117 research outputs found
The Tannakian Formalism and the Langlands Conjectures
Let H be a connected reductive group over an algebraically closed field of
characteristic zero, and let G be an abstract group. In this note we show that
every homomorphism from the Grothendieck semiring of H to that of G which maps
irreducible representations to irreducibles, comes from a group homomorphism
from G to H. We also connect this result with the Langlands conjectures.Comment: 15 page
Variance Analysis for Monte Carlo Integration: A Representation-Theoretic Perspective
In this report, we revisit the work of Pilleboue et al. [2015], providing a
representation-theoretic derivation of the closed-form expression for the
expected value and variance in homogeneous Monte Carlo integration. We show
that the results obtained for the variance estimation of Monte Carlo
integration on the torus, the sphere, and Euclidean space can be formulated as
specific instances of a more general theory. We review the related
representation theory and show how it can be used to derive a closed-form
solution
Non-coherence of arithmetic hyperbolic lattices
We prove, under the assumption of the virtual fibration conjecture for
arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n>
4, and different from 7, are non-coherent. We also establish noncoherence of
uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of
uniform lattices in SU(2,1) which have infinite abelianization.Comment: 26 pages, 3 figure
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