Related papers: Lecture on Branched Polymers and Dimensional Reduc…
We review recent progress concerning the analysis of Lagrangians on immersions into $\mathbb{R}^d$ depending on the first and second fundamental forms and their covariant derivatives.
This article will review recent results on dimensional reduction for branched polymers, and discuss implications for critical phenomena. Parisi and Sourlas argued in 1981 that branched polymers fall into the universality class of the…
Branched polymers can be classified into two categories that obey the different formulae: \begin{equation} \nu= \begin{cases} \hspace{1mm}\displaystyle\frac{2(1+\nu_{0})}{d+2} & \hspace{3mm}\mbox{for polymers…
In this paper we study the Grassmannian of submodules of a given dimension inside a finitely generated projective module $P$ for a finite dimensional algebra $\Lambda$ over an algebraically closed field. The orbit of such a submodule $C$…
These are lecture notes for a mini-course given at the St. Petersburg School in Probability and Statistical Physics in June 2012. Topics include integrable models of random growth, determinantal point processes, Schur processes and Markov…
This paper is an introduction to polarizations in the symplectic and orthogonal settings. They arise in association to a triple of compatible structures on a real vector space, consisting of an inner product, a symplectic form, and a…
We exhibit isomorphisms of Grassmann spaces and their relationship with collineations and embeddings of the underlying projective spaces.
We study a partial differential relation that arises in the context of the Born-Infeld equations (an extension of the Maxwell's equations) by using Gromov's method of convex integration in the setting of divergence free fields.
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the…
Modern machine learning algorithms have been adopted in a range of signal-processing applications spanning computer vision, natural language processing, and artificial intelligence. Many relevant problems involve subspace-structured…
We show that correlation functions for branched polymers correspond to those for $\phi^3$ theory with a single mass insertion, not those for the $\phi^3$ theory themselves, as has been widely believed. In particular, the two-point function…
This work presents an approach towards the representation theory of the braid groups $B_n$. We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids,…
We discuss the extension of the empirical equation: $\left\langle s_{N}^{2}\right\rangle_{0}\propto g\,l^{2}$, where the subscript 0 denotes the ideal value with no excluded volume and $g$ the generation number from the root to the youngest…
These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the (q,t)-deformation of those leading to…
In this revised version, we add some expository material and references and make some minor corrections.
This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving…
We completely describe by inequalities the set of boundary correlation matrices of planar Ising networks embedded in a disk. Specifically, we build on a recent result of M.~Lis to give a simple bijection between such correlation matrices…
We establish an exact relation between self-avoiding branched polymers in D+2 continuum dimensions and the hard-core continuum gas at negative activity in D dimensions. We review conjectures and results on critical exponents for D+2 = 2,3,4…
We briefly review the recent advances in the rheology of entangled polymers and identify emerging research trends and outstanding challenges, especially with respect to branched polymers. Emphasis is placed on the role of well-characterized…
These notes contain a survey of some aspects of the theory of differential modules and complexes as well as of their generalization, that is, the theory of $N$-differential modules and $N$-complexes. Several applications and examples coming…