Related papers: On the Mathieu Conjecture for $SU(N)$ and $SO(N)$

This paper proves a conjecture proposed by Ren and Li (2015: 393, \emph{Journal of Inequalities and Applications}). Our result eliminates the constraints on the parity and size of $m$, as well as the restriction $x > 1$, required in Ren and…

This talk, given at several conferences and meetings, explains the background leading to the formulation of the exact electromagnetic duality conjecture believed to be valid in N=4 supersymmetric SU(2) gauge theory.

Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A \subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible…

Let $\mathrm{SO}(3,\mathbb{R})$ be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure $\mu$. Resolving a problem by Breuillard and Green, we show that if $A \subseteq \mathrm{SO}(3,\mathbb{R})$ is…

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson constant term identity. The non-zero part of Kadell's conjecture is a constant term identity indexed by a weak…

In 2021, Navarro and Tiep proposed a conjecture on character fields of finite quasi-simple groups. We develop some theory on sums of roots of unity and use this theory to prove the conjecture for some infinite families of finite…

We discuss electric-magnetic duality in two new classes of supersymmetric Yang-Mills theories. The models have gauge group $Sp(2\nc)$ or $SO(\nc)$ with matter in both the adjoint and defining representations. By perturbing these theories…

We prove a result about reducibility behaviour of Thue polynomials over the rationals that was conjectured by M\"uller. More precisely, we show that, apart from few explicitly given exceptions, these polynomials have only finitely many…

We present an explicit integration formula for the Haar integral on a compact connected Lie group. This formula relies on a known decomposition of a compact connected simple Lie group into symplectic leaves, when one views the group as a…

In this short note, we state a stable and a $\tau$-reduced version of the second Brauer-Thrall Conjecture. The former is a slight strengthening of a brick version of the second Brauer-Thrall Conjecture raised by Mousavand and…

Let $G$ be the simple group ${\rm PSU}(2,2^{2^n})$, $n>0$. For any subgroup $H$ of $G$, we compute the M\"obius function $\mu_L(H,G)$ of $H$ in the subgroup lattice $L$ of $G$, and the M\"obius function $\mu_{\bar L}([H],[G])$ of $[H]$ in…

We prove two conjectures in this paper. The first conjecture is by Lund, Pham and Thu: Given a Borel set $A\subset \mathbb{R}^n$ such that $\dim A\in (k,k+1]$ for some $k\in\{1,\dots,n-1\}$. For $0<s<k$, we have \[ \text{dim}(\{y\in…

We prove that $1-o(1)$ fraction of all $k$-SAT functions on $n$ Boolean variables are unate (i.e., monotone after first negating some variables), for any fixed positive integer $k$ and as $n \to \infty$. This resolves a conjecture by…

Friedlander and Mazur proposed a conjecture of hard Lefschetz type on Lawson homology. We shall relate this conjecture to Suslin conjecture on Lawson homology. For abelian varieties, this conjecture is shown to be equivalent to a vanishing…

In this paper, we prove the conjecture of Yui and Zagier concerning the factorization of the resultants of minimal polynomials of Weber class invariants. The novelty of our approach is to systematically express differences of certain Weber…

The Tate conjecture has two parts: an assertion (S) about semisimplicity of Galois representations, and an assertion (T) which says that every Tate class is algebraic. We show that in characteristic 0, (T) implies (S). In characteristic p…

In [7], G. Navarro proposed a refinement of the McKay conjecture involving a special class of Galois automorphisms. In [6] this new conjecture was verified by the author for the alternating groups A(n) when p=2. In this note the Navarro…

The celebrated Mason's conjecture states that the sequence of independent set numbers of any matroid is log-concave, and even ultra log-concave. The strong form of Mason's conjecture was independently solved by Anari, Liu, Oveis Gharan and…

We prove a Euler-Poincar\'e reduction theorem for stochastic processes taking values in a Lie group and we show examples of its application to SO(3) and to the group of diffeomorphisms.

The goal of this notice is to present a proof of Bachet's conjecture based exclusively on the fundamental theorem of arithmetic. The novelty of this proof consists in its introduction of a partial order on rational integers through the…