Donaldson's theorem,Heegaard Floer homology,and knots with unknotting number one

We establish an obstruction to unknotting an alternating knot by a single crossing change. The obstruction is lattice-theoretic in nature, and combines Donaldson's diagonalization theorem with an obstruction developed by Ozsváth and Szabó using Heegaard Floer homology. As an application, we enumerate the alternating 3-braid knots with unknotting number one, and show that each has an unknotting crossing in its standard alternating diagram.     

Restriction to compact subgroups in the cyclic homology of reductive p-adic groups

Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.     

When does the top homology of a random simplicial complex vanish?

Several years ago Linial and Meshulam (Combinatorica 26 (2006) 457–487) introduced a model called of random n‐vertex d‐dimensional simplicial complexes. The following question suggests itself very naturally: What is the threshold probability at which the d‐dimensional homology of such a random d‐complex is, almost surely, nonzero? Here we derive an upper bound on this threshold. Computer experiments that we have conducted suggest that this bound may coincide with the actual threshold, but this remains an open question. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 26–35, 2015     

Internet resources in GPCR modelling1

G-Protein coupled receptors (GPCRs), one of the most important families of drug targets, belong to the super family of integral membrane proteins characterized by seven transmembrane helices. Because they are difficult to crystallize, the three dimensional structure of these receptors have not yet been determined by X-ray crystallography, except one. In the absence of a 3-D structure, in-silico approaches for solving the structure of this class of proteins are widely used and provide valuable information for structure based drug design. There are several web servers and computer programs available that automate the modelling process of GPCRs. Some of these include Modeller, Swiss-Model server, Homer, etc. Using these tools reliable homology models of human histamine H1 receptor (HRH1) and thrombin receptor (PAR-1) have been generated which explain the binding mode of the standard antagonists of these receptors and may be useful in designing their novel antagonists.     

Importance of CH/π hydrogen bonds in recognition of the core motif in proline-recognition domains: an ab initio fragment molecular orbital study

We examined CH/π hydrogen bonds in protein/ligand complexes involving at least one proline residue using the ab initio fragment molecular orbital (FMO) method and the program CHPI. FMO calculations were carried out at the Hartree–Fock (HF)/6‐31G*, HF/6‐31G**, second‐order Møller–Plesset perturbation (MP2)/6‐31G*, and MP2/6‐31G** levels for three Src homology 3 (SH3) domains and five proline‐recognition domains (PRDs) complexed with their corresponding ligand peptides. PRDs use a conserved set of aromatic residues to recognize proline‐rich sequences of specific ligands. Many CH/π hydrogen bonds were identified in these complexes. CH/π hydrogen bonds occurred, in particular, in the central part of the proline‐rich motifs. Our results suggest that CH/π hydrogen bonds are important in the recognition of SH3 and PRDs by their ligand peptides and play a vital role in the signal transduction system. Combined use of the FMO method and CHPI analysis is a valuable tool for the study of protein/protein and protein/ligand interactions and may be useful in rational drug design. © 2011 Wiley Periodicals, Inc. J Comput Chem 2011     

Homology of Linear Groups with Coefficients in the Adjoint Action and K-Theory

The main purpose of this paper is to compute some homology groups of linear groups with coefficients in the adjoint action which have appeared in different works. Our method uses various weight decompositions in group homology, algebraic K–theory, and Hochschild homology. This paper originated in a question posed by Cathelineau which is related to Hilbert's third problem on the scissors congruence of polyhedra.     

Whitney Homology of Semipure Shellable Posets

We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d. We derive a general formula for the representation of the symmetric group on the homology of posets of partitions whose block sizes are congruent to k mod d for any k and d. This formula reduces to the Calderbank-Hanlon-Robinson formulas when k = 0, 1 and to formulas of Sundaram for the virtual representation on the alternating sum of homology. Our results apply to restricted block size partition posets even more general than the k mod d partition posets. These posets include the lattice of partitions whose block sizes are bounded from below by some fixed k. Our main tools involve the new theory of nonpure shellability developed by Björner and Wachs and a generalization of a technique of Sundaram which uses Whitney homology to compute homology representations of Cohen-Macaulay posets. An application to subspace arrangements is also discussed.     

Transfer in Hochschild Cohomology of Blocks of Finite Groups

We develop the notion of a cohomology ring of blocks of finite groups and study its basic properties by means of transfer maps between the Hochschild cohomology rings of symmetric algebras associated with bounded complexes of finitely generated bimodules which are projective on either side.     

On the equivalence of multiplicative structures in floer homology and quantum homology

In this paper, we will prove that Floer homology equipped with either the intrinsic or exterior product is isomorphic to quantum homology as a ring. We will also prove that GW-invariants in Floer homology and quantum homology are equivalent.     

Deformations of Algebras Constructed using Quantum Stochastic Calculus

Deformations of associative algebras in which time is the deformation parameter are constructed using quantum stochastic flows in which the usual multiplicativity requirement is replaced by multiplicativity with respect to the deformed multiplication. The theory is restricted by a commutativity requirement on the structure maps of the flow, but examples which can be constructed in this way include the noncommutative torus and the Weyl–Moyal deformation.