Koszul homology and extremal properties of Gin and Lex

For every homogeneous ideal in a polynomial ring and for every we consider the Koszul homology with respect to a sequence of of generic linear forms. The Koszul-Betti number is, by definition, the dimension of the degree part of . In characteristic , we show that the Koszul-Betti numbers of any ideal are bounded above by those of the gin-revlex of and also by those of the Lex-segment of . We show that iff is componentwise linear and that and iff is Gotzmann. We also investigate the set of all the gin of and show that the Koszul-Betti numbers of any ideal in are bounded below by those of the gin-revlex of . On the other hand, we present examples showing that in general there is no is such that the Koszul-Betti numbers of any ideal in are bounded above by those of .

     

On restrictions of modular spin representations of symmetric and alternating groups

Let be an algebraically closed field of characteristic and be an almost simple group or a central extension of an almost simple group. An important problem in representation theory is to classify the subgroups of and -modules such that the restriction is irreducible. For example, this problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where is the Schur's double cover or .

     

Fundamental solutions for non-divergence form operators on stratified groups

We construct the fundamental solutions and for the non-divergence form operators and , where the 's are Hörmander vector fields generating a stratified group and is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of allow us to construct using both -saturation and approximation arguments.

     

-equivalence in adjoint classical groups over fields of virtual cohomological dimension

Let be a field of characteristic not whose virtual cohomological dimension is at most . Let be a semisimple group of adjoint type defined over . Let denote the normal subgroup of consisting of elements -equivalent to identity. We show that if is of classical type not containing a factor of type , . If is a simple classical adjoint group of type , we show that if and its multi-quadratic extensions satisfy strong approximation property, then . This leads to a new proof of the -triviality of -rational points of adjoint classical groups defined over number fields.

     

On Diophantine definability and decidability in some infinite totally real extensions of

Let be a number field, and a set of its non-Archimedean primes. Then let . Let be a finite set of prime numbers. Let be the field generated by all the -th roots of unity as and . Let be the largest totally real subfield of . Then for any 0$">, there exist a number field , and a set of non-Archimedean primes of such that has density greater than , and has a Diophantine definition over the integral closure of in .

     

The geometry of profinite graphs with applications to free groups and finite monoids

We initiate the study of the class of profinite graphs defined by the following geometric property: for any two vertices and of , there is a (unique) smallest connected profinite subgraph of containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition.

We define a pseudovariety of groups to be arboreous if all finitely generated free pro- groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties , a pro- analog of the Ribes and Zalesski product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions to the much studied pseudovariety equation .

     

Maps between non-commutative spaces

Let be a graded ideal in a not necessarily commutative graded -algebra in which for all . We show that the map induces a closed immersion between the non-commutative projective spaces with homogeneous coordinate rings and . We also examine two other kinds of maps between non-commutative spaces. First, a homomorphism between not necessarily commutative -graded rings induces an affine map from a non-empty open subspace . Second, if is a right noetherian connected graded algebra (not necessarily generated in degree one), and is a Veronese subalgebra of , there is a map ; we identify open subspaces on which this map is an isomorphism. Applying these general results when is (a quotient of) a weighted polynomial ring produces a non-commutative resolution of (a closed subscheme of) a weighted projective space.

     

Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets

Given a closed -rectifiable set embedded in Euclidean space, we investigate minimal weighted Riesz energy points on ; that is, points constrained to and interacting via the weighted power law potential , where is a fixed parameter and is an admissible weight. (In the unweighted case () such points for fixed tend to the solution of the best-packing problem on as the parameter .) Our main results concern the asymptotic behavior as of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution with respect to -dimensional Hausdorff measure on , our results provide a method for generating -point configurations on that are ``well-separated' and have asymptotic distribution as .

     

Sums of squares in real rings

Let be an excellent ring. We show that if the real dimension of is at least three then has infinite Pythagoras number, and there exists a positive semidefinite element in which is not a sum of squares in .

     

The periodic isoperimetric problem

Given a discrete group of isometries of , we study the -isoperimetric problem, which consists of minimizing area (modulo ) among surfaces in which enclose a -invariant region with a prescribed volume fraction. If is a line group, we prove that solutions are either families of round spheres or right cylinders. In the doubly periodic case we prove that for most rank two lattices, solutions must be spheres, cylinders or planes. For the remaining rank two lattices we show, among other results, an isoperimetric inequality in terms of the topology of the isoperimetric surfaces. Finally, we study the case where (the group of symmetries of the integer rank three lattice ) and other crystallographic groups of cubic type. We prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than , and we give an isoperimetric inequality for -invariant regions that, for instance, implies that the area (modulo ) of a surface dividing the three space in two -invariant regions with equal volume fractions, is at least (the conjectured solution is the classical Schwarz triply periodic minimal surface whose area is ). Another consequence of this isoperimetric inequality is that -symmetric surfaces (other than families of spheres) cannot be isoperimetric for the lattice group .

