The higher lower central series

The Baer invariants Γ _n (G) of a group are central extensions of the elementsγ _n (G) of the lower central series. We show that the inclusionsγ _n +1 ⊂γ _n can be lifted to functor morphisms Γ _n+1→Γ _n and a canonical Lie algebra, analogous to Lazard’s Lie algebra, can be constructed which is explicitly computable. This is applied in various ways.     

The Novikov Conjecture for Linear Groups

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C^*-algebra theory.     

2-Adic valuations of certain ratios of products of factorials and applications

We prove the conjecture of Falikman-Friedland-Loewy on the parity of the degrees of projective varieties of n×n complex symmetric matrices of rank at most k. We also characterize the parity of the degrees of projective varieties of n×n complex skew symmetric matrices of rank at most 2p. We give recursive relations which determine the parity of the degrees of projective varieties of m×n complex matrices of rank at most k. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of n×n skew symmetric real matrices and of m×n real matrices containing a nonzero matrix of rank at most k. The parity questions studied here are also of combinatorial interest since they concern the parity of the number of plane partitions contained in a given box, on the one hand, and the parity of the number of symplectic tableaux of rectangular shape, on the other hand.     

Analytic families of semigroups

The Cauchy Problem for linear partial differential equations with coefficients depending analytically on a parameter, motivates the study of analytic families of infinitesimal generators of semigroups. The relations between the three natural ways to understand “analyticity” of the family are clarified. This is then applied to obtain some general results on the stability of the analyticity of the family under perturbations of the generators.     

Thermal Effects and Structural Changes of Photosynthetic Reaction Centers Characterized by Wide Frequency Band Hydrophone: Effects of Carotenoids and Terbutryn

Photothermal characteristics and light‐induced structural (volume) changes of carotenoid‐containing and noncontaining photosynthetic reaction centers (RCs) were investigated by wide frequency band hydrophone. We found that the presence of carotenoid either does not play considerable role in the light‐induced conformational movements, or these rearrangements are too slow for inducing a photoacoustic (PA) signal. The kinetic component with a few tens of microseconds, exhibited by the carotenoid‐less RCs, appears to be similar to that of triplet state lifetimes, identified by other methods. The binding of terbutryn to the acceptor side is shown to affect the dynamics of the RC. Our results do not confirm large displacements or volume changes induced by the charge movements and by the charge relaxation processes in the RCs in few hundreds of microseconds time scale that accompanies the electron transfer between the primary and secondary electron acceptor quinones.     

On the vanishing of homology in random Čech complexes

We compute the homology of random ?ech complexes over a homogeneous Poisson process on the d‐dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erd?s ‐Rényi phase transition, where the ?ech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 14–51, 2017     

A polynomial oracle-time algorithm for convex integer minimization

In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. We extend these results to convex N-fold integer minimization problems and to convex 2-stage stochastic integer minimization problems. Finally, we present some applications of convex N-fold integer minimization problems for which our approach provides polynomial time solution algorithms.