Related papers: Trace map, Cayley transform and LS category of Lie…

We propose a new notion of `n-category with duals', which we call a Whitney n-category. There are two motivations. The first is that Baez and Dolan's Tangle Hypothesis is (almost) tautological when interpreted as a statement about Whitney…

The aim of this paper is to study the natural action of the real symplectic group, $\operatorname{Sp}(4, \mathbb{R})$, on the algebraic set of $4$-dimensional Lie algebras admitting symplectic structures and to give a complete…

This survey purports to be an elementary introduction to compactly presented groups, which are the analogue of finitely presented groups in the broader realm of locally compact groups. In particular, compact presentation is interpreted as a…

We detail an explicit construction of ordinary irreducible representations for the family of finite groups $SL_2({\mathbb Z} /p^n {\mathbb Z})$ for odd primes $p$ and $n\geq 2$. For $n=2$, the construction is a complete set of irreducible…

Let $S$ be a set of transpositions generating the symmetric group $S_n$. The transposition graph of $S$ is defined to be the graph with vertex set $\{1,\ldots,n\}$, and with vertices $i$ and $j$ being adjacent in $T(S)$ whenever $(i,j) \in…

We introduce the notion of a symmetric basis of a vector space equipped with a quadratic form, and provide a sufficient and necessary condition for the existence to such a basis. Symmetric bases are then used to study Cayley graphs of…

This work is motivated by an investigation into whether, and if so how, certain well known facts about Lie groups manifest in the context of group schemes over rings of integers of local fields. There are the following well-known relations…

We fill in the details in a procedure outlined by Serre for drawing pictures of compact real Lie groups. In the case of Sp($2n$), the picture generated by the method is connected with abelian varieties over a number field or a finite field.…

We construct explicit generating sets S_n and \tilde S_n of the for the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for all n. This…

We study Quillen model categories equipped with a monoidal skew closed structure that descends to a genuine monoidal closed structure on the homotopy category. Our examples are 2-categorical and include permutative categories and…

Topological transforms have been very useful in statistical analysis of shapes or surfaces without restrictions that the shapes are diffeomorphic and requiring the estimation of correspondence maps. In this paper we introduce two…

We give an explicit way of calculating the set of homotopy classes of morphisms from a Tamsamani n-category A to another one B. This calculation uses a Reedy-cofibrant cosimplicial resolution of A, using a new notion of ``free cofibration''…

The attempt is to give a formal concpet of system, and with this provide a definition of category, that will also satisfy the definition of a system. An axiomatic base is given, for constructing the group of integers. In the process, we…

In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce `higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously…

We incorporate nonlinear covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. This L-group is an extension of the absolute Galois group of a local or global field $F$ by a complex…

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to…

We study the representations of non-commutative universal lattices and use them to compute lower bounds for the \TauC for the commutative universal lattices $G_{d,k}= \SL_d(\Z[x_1,...,x_k])$ with respect to several generating sets. As an…

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the…

In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of…

Given an $\infty$-category with a set of weak equivalences which is stable under pullback, we show that the mapping spaces of the corresponding localization can be described as group completions of $\infty$-categories of spans. Furthermore,…