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

For an integer $t \geq 3$, let $\mathcal{L}(t)$ denote the linear equation $x_1 + x_2 + \cdots + x_{t-1} = x_t,$ where all variables are positive integers. For integers $k \geq 1$ and $t_0,t_1,\dots,t_{k-1} \geq 3$, the generalized Schur…

In this article, we prove that every compact simple Lie group $SO(n)$ for $n\geq 10$ admits at least $2\left([\frac{n-1}{3}]-2\right)$ non-naturally reductive left-invariant Einstein metrics.

We prove that if a time-periodic Tonelli Lagrangian on a closed manifold $M$ satisfies a strong version of the Differentiability Problem for Mather's $\beta$-function, then the Legendre transforms of rational homology classes are dense in…

Studying the supersymmetry enhancement mechanism of Aharony, Bergman, Jafferis and Maldacena, we find a simple condition on the gauge group generators for the matter fields. We analyze all possible compact Lie groups and their…

We prove that if all the simple groups involved in a finite group $G$ satisfy the `inductive Feit condition', then Walter Feit's conjecture from 1980 holds for $G$. In particular, this would solve Brauer's Problem 41 from 1963 in the…

In this note we discuss the classification of duality orbits of N = 8 gauged supergravity models. Using tensor classifiers, we show that there is a one-parameter family of inequivalent SO(8) gauged supergravity theories. We briefly discuss…

We prove Malle's conjecture for $G \times A$, with $G=S_3, S_4, S_5$ and $A$ an abelian group. This builds upon work of the fourth author, who proved this result with restrictions on the primes dividing $A$.

We generalize $N \leftrightarrow -N$ duality of dimension formulae of $SU(N)$ representations on a (class of) representations with $N$-dependent Young diagrams (which include the adjoint representation), and on eigenvalues of the Casimir…

We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C*-algebra $D$; we also explain how the known classification theorems for nuclear C*-algebras fit into this concept. As a new result…

We study Stanley's long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable…

For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that…

A conjecture of Boone and Higman from the 1970's asserts that a finitely generated group $G$ has solvable word problem if and only if $G$ can be embedded into a finitely presented simple group. We comment on the history of this conjecture…

Tuza (A conjecture, in Proceedings of the Colloquia Mathematica Societatis Janos Bolyai, 1981) conjectured that $\tau(G) \le 2\nu(G)$ for all graphs $G$, where $\tau(G)$ is the minimum size of an edge set whose removal makes $G$…

We formulate Aubry-Mather theory for Hamiltonians/Lagrangians defined on graphs and discuss its relationship with weak KAM theory developed in [24].

We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of L\"uck, Sauer and Wegner hold for treeable equivalence relations.

White's conjecture asserts that any two tuples of matroid bases that have the same multi-set union can be transformed from one to another by symmetric exchanges; it also implies that the toric ideals of matroids are generated by the…

Given a simple Lie algebra $\mathfrak{g}$ and an element $\mu\in\mathfrak{g}^*$, the corresponding shift of argument subalgebra of $\text{S}(\mathfrak{g})$ is Poisson commutative. In the case where $\mu$ is regular, this subalgebra is known…

Let $A$ be an abelian variety over a field finitely generated over $\mathbb{Q}$. We show that the finiteness of the $\ell$-primary torsion subgroup of the higher Brauer group is a sufficient criterion for the Tate conjecture to hold.…

We study the non-perturbative behavior of some N=1 supersymmetric product-group gauge theories with the help of duality. As a test case we investigate an SU(2)xSU(2) theory in detail. Various dual theories are constructed using known…

We prove a metrical result on a family of conjectures related to the Littlewood conjecture, namely the original Littlewood conjecture, the mixed Littlewood conjecture of de Mathan and Teuli\'e and a hybrid between a conjecture of Cassels…