On the Kashiwara–Vergne conjecture

Let G be a connected Lie group, with Lie algebra . In 1977, Duflo constructed a homomorphism of -modules , which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture on the Campbell-Hausdorff series, which (among other things) extends the Duflo theorem to germs of bi-invariant distributions on the Lie group G. The main results of the present paper are as follows. (1) Using a recent result of Torossian (2002), we establish the Kashiwara–Vergne conjecture for any Lie group G. (2) We give a reformulation of the Kashiwara–Vergne property in terms of Lie algebra cohomology. As a direct corollary, one obtains the algebra isomorphism , as well as a more general statement for distributions.     

On the Mumford–Tate conjecture for 1-motives

We show that the statement analogous to the Mumford–Tate conjecture for Abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge group and the unipotent part of the image of the absolute Galois group with the unipotent part of the motivic fundamental group.     

On the Lang–Trotter conjecture for two elliptic curves

The Ramanujan Journal - We extend the author’s earlier computation and give coefficient formulas for the (quasimodular) Poincaré square series of weight 3 / 2 and...     

On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.     

On a conjecture of the Randi? index

The Randi? index of a graph G is defined as , where d(u) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u, v. A conjecture on R(G) for connected graph G is as follows: R(G)≥r(G)−1, where r(G) denotes the radius of G. We proved that the conjecture is true for biregular graphs, connected graphs with order n≤10 and tricyclic graphs.     

On the Erd?s-Turán conjecture

We give equivalent formulations of the Erd?s-Turán conjecture on the unboundedness of the number of representations of the natural numbers by additive bases of order two of . These formulations allow for a quantitative exploration of the conjecture. They are expressed through some functions of reflecting the behavior of bases up to x. We examine some properties of these functions and give numerical results showing that the maximum number of representations by any basis is ?6.     

On the smooth transfer conjecture of Jacquet–Rallis for n=3

We obtain a relative Shalika germ expansion of orbital integrals appeared in the relative trace formulae Jacquet?CRallis when n=3. This is the first example where there are infinitely many nilpotent orbits. As an application we can prove the smooth transfer conjecture of Jacquet?CRallis for n=3.     

On the Bloch–Kato conjecture for elliptic modular forms of square-free level

Let $\kappa \ge 6$ be an even integer, $M$ an odd square-free integer, and $f \in S_{2\kappa -2}(\Gamma _0(M))$ a newform. We prove that under some reasonable assumptions that half of the $\lambda $ -part of the Bloch–Kato conjecture for the near central critical value $L(\kappa ,f)$ is true. We do this by bounding the $\ell $ -valuation of the order of the appropriate Bloch–Kato Selmer group below by the $\ell $ -valuation of algebraic part of $L(\kappa ,f)$ . We prove this by constructing a congruence between the Saito–Kurokawa lift of $f$ and a cuspidal Siegel modular form.     

On the compatibility between cup products,the Alekseev–Torossian connection and the Kashiwara–Vergne conjecture,II

We give a different proof of the famous result on compatibility between cup product (Kontsevich, 2003, [3, Section 8]) in cohomology of degree 0, for a finite-dimensional Lie algebra, from which we deduce an alternative way of re-writing Kontsevich?s star product by means of the Alekseev–Torossian connection (Alekseev and Torossian, 2010, [1]).     

Dade’s ordinary conjecture implies the Alperin–McKay conjecture

Archiv der Mathematik - We show that Dade’s ordinary conjecture implies the Alperin–McKay conjecture. We remark that some of the methods can be used to identify a canonical height zero...     

On the KŁR conjecture in random graphs

The K?R conjecture of Kohayakawa, ?uczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G _n,p , for sufficiently large p:= p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.     

On Ambrosetti–Malchiodi–Ni conjecture on two-dimensional smooth bounded domains

We consider the problem

$$\begin{aligned} \epsilon ^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad \text{ in }\quad \Omega , \quad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\quad \partial \Omega , \end{aligned}$$

where \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with smooth boundary, the exponent p is greater than 1, \(\epsilon >0\) is a small parameter, V is a uniformly positive, smooth potential on \(\bar{\Omega }\), and \(\nu \) denotes the outward unit normal of \(\partial \Omega \). Let \(\Gamma \) be a curve intersecting orthogonally \(\partial \Omega \) at exactly two points and dividing \(\Omega \) into two parts. Moreover, \(\Gamma \) satisfies stationary and non-degeneracy conditions with respect to the functional \(\int _{\Gamma }V^{\sigma }\), where \(\sigma =\frac{p+1}{p-1}-\frac{1}{2}\). We prove the existence of a solution \(u_\epsilon \) concentrating along the whole of \(\Gamma \), exponentially small in \(\epsilon \) at any fixed distance from it, provided that \(\epsilon \) is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by Ambrosetti et al. (Indiana Univ Math J 53(2), 297–329, 2004).