相关论文: Random surfaces enumerating algebraic curves
These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main…
We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of…
This is a submission to the Encyclopedia of Mathematical Physics (Elsevier, 2006) and conforms to its referencing guidelines.
These notes are loosely based on an introductory course in algebraic geometry given at Rutgers University in Spring of 2024. We introduce some relatively advanced topics at the expense of the technical details.
We describe smooth rational projective algebraic surfaces X, over an algebraically closed field of characteristic different from 2, having an even set of four disjoint (-2)-curves N_1,...,N_4, i.e. such that N_1+...+N_4 is divisible by 2 in…
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…
These are lecture notes that are based on the lectures from a class I taught on the topic of Spectral Graph Methods at UC Berkeley during the Spring 2015 semester.
In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many…
This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen.
This is an expanded version of my Shaw Prize Lecture delivered at the Chinese University of Hong Kong.
We present a number of questions in commutative algebra posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014 - Spring 2017.
Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance,…
We study the systole of a random surface, where by a random surface we mean a surface constructed by randomly gluing together an even number of triangles. We study two types of metrics on these surfaces, the first one coming from using…
Open problems from the 15th Annual ACM Symposium on Computational Geometry.
In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and…
In the present paper we study the problem of constructing a family of surfaces (surface pencils) from a given curve in 4-dimensional Euclidean space $\mathbb{E}^{4}$. We have shown that generalized rotation surfaces in $\mathbb{E}^{4}$ are…
A telegraphic survey of some of the standard results and conjectures about the set $C({\bf Q})$ of rational points on a smooth projective absolutely connected curve $C$ over ${\bf Q}$.
This is a write-up of the lectures given by the author during the Master Class "Categorification" at {\AA}rhus University, Denmark in October 2010.
In this article, we study the geometry of plane curves obtained by three sections and another section given as their sum on certain rational elliptic surfaces. We make use of Mumford representations of semi-reduced divisors in order to…
These are (not updated) notes from the lectures I gave at the NATO ASI ``Symmetric Functions 2001'' at the Isaac Newton Institute in Cambridge (June 25 -- July 6, 2001). Their goal is an informal introduction to asymptotic combinatorics…