Search for: All records

We prove that $$\omega \Delta ^{\prime}_{e_{k}}e_{n}|_{t=0}$$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all $$\Delta $$-Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Schützenberger cocharge statistic on a new combinatorial object that we call a battery-powered tableau. Our proof is geometric, and shows that the $$\Delta $$-Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $$t$$ and $$t^{2}$$ coefficients of $$\omega \Delta ^{\prime}_{e_{k}}e_{n}$$. 

Ecological restoration often targets plant community recovery, but restoration success may depend on the recovery of a complex web of biotic interactions to maintain biodiversity and promote ecosystem services. Specifically, management that drives resource availability, such as seeding richness and provenance, may alter species interactions across multiple trophic levels. Using experimentally seeded prairies, we examine three key groups – plants, pollinators, and goldenrod crab spiders (Misumena vatia, predators of pollinators) – to understand the effects of species richness and admixture seed sourcing of restoration seed mixtures on multi-trophic interactions. Working with prairie plants, we experimentally manipulated seed mix richness and the number of seed source regions (single-source region or admixture seed sourcing). In each experimental prairie, we surveyed floral abundance and richness, pollinator visitation, and plant-M. vatia interactions. A high-richness seed mix increased floral abundance when seeds were sourced from a single geographic region, and floral abundance strongly increased pollinator visitation, M. vatia abundance, and prey capture. Seeding richness and admixture seed sourcing of the seed mixture did not affect floral species richness, but floral species richness increased pollinator visitation. Pollinators interacted with different floral communities across seeding treatments, indicating a shift in visited floral species with restoration practices. Synthesis and applications. Long-term success in prairie restoration requires the restoration of plant-arthropod interactions. We provide evidence that seed mix richness and admixture seed sourcing affect arthropod floral associations, but effective restoration of plant-arthropod interactions should consider total floral resource availability. Incorporating a food web perspective in restoration will strengthen approaches to whole ecosystem restoration. 

We conjecture a simple combinatorial formula for the Schur expansion of the Frobenius series of the Sn-modules Rn,λ,s, which appear as the cohomology rings of the “∆-Springer” varieties. These modules interpolate between the Garsia-Procesi modules Rµ (which are the type A Springer fiber cohomology rings) and the rings Rn,k defined by Haglund, Rhoades, and Shimozono in the context of the Delta Conjecture. Our formula directly generalizes the known cocharge formula for Garsia-Procesi modules and gives a new cocharge formula for the Delta Conjecture at t = 0, by introducing battery-powered tableaux that “store” extra charge in their battery. Our conjecture has been verified by computer for all n ≤ 10 and s ≤ ℓ(λ)+2, as well as for n ≤ 8 and s ≤ ℓ(λ)+7. We prove it holds for several infinite families of n,λ,s. 

Life-history traits, which are physical traits or behaviours that affect growth, survivorship and reproduction, could play an important role in how well organisms respond to environmental change. By looking for trait-based responses within groups, we can gain a mechanistic understanding of why environmental change might favour or penalize certain species over others. We monitored the abundance of at least 154 bee species for 8 consecutive years in a subalpine region of the Rocky Mountains to ask whether bees respond differently to changes in abiotic conditions based on their life-history traits. We found that comb-building cavity nesters and larger bodied bees declined in relative abundance with increasing temperatures, while smaller, soil-nesting bees increased. Further, bees with narrower diet breadths increased in relative abundance with decreased rainfall. Finally, reduced snowpack was associated with reduced relative abundance of bees that overwintered as prepupae whereas bees that overwintered as adults increased in relative abundance, suggesting that overwintering conditions might affect body size, lipid content and overwintering survival. Taken together, our results show how climate change may reshape bee pollinator communities, with bees with certain traits increasing in abundance and others declining, potentially leading to novel plant–pollinator interactions and changes in plant reproduction. 

Abstract Ecological restoration often targets plant community recovery, but restoration success may depend on the recovery of a complex web of biotic interactions to maintain biodiversity and promote ecosystem services. Specifically, management that drives resource availability, such as seeding richness and provenance, may alter species interactions across multiple trophic levels. Using experimentally seeded prairies, we examine three key groups—plants, pollinators and goldenrod crab spiders (Misumena vatia, predators of pollinators)—to understand the effects of species richness and admixture seed sourcing of restoration seed mixtures on multitrophic interactions.Working with prairie plants, we experimentally manipulated seed mix richness and the number of seed source regions (single‐source region or admixture seed sourcing). In each experimental prairie, we surveyed floral abundance and richness, pollinator visitation and plant–M. vatiainteractions.A high richness seed mix increased floral abundance when seeds were sourced from a single geographic region, and floral abundance strongly increased pollinator visitation,M. vatiaabundance and prey capture. Seeding richness and admixture seed sourcing of the seed mixture did not affect floral species richness, but floral species richness increased pollinator visitation.Pollinators interacted with different floral communities across seeding treatments, indicating a shift in visited floral species with restoration practices.Synthesis and applications. Long‐term success in prairie restoration requires the restoration of plant–arthropod interactions. We provide evidence that seed mix richness and admixture seed sourcing affect arthropod floral associations, but effective restoration of plant–arthropod interactions should consider total floral resource availability. Incorporating a food web perspective in restoration will strengthen approaches to whole ecosystem restoration. 

An open problem of Manuel Abellanas asks whether every set of disjoint closed unit disks in the plane can be connected by a conveyor belt, which means a tight simple closed curve that touches the boundary of each disk, possibly multiple times. We prove three main results: For unit disks whose centers are both $$x$$-monotone and $$y$$-monotone, or whose centers have $$x$$-coordinates that differ by at least two units, a conveyor belt always exists and can be found efficiently. It is NP-complete to determine whether disks of arbitrary radii have a conveyor belt, and it remains NP-complete when we constrain the belt to touch disks exactly once. Any disjoint set of $$n$$ disks of arbitrary radii can be augmented by $O(n)$ "guide" disks so that the augmented system has a conveyor belt touching each disk exactly once, answering a conjecture of Demaine, Demaine, and Palop. 

Abstract In order to both combat the decline of biodiversity and produce food, fuel, and fiber for a growing human population, current agricultural landscapes must transition into diversified, multifunctional systems. Perennial cellulosic biofuel crops have potential to meet both of these challenges, acting as multifunctional systems that can enhance biodiversity. What is not well understood, and what we test here, are the tradeoffs among different perennial crops in their performance as biofuels and in biodiversity conservation. Working in an established bioenergy experiment with four native, perennial, cellulosic biofuel crop varieties—ranging from monoculture to diverse restoration planting—we tested the effect of biofuel crop management on flower communities, pollinator communities, and crop yield. The greatest abundance and diversity of pollinators and flowers were in treatments that were successional (unmanaged), followed by restored prairie (seeded mix of native grasses and forbs), switchgrass, and a mix of native grasses. However, biofuel crop yield was approximately the inverse, with native grasses having the highest yield, followed by switchgrass and prairie, then successional treatments. Restored prairie was the optimal biofuel crop when both pollinator conservation and crop yield are valued similarly. We add to mounting evidence that policy is needed to create sustainable markets that value the multifunctionality of perennial biofuel systems in order to achieve greater ecosystem services from agricultural landscapes.