogue of the period-index conjecture in each dimension away from a small set of primes.

We prove an infinite family of lacunary recurrences for the Lucas numbers using combinatorial means.

We prove a result of cohomology and base change for families of coherent systems over a curve. We use that in order to prove the existence of (non-split, non-degenerate) universal families of extensions for families of coherent systems (in…

We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive…

We use symplectic techniques to obtain partial results on Mahler's conjecture about the product of the volume of a convex body and the volume of its polar. We confirm the conjecture for hyperplane sections or projections of $\ell_p$-balls…

Some conjectures about Heegaard genera and ranks of fundamental groups of 3-manifolds are formulated, and it is shown that they imply new statements about hyperbolic volume.

This is (mainly) a survey of recent results on the problem of the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb flows. We focus on the Conley conjecture, proved for a broad class of closed symplectic…

We introduce a class of polytopes that we call chainlink polytopes and which allow us to construct infinite families of pairs of non isomorphic rational polytopes with the same Ehrhart quasi-polynomial. Our construction is based on circular…

In this paper, we find a more straightforward problem that is equivalent to one of the major challenges in knot theory: the classification of links in the 3-sphere. More precisely, we provide a simpler braid description for all links in the…

We disprove a conjecture of Simon for higher-order Szego theorems for orthogonal polynomials on the unit circle and propose a modified version of the conjecture.