unramified cohomology of quadrics i given a quadric x over a field f of characteristic 2 we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 galois cohomology of f to the unramified mod 2 cohomology of f x when dim x 10 and in several smaller dimensional cases applications of these results to real quadrics and to the unramified witt ring are given

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

the quaternionic second weighted zeta function of a graph and the study determinant abstract we establish a generalization of the second weighted zeta function of a graph to the case of quaternions for an arc weighted graph whose weights are quaternions we define the second weighted zeta function by using the study determinant that is a quaternionic determinant for quaternionic matrices defined by study this definition is regarded as a quaternionic analogue of the determinant expression of hashimoto type for the ihara zeta function of a graph we derive the study determinant expression of bass type and the euler product for the quaternionic second weighted zeta function

the arason invariant and mod 2 algebraic cycles introduction 73 1 review of the arason invariant 75 2 the special clifford group 76 3 k cohomology of split reductive algebraic groups 78 4 k cohomology of bg 86 5 gl n and cliff n n 92 6 two invariants for clifford bundles 95 7 snaking a bloch ogus differential 100 8 proof of theorem 1 101 9 application to quadratic forms 102 appendix a toral descent 104 appendix b the rost invariant 108 appendix c an amusing example 113 acknowledgements 116 references 116

degree three unramified cohomology of adjoint semisimple groups we compute degree 3 unramified cohomology of adjoint semisimple group new classes of non rational adjoint semisimple groups are found

sparse partition regularity our aim in this paper is to prove deuber s conjecture on sparse partition regularity that for every m p and c there exists a subset of the natural numbers whose m p c sets have high girth and chromatic number more precisely we show that for any m p c k and g there is a subset s of the natural numbers that is sufficiently rich in m p c sets that whenever s is k coloured there is a monochromatic m p c set yet is so sparse that its m p c sets do not form any cycles of length less than g our main tools are some extensions of ne etril rodl amalgamation and a ramsey theorem of bergelson hindman and leader as a sideline we obtain a ramsey theorem for products of trees that may be of independent interest

unramified degree three invariants of reductive groups abstract we prove that if g is a reductive group over an algebraically closed field f then for a prime integer p char f the group of unramified galois cohomology h nr 3 f bg q p z p 2 is trivial for the classifying space bg of g if p is odd or the commutator subgroup of g is simple

weight two motivic cohomology of classifying spaces for semisimple groups let f x to y be a torsor for a semisimple group g with y a smooth and geometrically irreducible variety over an arbitrary field we relate the etale motivic cohomology of weight two for x y and g we also compute the etale motivic cohomology groups of degree at most 4 for the classifying space of g this result was used in an another work by the author for the computation of the group of degree 3 cohomological invariants of semisimple groups with coefficients in bbb q bbb z 2

the kernel of the rost invariant serre s conjecture ii and the hasse principle for quasi split groups 3 6d4 e6 e7 we prove that for a simple simply connected quasi split group of type 3 6d4 e6 e7 defined over a perfect field f of characteristic 2 3 the rost invariant has trivial kernel in certain cases we give a formula for the rost invariant it follows immediately from the result above that if cd f 2 resp vcd f 2 then serre s conjecture ii resp the hasse principle holds for such a group for a c2 field in particular x y we prove the stronger result that serre s conjecture ii holds for all not necessary quasi split exceptional groups of type 3 6d4 e6 e7

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

motives of reductive groups our aim is to prove that the geometric motive with rational coefficients of a split reductive group g over a perfect field decomposes as the symmetric co algebra object of certain split mixed tate motives which are computable in terms of the root data of g moreover the isomorphism respects the hopf algebra structures in a weight graded sense we will also provide explicit decompositions in the case of classical groups

sur l inegalite de turan kubilius friable the exact value of the friable turan kubilius constant is determined when u log x log y tends to a finite limit for the set of y friable integers not exceeding x

the construction of weight two arithmetic cohomology let x be a regular noetherian scheme in a letter to soul6 be beilinson conjectured for each non negative integer n the existence of complexes f n x of zariski sheaves on x whose hypercohomology should be in the same relation to algebraic k theory as classical cohomology is to topological ktheory and proposed certain properties that these complexes should satisfy this investigation was continued in l where the conjectured complexes were moved to the a at other points we work only up to 2torsion or up to torsion involving primes dividing the residue field characteristics there seems good reason to expect our complex f 2 x to in fact exactly satisfy all the axioms we now sketch the construction of f 2 x and recall the axioms let a be a regular noetherian ring let w s p e c a t z s p e c a t t t 1 let b bl b 2 b be a finite sequence of exceptional units of a i e b i and

quantum chaos on discrete graphs adapting a method developed for the study of quantum chaos on quantum metric graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76 spectral functions and trace formulae for discrete laplacians on graphs are derived this is achieved by expressing the spectral secular equation in terms of the periodic orbits of the graph and obtaining functions which belong to the class of functions proposed originally by ihara 1966 j mat soc japan 18 219 and expanded by subsequent authors stark and terras 1996 adv math 121 124 kotani and sunada 2000 j math sci univ tokyo 7 7 finally a model of classical dynamics on the discrete graph is proposed it is analogous to the corresponding classical dynamics derived for quantum graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76

a sparse graham rothschild theorem the main result of this paper is a sparse version of the grahamrothschild partition theorem for n parameter sets r l graham and b l rothschild ramsey s theorem for n parameter sets trans amer math soc 159 1971 257 292 in particular a sparse version of hales jewett s theorem is proved we give several applications e g for arithmetic progressions and finite sums of integers confirming conjectures of j spencer and of j nesettil and v rodl we also consider graphs defined on parameter sets and prove a sparse and restricted induced partition theorem for such graphs extending results from h j promel induced partition properties of combinatorial cubes j combin theory ser a 39 1985 177 208 and p frankl r l graham and v rodl induced restricted ramsey theorems for spaces j combin theory ser a 44 1987 120 128

gerbal central extensions of reductive groups by k3 we classify central extensions of a reductive group g by mathcal k 3 and b mathcal k 3 the sheaf of third quillen k theory groups and its classifying stack these turn out to be parametrized by the group of weyl invariant quadratic forms on the cocharacter lattice valued in k times and the group of integral weyl invariant cubic forms on the cocharacter lattice respectively