A geometric version of the circle method

We develop a geometric version of the circle method and use it to compute the compactly supported cohomology of the space of rational curves through a point on a smooth affine hypersurface of sufficiently low degree.Comment: 49 pages. Final versio

Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums

We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums with characters modulo a prime $q$ which, for both variables of length $M$, are non-trivial as soon as $M\geq q^{3/8+\delta}$ for any $\delta>0$. This range, which matches Burgess's range, is identical with the best results previously known only for simpler exponentials of monomials. The proof combines refinements of the analytic tools from our previous paper and new geometric methods. The key geometric idea is a comparison statement that shows that even when the "sum-product" sheaves that appear in the analysis fail to be irreducible, their decomposition reflects that of the "input" sheaves, except for parameters in a high-codimension subset. This property is proved by a subtle interplay between \'etale cohomology in its algebraic and diophantine incarnations. We prove a first application concerning the first moment of a family of $L$-functions of degree $3$.Comment: 58 pages; minor cosmetic corrections; to appear in Annali della Scuola Normale Superiore di Pis

Quantitative sheaf theory

We introduce a notion of complexity of a complex of ell-adic sheaves on a quasi-projective variety and prove that the six operations are "continuous", in the sense that the complexity of the output sheaves is bounded solely in terms of the complexity of the input sheaves. A key feature of complexity is that it provides bounds for the sum of Betti numbers that, in many interesting cases, can be made uniform in the characteristic of the base field. As an illustration, we discuss a few simple applications to horizontal equidistribution results for exponential sums over finite fields.Comment: v3, 68 pages; the key ideas of this paper are due to W. Sawin; A. Forey, J. Fres\'an and E. Kowalski drafted the current version of the text; revised after referee report

THE INFLUENCE OF BREEDING EXPERIENCE ON NEST SUCCESS IN RED-WINGED BLACKBIRD

The Red-winged Blackbird (Agelaius phoeniceus) is a polygynous species, and females are typically responsible for the majority of parental care. Despite their limited involvement, males can contribute to reproduction through nest defense and the feeding of nestlings. Some aspects of nest defense may be learned, and older males are more likely to feed young, suggesting that males with previous breeding experience may enjoy higher nest success than their naïve counterparts. We manipulated territory ownership on 10 wetlands in central North Dakota in order to examine the influence of breeding experience on reproductive success. We found no evidence that breeding experience increased nest survival (x– = 23.0%) or increased the number of fledglings per nest (x– = 2.2). In this population, territory quality may contribute more to nesting success than male parental contributions

Precambrian nomenclature in Kansas

The informal stratigraphic term “Precambrian” is replaced by formal nomenclature—Proterozoic and Archean Eonothems/Eons—and the informal term Hadean. The Phanerozoic Eonothem/Eon, representing all rocks younger than the Proterozoic, is added. The Proterozoic is further divided into Paleoproterozoic, Mesoproterozoic, and Neoproterozoic Erathems/Eras. The name Rice Formation (Scott, 1966) is abandoned, and the use of the informal term “Rice unit” is recommended. The proposed name Rice Series (Berendsen, 1994) is not accepted. These changes are adopted by the Kansas Geological Survey (KGS) and the stratigraphic nomenclature of Zeller (1968) has been revised accordingly

Clarification and Changes in Permian Stratigraphic Nomenclature in Kansas

This paper outlines Permian nomenclature changes to Zeller (1968) that have been adopted by the Kansas Geological Survey. The Permian System/Period, Cisuralian Series/Epoch, and Asselian Stage/Age are established at the base of the Bennett Shale Member of the Red Eagle Limestone. Series/epoch names Wolfcampian, Leonardian, and Guadalupian are retained and usage of Gearyan, Cimarronian, and Custerian is abandoned. The repositioned Carboniferous-Permian boundary divides the Council Grove Group into Carboniferous (Upper Pennsylvanian Series/Epoch; Virgilian Stage/Age) and Permian (Wolfcampian Series/Epoch) segments

Carboniferous–Permian Boundary in Kansas, Midcontinent, U.S.A.

The placement of the Carboniferous (Pennsylvanian)-Permian boundary in Kansas has been debated since the rocks of this age were first described and named. With the ratification of the Global Stratotype Section and Point (GSSP) for the base of the Permian System in the southern Ural Mountains, the Carboniferous-Permian boundary in Kansas can now be confidently defined. Based on the identification of the first occurrence of the conodont Streptognathodus isolatus that definitively correlates the Kansas rock section to the basal Permian GSSP, the Carboniferous-Permian boundary in Kansas can be placed at the base of the Bennett Shale Member of the Red Eagle Limestone. The Kansas Geological Survey proposes that the Tuttle Creek Lake Spillway section, located in northeast Kansas, be considered for the Carboniferous-Permian boundary stratotype in Kansas. It is further suggested that the stratigraphic position of the Carboniferous-Permian boundary in the Tuttle Creek Lake Spillway section be considered as a potential North American stratotype. In addition to being a significant biostratigraphic boundary, the Carboniferous-Permian boundary and enclosing strata also have significance because they reflect important geologic events and changes that occurred on a regional and global scale

New Stratigraphic Rank for the Carboniferous, Mississippian, and Pennsylvanian in Kansas

A new classification for the Carboniferous System/Period is formally adopted by the Kansas Geological Survey (KGS), and Zeller (1968) is modified accordingly. The Carboniferous is the system/period between the Devonian and Permian, and the Mississippian and Pennsylvanian are subsystems/subperiods of the Carboniferous. The Mississippian is subdivided into Lower, Middle, and Upper Mississippian Series and the Pennsylvanian is subdivided into Lower, Middle, and Upper Pennsylvanian Series. Regional stage names remain unchanged

Mississippian Stratigraphic Nomenclature Revisions in Kansas

This paper reviews proposed Mississippian nomenclature changes in Kansas and outlines the changes to Zeller (1968) that have been adopted by the Kansas Geological Survey. The Sedalia Dolomite is changed to the Sedalia Formation and the Northview Shale is changed to Northview Formation due to lateral lithology changes. The Short Creek Oolite Member as originally defined and described by Smith and Siebenthal (1907) at the type section in Kansas is reinstated. The Cowley Formation as originally defined and described by Lee (1940) in Kansas is reinstated. The Ste. Genevieve Limestone is placed as the basal formation of the Chesteran Stage