     

Positive laws in fixed points

Let be an elementary abelian group of order at least acting on a finite -group in such a manner that satisfies a positive law of degree for any . It is proved that the entire group satisfies a positive law of degree bounded by a function of and only.

     

Symmetrization, symmetric stable processes, and Riesz capacities

Let be a symmetric -stable process killed on exiting an open subset of . We prove a theorem that describes the behavior of its transition probabilities under polarization. We show that this result implies that the probability of hitting a given set in the complement of in the first exit moment from increases when and are polarized. It can also lead to symmetrization theorems for hitting probabilities, Green functions, and Riesz capacities. One such theorem is the following: Among all compact sets in with given volume, the balls have the least -capacity ( ).

     

Lattice invariants and the center of the generic division ring

Let be a finite group, let be a -lattice, and let be a field of characteristic zero containing primitive roots of 1. Let be the quotient field of the group algebra of the abelian group . It is well known that if is quasi-permutation and -faithful, then is stably equivalent to . Let be the center of the division ring of generic matrices over . Let be the symmetric group on symbols. Let be a prime. We show that there exist a split group extension of by a -elementary group, a -faithful quasi-permutation -lattice , and a one-cocycle in such that is stably isomorphic to . This represents a reduction of the problem since we have a quasi-permutation action; however, the twist introduces a new level of complexity. The second result, which is a consequence of the first, is that, if is algebraically closed, there is a group extension of by an abelian -group such that is stably equivalent to the invariants of the Noether setting .

     

Indecomposable modules of large rank over Cohen-Macaulay local rings

A commutative Noetherian local ring is called Dedekind-like provided is one-dimensional and reduced, the integral closure is generated by at most 2 elements as an -module, and is the Jacobson radical of . If is an indecomposable finitely generated module over a Dedekind-like ring , and if is a minimal prime ideal of , it follows from a classification theorem due to L. Klingler and L. Levy that must be free of rank 0, 1 or 2.

Now suppose is a one-dimensional Cohen-Macaulay local ring that is not Dedekind-like, and let be the minimal prime ideals of . The main theorem in the paper asserts that, for each non-zero -tuple of non-negative integers, there is an infinite family of pairwise non-isomorphic indecomposable finitely generated -modules satisfying for each .

     

Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc

Let be a holomorphic self-map of the unit disc . For every , there is a measure on (sometimes called Aleksandrov measure) defined by the Poisson representation . Its singular part measures in a natural way the ``affinity' of for the boundary value . The affinity for values inside is provided by the Nevanlinna counting function of . We introduce a natural measure-valued refinement of and establish that the measures are obtained as boundary values of the refined Nevanlinna counting function . More precisely, we prove that is the weak limit of whenever converges to non-tangentially outside a small exceptional set . We obtain a sharp estimate for the size of in the sense of capacity.

     

How to do a -descent on an elliptic curve

In this paper, we describe an algorithm that reduces the computation of the (full) -Selmer group of an elliptic curve over a number field to standard number field computations such as determining the (-torsion of) the -class group and a basis of the -units modulo th powers for a suitable set of primes. In particular, we give a result reducing this set of `bad primes' to a very small set, which in many cases only contains the primes above . As of today, this provides a feasible algorithm for performing a full -descent on an elliptic curve over , but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of is favorable, simplifications are possible and -descents for larger are accessible even today. To demonstrate how the method works, several worked examples are included.

     

Quantum symmetric derivatives

For , a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For , symmetrization holds, that is, whenever the th Peano derivative exists at a point, all of these derivatives of order also exist at that point. The main result, desymmetrization, is that conversely, for , each symmetric quantum derivative is a.e. equivalent to the Peano derivative of the same order. For and , each th symmetric quantum derivative coincides with both corresponding th Riemann symmetric quantum derivatives, so, in particular, for and , both th Riemann symmetric quantum derivatives are a.e. equivalent to the Peano derivative.

     

On the non-vanishing of cubic twists of automorphic -series

Let be a normalised new form of weight for over and , its base change lift to . A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the -function of . There is an algorithm to check the condition for any given form. The new form of level is used to illustrate our method.