Cohomology of the minimal nilpotent orbit

We compute the integral cohomology of the minimal non-trivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the sub-root system generated by the long simple roots. The modulo $\ell$ reduction of the Springer correspondent representation involves the sign representation exactly when $\ell$ divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes.Comment: 29 pages, v2 : Leray-Serre spectral sequence replaced by Gysin sequence only, corrected typo

Arbitrarily large families of spaces of the same volume

In any connected non-compact semi-simple Lie group without factors locally isomorphic to SL_2(R), there can be only finitely many lattices (up to isomorphism) of a given covolume. We show that there exist arbitrarily large families of pairwise non-isomorphic arithmetic lattices of the same covolume. We construct these lattices with the help of Bruhat-Tits theory, using Prasad's volume formula to control their covolumes.Comment: 9 pages. Syntax corrected; one reference adde

Final Report Buffalo National River Ecosystems Part IV

Sampling point locations and analytical procedures remained unchanged from those outlined in previous Buffalo National River Ecosystem reports. The only significant change in analytical procedures was a reversion to the glass fiber filter method for collection and extraction of samples for chlorophyll analysis. This change was neeessitated by a need for filtering a larger volume to obtain enough chlorophyll for an accurate measurement. Samples were collected monthly from April 9 through December 30. No samples were taken in January or February due to the extremely uncertain traveling conditions caused by the frequent snows. Prior research indicates that the December 30 sample is sufficiently reflective of stable winter conditions to obviate the need for more winter samples (see previous reports)

Shear sum rules at finite chemical potential

We derive sum rules which constrain the spectral density corresponding to the retarded propagator of the T_{xy} component of the stress tensor for three gravitational duals. The shear sum rule is obtained for the gravitational dual of the N=4 Yang-Mills, theory of the M2-branes and M5-branes all at finite chemical potential. We show that at finite chemical potential there are additional terms in the sum rule which involve the chemical potential. These modifications are shown to be due to the presence of scalars in the operator product expansion of the stress tensor which have non-trivial vacuum expectation values at finite chemical potential.Comment: The proof for the absence of branch cuts is corrected.Results unchange

Sum rules and three point functions

Sum rules constraining the R-current spectral densities are derived holographically for the case of D3-branes, M2-branes and M5-branes all at finite chemical potentials. In each of the cases the sum rule relates a certain integral of the spectral density over the frequency to terms which depend both on long distance physics, hydrodynamics and short distance physics of the theory. The terms which which depend on the short distance physics result from the presence of certain chiral primaries in the OPE of two R-currents which are turned on at finite chemical potential. Since these sum rules contain information of the OPE they provide an alternate method to obtain the structure constants of the two R-currents and the chiral primary. As a consistency check we show that the 3 point function derived from the sum rule precisely matches with that obtained using Witten diagrams.Comment: 41 page

Buffalo National River Ecosystem - Part III

Samples for water quality analyses and phycological studies were taken from the nine standard sampling locations on the Buffalo River nine times during the period from March 1976 through February 1977. The April-June 1976 samples represent nearly identical conditions throughout the spring period; therefore, emphasis was placed on taxonomic research. As the early January sample was considered sufficient~y reflective of stable winter conditions, the December and February periods were. used for detailed microscopic examination of the rich and diverse diatom flora that was found in the river this year. A total of 273 taxa of diatoms were identified from the 75 samples collected, including 123 new additions to the diatom flora of the Buffalo River. Details of this study, including the breakdown of many species into varieties, will be presented in a separate paper. A list of the new species found and a general discussion are included in this report

Schubert varieties and generalizations

This contribution reviews the main results on Schubert varieties and their generalizations It covers more or less the material of the lectures at the Seminar These were partly expository introducing material needed by other lecturers In particular Section reviews classical material used in several of the other contribution