Related papers: On the three-dimensional Singer Conjecture for Cox…

In this third part, we make the following hypothesis: representation $R=R(\alpha,\beta,\gamma ;l)$ of $W(p,q,r)$ is reducible and there exist a $G$-invariant non-nulle bilinear form where $G=Im R$. With those conditions, we know the…

Let $\Omega\subset\mathbb{R}^n$ be a strictly convex domain with smooth boundary and diameter $D$. The fundamental gap conjecture claims that if $V:\bar\Omega\to\mathbb{R}$ is convex, then the spectral gap of the Schr\"odinger operator…

We give a sufficient condition to study the vanishing of certain Koszul cohomology groups for general pairs $(X,L)\in W^r_{g,d}$ by induction. As an application, we show that to prove the Maximal Rank Conjecture (for quadrics), it suffices…

We study supervertices in six dimensional (2,0) supergravity theories, and derive supersymmetry non-renormalization conditions on the 4- and 6-derivative four-point couplings of tensor multiplets. As an application, we obtain exact…

The spherical harmonics $Y_\ell^m$ fall into three families -- sectoral ($\ell = |m|$), tesseral ($\ell > |m| > 0$), and zonal ($m = 0$) -- which exhibit fundamentally different behaviour under analytic continuation to non-integer…

We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus $\Sigma$, and moreover determine the ranks of the nontrivial homology…

We study Liouville-type theorem for polyharmonic H\'enon-Lane-Emden system $(-\Delta)^mu=|x|^av^p,\; (-\Delta)^mv=|x|^bu^q$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions…

Classical variational Hodge structure theory characterizes the algebraicity of Hodge classes by studying the transversality of period mappings under geometric deformations. However, when algebraic varieties lack appropriate deformation…

The Wehrl entropy conjecture for coherent (highest weight) states in representations of the Heisenberg group, which was proved in 1978 and recently extended by us to the group $SU(2)$, is further extended here to symmetric representations…

I. Hambleton, L. Taylor and B. Williams conjectured a general formula in spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group $G$ and noetherian ring $R.$ The conjectured decomposition was shown to hold for some…

Let $\mathbf{\Sigma}=(\Sigma,M,O)$ be a surface with marked points and order-2 orbifold points which is either unpunctured or once-punctured closed, and $\omega:O\rightarrow\{1,4\}$ a function. For each triangulation $\tau$ of…

In this paper, we give a new class of rigid Coxeter groups. Let $(W,S)$ be a Coxeter system. Suppose that (0) for each $s,t\in S$ such that $m(s,t)$ is even, $m(s,t)=2$, (1) for each $s\neq t\in S$ such that $m(s,t)$ is odd, $\{s,t\}$ is a…

We study different problems related to the Solomon's descent algebra $\Sigma(W)$ of a finite Coxeter group $(W,S)$: positive elements, morphisms between descent algebras, Loewy length... One of the main result is that, if $W$ is irreducible…

We propose a generalization of Haiman's conjecture on the diagonal coinvariant rings of real reflection groups to the context of irreducible quaternionic reflection groups (also known as symplectic reflection groups). For a reflection group…

In 1989, D. Happel pointed out for a possible connection between the global dimension of a finite-dimensional algebra and its Hochschild cohomology: is it true that the vanishing of Hochschild cohomology higher groups is sufficient to…

If E is an elliptic curve over Q and K is an imaginary quadratic field, there is an Iwasawa main conjecture predicting the behavior of the Selmer group of E over the anticyclotomic Z_p-extension of K. The main conjecture takes different…

In this note, we confirm a conjecture of Larson that arises in the Adams--Novikov spectral sequence (ANSS) for the stable homotopy groups of spheres and, specifically, in Behrens' program on explicit modular forms detecting $v_2$--periodic…

We study the C_p-equivariant Tate construction on the topological Hochschild homology THH(B) of a symmetric ring spectrum B by relating it to a topological version R_+(B) of the Singer construction, extended by a natural circle action. This…

We put into a general setting a technique of Rene' David (see "A Very Absolute Pi^1_2 Singleton, Annals of Pure and Applied Logic, 1982) to show that for S a Sigma^1_1 statement quantifying over subclasses of V of a special form, there is a…

Alternating-sign Hopf plumbing along a tree yields fibered alternating links whose homological monodromy is, up to a sign, conjugate to some alternating-sign Coxeter transformation. Exploiting this tie, we obtain results about the location…