Nilpotency of Bocksteins, Kropholler's hierarchy and a conjecture of Moore

We show that the class of pairs (Γ,H) of a group and a finite index subgroup which verify a conjecture of Moore about projectivity of modules over ZΓ satisfy certain closure properties. We use this, together with a result of Benson and Goodearl, in order to prove that Moore's conjecture is valid for groups which belongs to Kropholler's hierarchy LHF. For finite groups, Moore's conjecture is a consequence of a theorem of Serre, about the vanishing of a certain product in the cohomology ring (the Bockstein elements). Using our result, we construct examples of pairs (Γ,H) which satisfy the conjecture without satisfying the analog of Serre's theorem.     

A theorem on smoothness- Bass-Quillen, Chow groups and intersection multiplicity of Serre

We describe here an inherent connection of smoothness among the Bass-Quillen conjecture, the Chow-group problem and Serre's Theorem on Intersection Multiplicity. Extension of a theorem of Lindel on smoothness plays a key role in our proof of the Serre-multiplicity theorem in the geometric (resp. unramified) case. We reduce the complete case of the theorem to the above case by using Artin's Approximation. We do not need the concept of ``complete Tor'. Similar proofs are sketched for Quillen's theorem on Chow groups and its extension due to Gillet and Levine.

     

A Positivity Property of a Quantum Anharmonic Oscillator Suggested by the BMV Conjecture

In this work an observation concerning a positivity property of the quantum anharmonic oscillator is made. This positivity property is suggested by the BMV conjecture.     

Profinite groups, profinite completions and a conjecture of Moore

Let R be any ring (with 1), Γ a group and RΓ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an RΓ-module, whose restriction to RH is projective.Moore's conjecture (J. Pure Appl. Algebra 7(1976)287): Assume for every nontrivial element x in Γ, at least one of the following two conditions holds:

(M1)

〈x〉∩H≠{e} (in particular this holds if Γ is torsion free)

(M2)

ord(x) is finite and invertible in R.

Then M is projective as an RΓ-module.More generally, the conjecture has been formulated for crossed products R*Γ and even for strongly graded rings R(Γ). We prove the conjecture for new families of groups, in particular for groups whose profinite completion is torsion free.The conjecture can be formulated for profinite modules M over complete groups rings [[RΓ]] where R is a profinite ring and Γ a profinite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.     

Serre''s contribution to the development of algebraic topology

We describe in detail Serre's application of spectral sequence theory to the study of the relations between the homology of total space, base space and fibre in a Serre fibration; and we apply the results to establish that a 1-connected space X has homology groups (in positive dimension) in a Serre class C if and only if its homotopy groups are in C.

We include in this paper some personal reflections on the contact the author had with Serre during the decade of the 1950's when Serre's revolutionary work in homotopy theory was completely changing the face of algebraic topology.     

Proof of the Kontsevich non-commutative cluster positivity conjecture

We extend the Lee–Schiffler Dyck path model to give a proof of the Kontsevich non-commutative cluster positivity conjecture with unequal parameters.     

Lattices in rank one Lie groups over local fields

We prove that if is theK-rational points of aK-rank one semisimple group over a non archimedean local fieldK, thenG has cocompact non-arithmetic lattices and if char(K)>0 also non-uniform ones. We also give a general structure theorem for lattices inG, from which we confirm Serre's conjecture that such arithmetic lattices do not satisfy the congruence subgroup property.Partially supported by a grant from the Bi-national Science Foundation U.S.-Israel.     

Projective modules in the intersection cohomology of Deligne–Lusztig varieties

We formulate a strong positivity conjecture on characters afforded by the Alvis–Curtis dual of the intersection cohomology of Deligne–Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of unipotent ?-blocks of finite reductive groups.     

Strong openness of multiplier ideal sheaves and optimal <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> extension

In this paper, we reveal that our solution of Demailly’s strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Kollár and Jonsson-Mustat? implies the truth of twisted versions of the strong openness conjecture; our optimal L ² extension implies Berndtsson’s positivity of vector bundles associated to holomorphic fibrations over a unit disc.     

