We initiate the program of extending to higher-rank graphs (k-graphs) the geometric classification of directed graph -algebras, as completed in Eilers et al. (2016, Preprint). To be precise, we identify four “moves,” or modifications, one can perform o…
The Parisi formula is a self-contained description of the infinite-volume limit of the free energy of mean-field spin glass models. We showthat this quantity can be recast as the solution of a Hamilton–Jacobi equation in the Wasserstein space of probab…
Let A be a two-dimensional abelian variety defined over a number field K. Fix a prime number
and suppose
for a set of primes
of density 1. When
Serre has shown that there does not necessarily exist a K-isogenous
such…
Let
,
, and
be distinct reduced residues modulo q satisfying the congruences
. We conditionally derive an asymptotic formula, with an error term that has a power savings in q, for the logarithmic density of the set of real nu…
We construct a family of compactifications of the affine cone of the Grassmannian variety of
-planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton–Okounkov bodies f…
Let
be the set of real numbers that are normal to base b. A well-known result of Ki and Linton [19] is that
is
-complete. We show that the set
of reals, which preserve
under addition, is also
-complete. We use the ch…
In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let
be a minimal or next-to-minimal automorphic representation of G. We prove that any
is completely determined…
We present a new approach to graph limit theory that unifies and generalizes the two most well-developed directions, namely dense graph limits (even the more general
limits) and Benjamini–Schramm limits (even in the stronger local-global setti…
We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a Möbius automorphism group of dimension at least two. Our theorem generalizes the cl…
The asymptotic behavior of solutions to a family of Dirichlet boundary value problems, involving differential operators in divergence form, on a domain equipped with a Finsler metric is investigated. Solutions are shown to converge uniformly to the d…