Arithmetic harmonic analysis on character and quiver varieties

We present a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a Riemann surface of genus g to GL_n(C) with fixed generic semi-simple conjugacy classes at k punctures. Using the character table of GL_n(F_q) we calculate the E-polynomial of these character varieties and confirm that it is as predicted by our main conjecture. Then, using the character table of gl_n(F_q), we calculate the E-polynomial of certain associated comet-shaped quiver varieties, the additive analogues of our character variety, and find that it is the pure part of our conjectured mixed Hodge polynomial. Finally, we observe that the pure part of our conjectured mixed Hodge polynomial also equals certain multiplicities in the tensor product of irreducible representations of GL_n(F_q). This implies a curious connection between the representation theory of GL_n(F_q) and Kac-Moody algebras associated with comet-shaped, typically wild, quivers.Comment: To appear in Duke Math. Journal + a section with examples is adde

Topology of character varieties and representations of quivers

In arXiv:0810.2076 we presented a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the representation varieties of Riemann surfaces with semi-simple conjugacy classes at the punctures. We proved several results which support this conjecture. Here we announce new results which are consequences of those of arXiv:0810.2076

Modelación del frijol en Latinoamérica: Estado del arte y base de datos para parametrización

Frijol común (Phaseolus vulgaris L.) es la leguminosa de grano para consumo humano de mayor producción en el mundo, y es nativo de las Américas, donde juega un papel importante en la dieta. Los ambientes donde se cultiva el frijol varían desde zonas tropicales hasta alta montaña, con diversos hábitos de crecimiento (arbustivo determinado, arbustivo indeterminado, y voluble), y en sistemas de producción desde los tradicionales hasta los altamente tecnificados. Los esfuerzos en modelaje, por tanto, deben empezar desde identificar el tipo de frijol y el sistema objetivo. El frijol es muy sensible al estrés abiótico, hecho que ha animado el modelaje de su posible respuesta bajo escenarios de cambio climático. Se llevó a cabo una revisión de literatura para identificar quince ejercicios de modelaje ejecutados en América Latina, abarcando estudios de crecimiento (tazas de producción de nudos y área foliar), fenología, y de rendimiento. Los modelos empleados en dichos estudios incluyen EcoCrop, CROPGRO-DRYBEAN (implementado en la plataforma DSSAT), y en un caso cada uno, Maxent y CLIMEX. Se describen cuatro estudios en detalle: en los dos países de mayor producción en el mundo (Brasil y México), y en Centroamérica como región altamente vulnerable al cambio climático. Estos estudios concuerdan que la productividad del frijol podría sufrir serios efectos negativos en el transcurso del Siglo XXI a raíz del cambio climático. Finalmente, se informa sobre un ejercicio reciente de recopilar datos históricos de ensayos de frijol en Latinoamérica para alimentar futuros esfuerzos de modelaje. Common bean (Phaseolus vulgaris L.) is the grain legume of greatest volume of production for direct human consumption, and is native to the Americas where it plays an important role in the diet of consumers. Bean is cultivated in environments from lowland tropical areas to high mountainous zones; with growth habits ranging from determinate bush, to indeterminate bush, to climbing types; and in production systems varying from traditional low input agriculture to highly technified systems. As such, efforts at modelling should focus on a specific plant type and production system. Bean is very sensitive to abiotic stress, a fact that has motivated modelling of its response in light of the dangers of climate change. A review of literature was carried out revealing fifteen studies in Latin America considering different aspects of plant growth (rates of node and leaf area production), phenology, and yield. Models employed include EcoCrop, CROPGRO (a module within the DSSAT cropping system model), and in one instance each, Maxent and CLIMEX. Three studies in particular are detailed: in the two countries of greatest production in the world (Brazil and Mexico), and one study in Central America as a region under direct threat of climate change. These three studies confirm that bean productivity will likely suffer severe negative effects in the course of the 21st century, as a result of climate change. A recent effort has compiled data from historical yield trials in Latin America as a resource for future modelling efforts.JRC.D.5-Food Securit