Total positivity of Narayana matrices

We prove the total positivity of the Narayana triangles of type $A$ and type $B$, and thus affirmatively confirm a conjecture of Chen, Liang and Wang and a conjecture of Pan and Zeng. We also prove the strict total positivity of the Narayana squares of type $A$ and type $B$.     

Two positivity conjectures for Kerov polynomials

Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the “free cumulants” of the associated Young diagram. We present two positivity conjectures for their coefficients. The latter are stronger than the positivity conjecture of Kerov–Biane, recently proved by Féray.     

On a sufficient condition for a Fano manifold to be covered by rational N-folds

In this paper, we prove a conjecture by T. Suzuki, which says if a smooth Fano manifold satisfies some positivity condition on its Chern characters, then it can be covered by rational N-folds. We prove this conjecture by using purely combinatorial properties of Bernoulli numbers.     

On the number of even and odd strings along the overpartitions of n

Recently, Andrews, Chan, Kim, and Osburn introduced the even strings and the odd strings in the overpartitions. We show that their conjecture $$A_k (n) \geq B_k (n)$$ holds for large enough positive integers n, where A _k (n) (resp. B _k (n)) is the number of odd (resp. even) strings along the overpartitions of n. We introduce m-strings and show how this new combinatorial object is related with another positivity conjecture of Andrews, Chan, Kim, and Osburn. Finally, we confirm that the positivity conjecture is also true for large enough integers.     

Schur positivity and log-concavity related to longest increasing subsequences

Chen proposed a conjecture on the log-concavity of the generating function for the symmetric group with respect to the length of longest increasing subsequences of permutations. Motivated by Chen’s log-concavity conjecture, Bóna, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subsets. They obtained the log-concavity of the corresponding generating functions for these subsets by using the hook-length formula. In this paper, we generalize and prove their results by establishing the Schur positivity of certain symmetric functions. This also enables us to propose a new approach to Chen’s original conjecture.     

Positivity and tameness in rank 2 cluster algebras

We study the relationship between the positivity property in a rank 2 cluster algebra, and the property of such an algebra to be tame. More precisely, we show that a rank 2 cluster algebra has a basis of indecomposable positive elements if and only if it is of finite or affine type. This statement disagrees with a conjecture by Fock and Goncharov.     

Asymptotic syzygies of algebraic varieties

We study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows. The main result asserts that the syzygy modules are non-zero in almost all degrees allowed by Castelnuovo?CMumford regularity. We also give an effective statement for Veronese varieties that we conjecture to be optimal.     

On some p-adic properties of Siegel-Eisenstein series

We introduce a formula for the p-adic Siegel-Eisenstein series which demonstrates a connection with the genus theta series and the twisted Eisenstein series with level p. We then prove a generalization of Serre's formula in the elliptic modular case.     

On a conjecture of B. Berndt and B. Kim

We prove a recent conjecture of B. Berndt and B. Kim regarding the positivity of the coefficients in the asymptotic expansion of a class of partial theta functions. This generalizes results found in Ramanujan’s second notebook and recent work of Galway and Stanley.     

Saddle points and overdetermined problems for the Helmholtz equation

In a seminal 1971 paper, James Serrin showed that the only open, smoothly bounded domain in ⁿ on which the positive Dirichlet eigenfunction of the Laplacian has constant (nonzero) normal derivative on the boundary, is then-dimensional ball. The positivity of the eigenfunction is crucial to his proof. To date it is an open conjecture that the same result is true for Dirichlet eigenvalues other than the least. We show that for simply connected, plane domains, the absence of saddle points is a condition sufficient to validate this conjecture. This condition is also sufficient to prove Schiffer's conjecture: the only simply connected planar domain, on the boundary of which a nonconstant Neumann eigenfunction of the Laplacian can take constant value, is the disc.     

An Alternative Proof of a Weak Form of Serre's Theorem

A proof of a weak form of Serre's theorem on the finiteness of stable homotopy groups of spheres is suggested, which uses neither loop spaces nor spectral sequences.