Discrete mathematics: methods and challenges

离散数学

Discrete Mathematics

本研究提供一套采用Rota方法的初等计数代数组合学入门课程教学讲义，以组合构造和两种初等原理推导结果，内容自洽仅需高中数学基础，兼具组合与概率视角。

Limit theory of discrete mathematics problems

We show a general problem-solving tool called limit theory. This is an advanced version of asymptotic analysis of discrete problems when some finite parameter tends to infinity. We will apply it on three closely related problems. Alpern's Caching Game (for 2 nuts) is defined as follows. The hider caches 2 nuts into one or two of $n$ potential holes by digging at most 1 depth in total. The goal of the searcher is to find both nuts in a limited time $h$, otherwise the hider wins. We will show that if $h$ and $n/h$ are large enough, then very counterintuitively, any optimal hiding strategy should…

On the Reconstruction of Static and Dynamic Discrete Structures

We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the `classical' task of reconstructing finite point sets in $\mathbb{R}^d$). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast…

Tanglegrams: a reduction tool for mathematical phylogenetics

Many discrete mathematics problems in phylogenetics are defined in terms of the relative labeling of pairs of leaf-labeled trees. These relative labelings are naturally formalized as tanglegrams, which have previously been an object of study in coevolutionary analysis. Although there has been considerable work on planar drawings of tanglegrams, they have not been fully explored as combinatorial objects until recently. In this paper, we describe how many discrete mathematical questions on trees "factor" through a problem on tanglegrams, and how understanding that factoring can simplify analysis…

Marches aléatoires dans un cône et fonctions discrètes harmoniques

Random walks in cones have the double interest of being at the heart of many probabilistic problems and of being related to many mathematical fields, such as spectral theory, combinatorics, or discrete complex analysis. In this article, we present some key ideas associated with these processes: we will discuss their definition, the link with Brownian motion in cones, as well as some recent research topics such as the construction of discrete harmonic functions.

On discrete field theory properties of the dimer and Ising models and their conformal field theory limits

We study various mathematical aspects of discrete models on graphs, specifically the Dimer and the Ising models. We focus on proving gluing formulas for individual summands of the partition function. We also obtain partial results regarding conjectured limits realized by fermions in rational conformal field theories.

A survey of discrete methods in (algebraic) statistics for networks

Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view of a particular family of statistical models for networks called exponential random graph models. The problems and underlying constructions are also related to well-known concepts in commutative algebra and graph-theoretic concepts in computer science. We outline a few lines of recent work that highlight the natural connection between these fields and unify t…

Dominic Welsh：他的工作与影响

Dominic Welsh: his work and influence

本文回顾了Dominic Welsh在离散数学领域的广泛贡献，重点强调其在拟阵理论和Tutte-Whitney多项式方面的工作及深远影响。