Related papers: Independence Tuples and Deninger's Problem

Given a finite group $G$ and a subgroup $K$, we study the commutant of $\text{Ind}_K^G\theta$, where $\theta$ is an irreducible $K$-representation. After a careful analysis of Frobenius reciprocity, we are able to introduce an orthogonal…

A contractive $n$-tuple $A=(A_1,...,A_n)$ has a minimal joint isometric dilation $S=(S_1,...,S_n)$ where the $S_i$'s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$…

Let $G=\{e^{tA}:t\in\mathbb{R}\}$ be a closed one-parameter subgroup of the general linear group of matrices of order $n$ acting on $\mathbb{R}^{n}$ by matrix-vector multiplications. We assume that all eigenvalues of $A$ are rationally…

Let $p$ be a prime, $k$ a finite extension of $\mathbf{F}_p$ of cardinal $q$, $l$ a finite extension of $k$ of group $\Sigma=\mathrm{Gal}(l|k)$, and $T$ a subgroup of $l^\times$. Using the method of "little groups", we classify irreducible…

We define a new notion of entropy for operators on Fock spaces and positive definite multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (multi-Toeplitz, multi-analytic, etc.) operators…

Entropic independence is a structural property of measures that underlies modern proofs of functional inequalities, notably (modified) log-Sobolev inequalities, via ``annealing'' or local-to-global schemes. Existing sufficient criteria for…

We define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is $d^p$-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces and we…

Free independence is an important tool for studying the structure of operator algebras. It is natural to ask from the model-theoretic standpoint whether free independence is captured well in first-order model theory via the notion of a…

For $p \in \mathbb{Q}_+ \smallsetminus \{ 1 \}$ a positive rational number different from one, we say that the Puisseux series $f \in \mathbb{C}((t))^\text{alg}$ is $p$-Mahler of non-exceptional polynomial type if there is a polynomial $P…

We produce infinite families of knots $\{K^i\}_{i\geq 1}$ for which the set of cables $\{K^i_{p,1}\}_{i,p\geq 1}$ is linearly independent in the knot concordance group. We arrange that these examples lie arbitrarily deep in the solvable and…

We calculate the Fuglede-Kadison determinant for operators of the form $\sum_{i=1}^n M_{f_i}L_{g_i}$ where $L_{g_i}$ are unitaries or partial isometries coming from Borel (partial) isomorphisms $g_i$ on a probability space which generate an…

We extend L\"uck's determinant conjecture from groups to invariant random subgroups (IRS) of free groups, a framework generalizing groups where a non-sofic object is known to exist. For every free group, we prove the existence of an IRS…

In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings.…

We establish a "matrix simultaneous diagonalization theorem" for disconnected reductive groups which relaxes both the semisimplicity condition and the commutativity condition. As an application, we prove the following basic results…

We impose rank one constraints on marginalizations of a tensor, given by a simplicial complex. Following work of Kirkup and Sullivant, such marginal independence models can be made toric by a linear change of coordinates. We study their…

Let A be a commutative noetherian ring. Call a functor <<commutative A-algebras>> --> <<sets>> coherent if it can be built up (via iterated finite limits) from functors of the form B \mapsto M tensor_A B, where M is a f.g. A-module. When…

Let f(x) be a differentiable function on the real line R, and let P be a point not on the graph of f(x). Define the illumination index of P to be the number of distinct tangents to the graph of f which pass thru P. We prove that if f '' is…

The Deligne-Simpson problem (DSP) (resp. the weak DSP) is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ or $c_j\subset gl(n,{\bf C})$ so that there…

In the restricted setting of product phase space lattices, we give an alternate proof of P. Linnell's theorem on the finite linear independence of lattice Gabor systems in $L^2(\mathbb R^d)$. Our proof is based on a simple argument from the…

In this paper, we prove that for every irreversible Finsler $n$-dimensional real projective space $(\mathbb{R}P^n,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $\frac{16}{9}\left(\frac{\lambda}{1+\lambda}\right)^2<K\le…