The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

In this chapter, we derive the currently best known bounds on the constants in the Lieb–Thirring inequality following Hundertman–Laptev–Weidl and Frank–Hundertmark–Jex–Nam. These arguments proceed by proving bounds for one-dimensional Schrödinger operators with matrix-valued potentials and then using the method of "lifting in dimension." In the final section, we summarize the results in the book and provide an overview of what is known about the sharp constants in the Lieb–Thirring and Cwikel–Lieb–Rozenblum inequalities.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

We discuss the naive duality theory of coherent, torsion free, S2 sheaves on schemes.

It is our great pleasure and honour to say Happy Birthday to our friend Miles Reid, for your 70th birthday and indeed a few subsequent ones; it takes a long time to make a big pot, as they say, perhaps especially when there are so many potters.

We prove that Q-Fano threefolds of Fano index ≥ 8 are rational.

We consider semi-orthogonal decompositions of derived categories for 3-dimensional projective varieties in the case when the varieties have ordinary double points.

We classify minimal projective 3-folds of general type with pg = 2 by studying the birationality of their 6-canonical maps.

We explain a classical construction of a del Pezzo surface of degree d = 4 or 5 as a smooth order 2 congruence of lines in P3 whose focal surface is a quartic surface X20-d with 20-d ordinary double points. We also show that X15 can be realized as a hyperplane section of the Castelnuovo– Richmond–Igusa quartic hypersurface in P4. This leads to the proof of rationality of the moduli space of 15-nodal quartic surfaces. We discuss some other birational models of X15: quartic symmetroids, 5-nodal quartic surfaces, 10-nodal sextic surfaces in P4 and nonsingular surfaces of degree 10 in P6. Finally we study some birational involutions of a 15- nodal quartic surface which, as it is shown in Part II of the paper jointly with I. Shimada [DS20], belong to a finite set of generators of the group of birational automorphisms of a general 15 nodal quartic surface.

We introduce and study the notion of ‘surface decomposable’ variety, and discuss the possibility that any projective hyper-Käahler manifold is surface decomposable, which would produce new evidence for Beauville’s weak splitting conjecture.

A number of years ago, one of us (Mark Gross) was giving a lecture at the University of Warwick on the material on scattering diagrams from [GPS10]. Of course, Miles was in the audience, and he asked (paraphrasing as this was many years ago) whether, at some point, the lecturer would come back down to earth. The goal of this note is to show, in fact, we have not left the planet by considering a particularly beautiful example of the mirror symmetry construction of [GHK15b], namely the mirror to a cubic surface.

We consider that kernels of inflation maps associated with extraspecial p-groups in stable group cohomology are generated by their degree-2 components. This turns out to be true if the prime is large enough compared to the rank of the elementary abelian quotient, but false in general.