Related papers: On $p$-adic Transference Theorem
The lattice definition of the two-dimensional topological quantum field theory [Fukuma, {\em et al}, Commun.~Math.~Phys.\ {\bf 161}, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that…
We divide the circular boundary of a hyperbolic lattice into four equal intervals, and study the probability of a percolation crossing between an opposite pair, as a function of the bond occupation probability p. We consider the {7,3}…
A partial algebra construction of Gr\"atzer and Schmidt from "Characterizations of congruence lattices of abstract algebras" (Acta Sci. Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to a well-known fact that…
We continue our work on the model theory of free lattices, solving two of the main open problems from our first paper on the subject. Our main result is that the universal (existential) theory of infinite free lattices is decidable. Our…
Motivated by team semantics and existential second-order logic, we develop a model-theoretic framework for studying second-order objects such as sets and relations. We introduce a notion of abstract elementary team categories that…
It is known from Grzegorczyk's paper \cite{grze-1951} that the lattice of real semi-algebraic closed subsets of ${\mathbb R}^n$ is undecidable for every integer $n\geq 2$. More generally, if $X$ is any definable set over a real or…
The self-destructive percolation model is defined as follows: Consider percolation with parameter $p > p_c$. Remove the infinite occupied cluster. Finally, give each vertex (or, for bond percolation, each edge) that at this stage is vacant,…
We generalize Banaszczyk's seminal tail bound for the Gaussian mass of a lattice to a wide class of test functions. From this we obtain quite general transference bounds, as well as bounds on the number of lattice points contained in…
The spatial distribution of fields and currents in confining theories can give direct evidence of dual superconductivity. We review the behavior of vortices in the lattice Higgs effective theory. We discuss the techniques for finding these…
This paper studies the differential lattice, defined to be a lattice $L$ equipped with a map $d:L\to L$ that satisfies a lattice analog of the Leibniz rule for a derivation. Isomorphic differential lattices are studied and classifications…
A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a…
We prove the existence of two-dimensional good lattice points in thick multiplicative subgroups modulo $p$.
In this paper we prove complete $p$-adic analogues of Kleinbock's theorems \cite{Kleinbock-extremal, Kleinbock-exponent} on inheritance of Diophantine exponents for affine subspaces. In particular, we answer in the affirmative (and in a…
A new result of G. Cz\'edli states that for an ordered set $P$ with at least two elements and a group $G$, there exists a bounded lattice $L$ such that the ordered set of principal congruences of $L$ is isomorphic to $P$ and the…
Updated version of 2013 Arizona WInter School notes on modularity lifting theorems for for two-dimensional p-adic representations, using wherever possible arguments that go over to the n-dimensional (self-dual) case.
Let $\mathbf{K}$ be an algebraically closed field of arbitrary characteristic, complete with respect to a non-archimedean absolute value $|\,|$. We establish a Second Main Theorem type estimate for analytic map $f\colon…
A new "bond-algebraic" approach to duality transformations provides a very powerful technique to analyze elementary excitations in the classical two-dimensional XY and $p$-clock models. By combining duality and Peierls arguments, we…
We give a new existence proof for the rank 2^d even lattices usually called the Barnes-Wall lattices, and establish new results on uniqueness, structure and transitivity of the automorphism group on certain kinds of sublattices. Our proofs…
In this paper, we give a new proof of an arithmetic analogue of the Riemann-Roch Theorem, due originally to Serge Lang. Lang's result was first proved using the lattice point geometry of Minkowski. By contrast, our proof is completely…
We establish a "matrix simultaneous diagonalization theorem" for disconnected reductive groups which relaxes both the semisimplicity condition and the commutativity condition. As an application, we prove the following basic results…