on the classification of unitary representations of reductive lie groups each subset is identified conjecturally conjecture 0 6 with a collection of unitary representations of a certain subgroup g u of g we will give strong evidence and partial results for this conjecture in this way the problem of classifying u g would be reduced by induction on the dimension of g to the case g u g before considering the general program in more detail we describe it in the familiar case g sl 2 r this example will be treated more com

effectively categorical abelian groups abstract we study effective categoricity of computable abelian groups of the form i h where h is a subgroup of q such groups are called homogeneous completely decomposable it is well known that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite we study n 0 categoricity in this class of groups for n 1 we introduce a new algebraic concept of s independence which is a generalization of the well known notion of p independence we develop the theory of s independent sets we apply these techniques to show that every homogeneous completely decomposable group is 3 0 categorical we prove that a homogeneous completely decomposable group of infinite rank is 2 0 categorical if and only if it is isomorphic to the free module over the localization of z by a computably enumerable set of primes p with the semi low complement within the set of all primes we apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers including the free abelian group

on the dirac series of u p q this paper computes the dirac index of all the weakly fair a mathfrak q lambda modules of u p q we find counter examples to a conjecture of vogan on the unitary dual of u p q which was phrased by trapa in 2001 however we still believe that any irreducible unitary representation of u p q with non zero dirac cohomology must be a weakly fair a mathfrak q lambda module

the random k cycle walk on the symmetric group we study the random walk on the symmetric group s n generated by the conjugacy class of cycles of length k we show that the convergence to uniform measure of this walk has a cut off in total variation distance after frac n k log n steps uniformly in k o n as n rightarrow infty the analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles

dirac cohomology for graded affine hecke algebras we define an analogue of the casimir element for a graded affine hecke algebra mathbb h and then introduce an approximate square root called the dirac element using it we define the dirac cohomology hd x of an mathbb h module x and show that hd x carries a representation of a canonical double cover of the weyl group widetilde w our main result shows that the widetilde w structure on the dirac cohomology of an irreducible mathbb h module x determines the central character of x in a precise way this can be interpreted as p adic analogue of a conjecture of vogan for harish chandra modules we also apply our results to the study of unitary representations of mathbb h

computable torsion abelian groups abstract we prove that c c torsion abelian groups can be described by a 4 0 predicate that describes the failure of a brute force diagonalisation attempt on such groups we show that there is no simpler description since their index set is 4 0 complete the results can be viewed as a solution to a 60 year old problem of mal cev in the case of torsion abelian groups we prove that a computable torsion abelian group has one or infinitely many computable copies up to computable isomorphism the result confirms a conjecture of goncharov from the early 1980s for the case of torsion abelian groups

dirac series for some real exceptional lie groups abstract up to equivalence this paper classifies all the irreducible unitary representations with non zero dirac cohomology for the following simple real exceptional lie groups ei e 6 6 eiv e 6 26 fi f 4 4 fii f 4 20 along the way we find an irreducible unitary representation of f 4 4 whose dirac index vanishes while its dirac cohomology is non zero this disproves a conjecture raised in 2015 asserting that there should be no cancellation between the even part and the odd part of the dirac cohomology

characters of symmetric groups sharp bounds and applications we provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible it follows from our general bound that if a permutation s n has at most n o 1 cycles of length m then 1 1 m o 1 for all irreducible characters of s n this is a far reaching generalization of a result of fomin and lulov

recursive labelling systems and stability of recursive structures in hyperarithmetical degrees we show that under certain assumptions of recursiveness in 2t the recursive structure 2t is ao stable for ca w we need to deal with limit ordinals and we frame an analogous result proposition 2 for yn systems where yn is an increasing sequence of ordinals propositions 1 and 2 are then established by the

characterization of recursively enumerable sets with supersets effectively isomorphic to all recursively enumerable sets we show that the lattice of supersets of a recursively enumerable r e set a is effectively isomorphic to the lattice of all r e sets if and only if the complement a of a is infinite and e wie n a finite i 0 i e a is semilow 5 it is obvious that the condition a semilow 5 is necessary for the other direction a certain uniform splitting property the outer splitting property is derived from semilow 5 and this property is used in an extension of soare s automorphism machinery for the construction of the effective isomorphism since this automorphism machinery is quite complicated we give a simplified proof of soare s extension theorem before we add new features to this argument

the isomorphism problem for torsion free abelian groups is analytic complete we prove that the isomorphism problem for torsion free abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy namely 1 complete

dirac cohomology of cohomologically induced modules for reductive lie groups we extend the setting and a proof of the vogan s conjecture on dirac cohomology to a possibly disconnected real reductive lie group g in the harish chandra class we show that the dirac cohomology of cohomologically induced module mathcal l s z is completely determined by the dirac cohomology of the inducing module z more precisely we prove that if z is weakly good then the dirac cohomology h d mathcal l s z is equal to mathcal l widetilde k s h d z otimes bbb c rho frak p cap frak u an immediate application is a classification of tempered irreducible unitary representations with nonzero dirac cohomology

describing free groups we consider countable free groups of different ranks for these groups we investigate computability theoretic complexity of index sets within the class of free groups and within the class of all groups for a computable free group of infinite rank we consider the difficulty of finding a basis

on the analogy between real reductive groups and cartan motion groups contraction of irreducible tempered representations attached to any reductive lie group g is a cartan motion group g 0 a lie group with the same dimension as g but a simpler group structure a natural one to one correspondence between the irreducible tempered representations of g and the unitary irreducible representations of g 0 whose existence had been suggested by mackey in the 1970s has recently been described by the author in the present notes we use the existence of a family of groups interpolating between g and g 0 to realize the bijection as a deformation for every irreducible tempered representation pi of g we build in an appropriate fr echet space a family of subspaces and evolution operators that contract pi onto the corresponding representation of g 0