Coordination chemistry of amide-functionalised tetraazamacrocycles: structural, relaxometric and cytotoxicity studies

Three different tetraazamacrocyclic ligands containing four amide substituents that feature groups (namely allyl, styryl and propargyl groups) suitable for polymerisation have been synthesised. Gadolinium(III) complexes of these three ligands have been prepared as potential monomers for the synthesis of polymeric MRI contrast agents. To assess the potential of these monomers as MRI contrast agents, their relaxation enhancement properties and cytotoxicity have been determined. A europium(III) complex of one of these ligands (with propargyl substituents) is also presented together with its PARACEST properties. In addition, to gain further insight into the coordination chemistry of the tetra-propargyl substituted ligand, the corresponding zinc(II) and cadmium(II) complexes have been prepared. The X-ray crystal structures of the tetra-propargyl ligand and its corresponding gadolinium(III), zinc(II) and cadmium(II) complexes are also presented

Spanning tree generating functions and Mahler measures

We define the notion of a spanning tree generating function (STGF) $\sum a_n z^n$, which gives the spanning tree constant when evaluated at $z=1,$ and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form expressions for the spanning tree constants for all such lattices, which were previously largely unknown in all but one three-dimensional case. We show for all lattices that these can also be represented as Dirichlet $L$-series. Making the connection between spanning tree generating functions and lattice Green functions produces integral identities and hypergeometric connections, some of which appear to be new.Comment: 26 pages. Dedicated to F Y Wu on the occasion of his 80th birthday. This version has additional references, additional calculations, and minor correction

Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

Three-variable Mahler measures and special values of modular and Dirichlet LL-series

In this paper we prove that the Mahler measures of the Laurent polynomials $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k$, $(x+x^{-1})^2(y+y^{-1})^2(1+z)^3z^{-2}-k$, and $x^4+y^4+z^4+1+k^{1/4}xyz$, for various values of $k$, are of the form $r_1 L'(f,0)+r_2 L'(\chi,-1)$, where $r_1,r_2\in \mathbb{Q}$, $f$ is a CM newform of weight 3, and $\chi$ is a quadratic character. Since it has been proved that these Maher measures can also be expressed in terms of logarithms and $_5F_4$-hypergeometric series, we obtain several new hypergeometric evaluations and transformations from these results

Using Genetic Diversity in Deep Root Systems of Perennial Forage Grasses and Rice to Capture Carbon in Tropical Soils

Agricultural soils have the potential not only to be sinks of carbon dioxide (CO2) but also to mitigate the emissions of this gas to the atmosphere, thus, alleviating global warming. Perennial tropical grasses and rice upland and lowland varieties exhibit a large untapped genetic diversity in their root systems (e.g., deep rooting ability, exudation rates and chemical composition) that, if unlocked, could contribute to increased food production in crop-livestock systems while enhancing soil organic carbon (SOC) in tropical regions. Naturebased solutions that improve crop adaptation and SOC storage in tropical soils could help to remove CO2 from the atmosphere and thereby benefit the global climate system. With the launch of Future Seeds, one of the world’s largest repositories of tropical crop varieties, the Bezos Earth Fund (BEF) granted a major project within the Program of Future of Food. The focus of this BEF funded project is to: (i) develop novel high-throughput phenotyping methods to evaluate genetic diversity of root systems of tropical grasses and rice; (ii) unravel the potential of root systems in crop-livestock systems to replenish soil organic carbon (SOC) in human-intervened areas in tropical soils; (iii) identify and target hotspots/agroecological niches for SOC storage in tropical soils; and (iv) build capacity in conducting research on root systems and SOC storage towards carbon farming in tropical regions. Implementation of land-based SOC storage practices/projects (through carbon markets) based on deep rooting ability of perennial tropical forage grasses and rice cultivars in crop-pasture rotational systems could significantly reduce net emissions from tropical soils