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Let $k$ be a local field of characteristic 0, and let $G$ be a connected semisimple almost $k$-algebraic group. Suppose rank$_kG\geq 1$ and $\rho$ is an excellent representation of $G$ on a finite dimensional $k$-vector space $V$. We…

The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not fully investigated question. In the current…

Let G be a real Lie group and H a lattice or, more generally, a closed subgroup of finite covolume in G. We show that the unitary representation lambda_{G/H} of G on L^2(G/H) has a spectral gap, that is, the restriction of lambda_{G/H} to…

We express each Clebsch-Gordan (CG) coefficient of a discrete group as a product of a CG coefficient of its subgroup and a factor, which we call an embedding factor. With an appropriate definition, such factors are fixed up to phase…

We give a simple definition of property T for discrete quantum groups. We prove the basic expected properties: discrete quantum groups with property T are finitely generated and unimodular. Moreover we show that, for "I.C.C." discrete…

Let $G$ be a second-countable amenable group with a uniform $k$-approximate lattice $\Lambda$. For a projective discrete series representation $(\pi, \mathcal{H}_{\pi})$ of $G$ of formal degree $d_{\pi} > 0$, we show that $D^-(\Lambda) \geq…

Lipschitz constants are connected to many properties of neural networks, such as robustness, fairness, and generalization. Existing methods for computing Lipschitz constants either produce relatively loose upper bounds or are limited to…

We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of $PSL(2,{\bold C})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in…

We give very precise bounds for the congruence subgroup growth of arithmetic groups. This allows us to determine the subgroup growth of irreducible lattices of semisimple Lie groups. In the most general case our results depend on the…

If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here "almost" and "close" are in terms of the Hilbert-Schmidt norm. Using tracial 2-norm…

Deriving sharp and computable upper bounds of the Lipschitz constant of deep neural networks is crucial to formally guarantee the robustness of neural-network based models. We analyse three existing upper bounds written for the $l^2$ norm.…

We introduce and study certain asymptotic invariants associated with fusion algebras (equipped with a dimension function), which arise naturally in the representation theory of compact quantum groups. Our invariants generalise the analogous…

We show that computing the lattice programming gap of the group problems is NP-hard when the dimension is a part of input. We also obtain lower and upper bounds for the gap in terms of the cost vector and the determinant of the lattice.

For an abelian variety $A$ over a number field we study bounds depending only on the dimension of $A$ for the minimal degree $d(A)$ of a field extension over which $A$ acquires semi-stable reduction. We first compute $d(A)$ in terms of the…

The existence of a strong spectral gap for lattices in semi-simple Lie groups is crucial in many applications. In particular, for arithmetic lattices it is useful to have bounds for the strong spectral gap that are uniform in the family of…

Given a finite group with a generating subset there is a well-established notion of length for a group element given in terms of its minimal length expression as a product of elements from the generating set. Recently, certain quantities…

Let $G$ be a countable group. We introduce several equivalence relations on the set ${\rm Sub}(G)$ of subgroups of $G$, defined by properties of the quasi-regular representations $\lambda_{G/H}$ associated to $H\in {\rm Sub}(G)$ and compare…

A complex spherical code is a finite subset on the unit sphere in $\mathbb{C}^d$. A fundamental problem on complex spherical codes is to find upper bounds for those with prescribed inner products. In this paper, we determine the irreducible…

Spectral clustering is a well-known technique which identifies $k$ clusters in an undirected graph with weight matrix $W\in\mathbb{R}^{n\times n}$ by exploiting its graph Laplacian $L(W)$, whose eigenvalues $0=\lambda_1\leq \lambda_2 \leq…

We show some preservation results of amenably extending strongly Ulam stable groups under mild decay assumptions, including quantitative preservation of asymptotic bounds under the assumption that the modulus of stability is H\"older…