Related papers: Morse Spectra, Homology Measures and Parametric Pa…
This dissertation develops, for several families of statistical mechanical and random growth models, techniques for analyzing infinite-volume asymptotics. In the statistical mechanical setting, we focus on the low-temperature phases of spin…
We discuss multiloop MHV amplitudes in the N=4 SYM theory in terms of effective gravity in the momentum space with IR regulator branes as degrees of freedom. Kinematical invariants of external particles yield the moduli spaces of complex or…
The quantum mechanics of one degree of freedom exhibiting the exact conformal SL(2,R) symmetry is presented. The starting point is the classification of the unitary irreducible representations of the SL(2,R) group (or, to some extent, its…
We prove a recognition principle for motivic infinite P1-loop spaces over a perfect field. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of E-infinity-spaces. A framed motivic…
We introduce two new heuristic ideas concerning the spectrum of a Laplacian, and we give theorems and conjectures from the realms of manifolds, graphs and fractals that validate these heuristics. The first heuristic concerns Laplacians that…
We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in…
This paper presents a spectral calculus for computing the spectrum of a causal Lorentz invariant Borel complex measure on Minkowski space, thereby enabling one to compute the density for such a measure with respect to Lebesque measure. It…
This paper is built on the following observation: the purity of the mixed Hodge structure on the cohomology of Brown's moduli spaces is essentially equivalent to the freeness of the dihedral operad underlying the gravity operad. We prove…
We consider two models where the wave equation can be reduced to the effective Schr\"odinger equation whose potential contains both harmonic and the Coulomb terms, $\omega^{2}r^{2}-a/r$. The equation reduces to the biconfluent Heun's…
We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. Knopp…
We study some problems on the distribution of values of symmetric power $L$-functions at $s=1$ in both aspects of level and weight: bounds of these values, extreme values, Montgomery-Vaughan's conjecture and distribution functions. Our…
We present here some connections between the liberation process for projections $(P,Q)\mapsto(P,U_tQU_t^*)$ and its counterpart $(R,S)\mapsto(R,U_tSU_t^*)$ for symmetries when the projections $\{P,Q\}$ and the symmetries $\{R,S\}$ are…
Consider the algebra Q<<x_1,x_2,...>> of formal power series in countably many noncommuting variables over the rationals. The subalgebra Pi(x_1,x_2,...) of symmetric functions in noncommuting variables consists of all elements invariant…
In the article we consider Bach-flat metrics on four-manifolds with boundary, with conformally invariant boundary conditions. We show that such metrics arise naturally as critical points of the Weyl energy under a constraint. We then prove…
Recent conjectures of the c-theorem in four and higher dimensions have suggested that the coefficient of the Euler characteristic in the trace anomaly could measure the degrees of freedom in field theory and decrease along the…
We propose a definition of magnitude for a length space with a Borel measure, which involves integrals over the set of geodesics. This quantity agrees with the magnitude of finite metric spaces, up to re-scaling the metric to ensure the…
The quantum mechanical version of a classical model for studying the orientational degrees of freedom corresponding to a nematic liquid composed of biaxial molecules is presented. The effective degrees of freedom are described by operators…
The problem of duality symmetry in free field models is examined in details by performing a mode expansion of these fields which provides a mapping with the purely quantum mechanical example of a harmonic oscillator. By analysing the…
We show exact dimensionality of harmonic measures associated with random walks on groups acting on a hyperbolic space under finite first moment condition, and establish the dimension formula by the entropy over the drift. We also treat the…
Some relations of the quantum potential to Weyl geometry are indicated with applications to the Friedmann equations for a toy quantum cosmology. Osmotic velocity and pressure are briefly discussed in terms of quantum mechanics and…