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This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of…
Let W be the Weyl group of a crystallographic root system acting on the associated weight lattice by reflections. In the present notes we extend the notion of an exponent of the W-action introduced in [Baek-Neher-Zainoulline,…
The special linear groups, the mapping class groups of surfaces, the outer autormorphism groups of free groups appear in numerous domains. Their analogies, developped in particular in K. Vogtmann's work, have been written about a lot. In…
We investigate a family of rule-based logics. The focus is on very expressive languages. We provide a range of characterization results for the expressive powers of the logics and relate them with corresponding game systems.
I describe three geometric approaches to resolving variants of P v. NP, present several results that illustrate the role of group actions in complexity theory, and make a first step towards completely geometric definitions of complexity…
Causal models defined in terms of a collection of equations, as defined by Pearl, are axiomatized here. Axiomatizations are provided for three successively more general classes of causal models: (1) the class of recursive theories (those…
In this paper we indicate one method of construction of linear representations of groups and algebras with translation invariant (except, maybe , finite number) defining relationships. As an illustration of this method, we give one approach…
We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the…
Ergodic theory includes several notions of entropy for probability-preserving actions of countable groups. These include Kolmogorov--Sinai entropy based on F\o lner sequences for amenable groups, entropy defined using a random ordering of…
This paper is a set of lecture notes of my course "Special functions, KZ type equations, and representation theory" given at MIT during the spring semester of 2002. The notes do not contain new results, and are an exposition (mostly without…
Causal models defined in terms of a collection of equations, as defined by Pearl, are axiomatized here. Axiomatizations are provided for three successively more general classes of causal models: (1) the class of recursive theories (those…
Two groups are called isocategorical over a field $k$ if their respective categories of $k$-linear representations are monoidally equivalent. We classify isocategorical groups over arbitrary fields, extending the earlier classification of…
We bring a linkage from representation theory of Lie groups to homotopy theory for maps between flag manifolds. As applications we derive from representation theory abundant families of homotopy classes of maps between flag manifolds whose…
In this note we give an overview of some of our recent work on Anosov representations of discrete groups into higher rank semisimple Lie groups.
We consider the Ext-analogues of branching laws of representations for a group to its subgroups in the context of $p$-adic groups.
Extending earlier work of Guralnick and of Cai and Zhang, we classify the almost simple groups which have transitive permutation representations of prime power degree $p^k$, and those which have $p$-complements (stabilisers of order coprime…
This is a largely expository paper about how groups arise or are of interest in model theory. Included are the following topics: classifying groups definable in specific structures or theories and the relation to algebraic groups, groups…
The major concern in the study of categories of logics is to describe condition for preservation, under the a method of combination of logics, of meta-logical properties. Our complementary approach to this field is study the "global"…
We introduce a logic specifically designed to support reasoning about social choice functions. The logic includes operators to capture strategic ability, and operators to capture agent preferences. We establish a correspondence between…
We study several properties of expansive group actions on metric spaces and obtain relation between expansivity for subgroup and group actions. Through counter examples necessity of hypothesis are justified. We also study expansivity of…