Universality in the 2D Ising model and conformal invariance of fermionic observables

It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no mathematical proof has ever been given, and even physics arguments support (a priori weaker) M?bius invariance. We introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. We show that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, thus proving the universality and conformal invariance conjectures.     

Modular andp-adic cyclic codes

This paper presents some basic theorems giving the structure of cyclic codes of lengthn over the ring of integers modulop ^a and over thep-adic numbers, wherep is a prime not dividingn. An especially interesting example is the 2-adic cyclic code of length 7 with generator polynomialX ³ +X ² +(–1)X–1, where satisfies ² - + 2 = 0. This is the 2-adic generalization of both the binary Hamming code and the quaternary octacode (the latter being equivalent to the Nordstrom-Robinson code). Other examples include the 2-adic Golay code of length 24 and the 3-adic Golay code of length 12.     

Arrangements with one bounded component andp-adic integrals

In this article we study arrangementsA, such that ℝ ⁿ \A has exactly one bounded component. We obtain a result about their structure which gives us a method to construct all combinatorially different such arrangements in a given dimension. (A complete list for dimensions 1,2,3 and 4 is included). Furthermore we associate ap-adic integral to each such arrangement and proof that this integral can be written as a product ofp-adic beta functions. This is analogous to results of Varchenko and Loeser for integrals over ℝ and character sums over finite fields respectively.     

Proof of conformal invariance in the critical regime for models of Gross-Neveu type

It is shown that for all multiplicatively renormalizable fermionic models without derivativs in the interaction, in particular, for the Gross—Neveu model, conformal invariance of the Green's functions of the fields in the critical regime (massless model at fixed renormalization-group point) automatically follows from the scale invariance for arbitrary space dimension.St Petersburg State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 486–497, September, 1992.     

Asymptotic conformal invariance and asymptotic finiteness in grand unification theories in curved spacetime

Lenin Komsomol State Pedagogical Institute, Tomsk. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 3, pp. 399–405, June, 1990.     

The universal vectorial Bi-extension andp-adic heights

Oblatum 18-XII-1988 &; 1-VIII-1990     

Syntomic regulators andp-adic integration I: Rigid syntomic regulators

We construct a new version of syntomic cohomology, called rigid syntomic cohomology, for smooth schemes over the ring of integers of ap-adic field. This version is more refined than previous constructions and naturally maps to most of them. We construct regulators fromK-theory into rigid syntomic cohomology. We also define a “modified” syntomic cohomology, which is better behaved in explicit computations yet is isomorphic to rigid syntomic cohomology in most cases of interest.     

Semistable measures and limit theorems on real andp-adic groups

For any locally compact groupG, we show that any locally tight homomorphism from a real directed semigroup intoM ¹ (G) (semigroup of probability measures onG) has a shift which extends to a continuous one-parameter semigroup. IfG is ap-adic algebraic group then the above holds even iff is not locally tight. These results are applied to give sufficient conditions for embeddability of some translate of limits of sequences of the form {v _n ^kn } and M ¹ (G) such that ()= ^M , for somek>1 and AutG (cf. Theorems 2.1, 2.4, 3.7).     

Theorems of Kakutani and Dyson revisited

Two theorems of Kakutani and Dyson concerning real-valued functions on a sphere, dating from the middle of the last century, are presented as special cases of a result on stable cohomotopy Euler classes. A complex analogue is also proved and applied to establish a complex version of Dyson’s theorem. Dedicated to Felix Browder