Probabilistic and Extremal Combinatorics

该研究的大部分是在概率组合学领域，其中包括随机组合结构，随机算法和组合的概率存在性证明（即概率方法）的研究。 近年来，这一领域已经看到了一种技术的发展，用于证明离散随机过程的关键统计数据可能会保持接近其预期的轨迹，随着过程的演变。 这种方法被称为贪婪算法和随机图过程的微分方程方法。 研究人员最近扩展了这种技术来分析三角形自由的过程，一个“控制”的随机图形过程，产生一个有趣的分布在空间上的三角形自由图。 研究人员表明，从这个分布绘制的图可能在最小可能的常数乘法因子内具有独立数。 换句话说，无三角形过程产生拉姆齐R（3，t）图。 研究人员将继续发展这种方法，用于研究相关的“受控”随机过程，重点是从极值组合学的角度来看有趣的过程。 相关问题的动机是计算机科学。 在这里，研究者试图发展微分方程方法的应用，以证明随机算法的良好性能，而无需相关微分方程的显式解或解的数值近似（对这些信息的需要通常使这种方法的应用显着复杂化）。这个研究项目是对离散数学对象的调查，如网络或代码，其随时间的演变由一系列随机选择决定。 我们对不同轮中所做的选择之间存在依赖性的过程特别感兴趣。 虽然这样的过程可以很好地模拟真实世界现象的动力学，比如疾病在网络中的传播或材料中的相变，但这项研究的重点是生成有趣的数学对象的过程。 长期以来，随机性在复杂组合对象的构建中扮演着重要角色，就像随机性在计算机科学中的许多算法中扮演着核心角色一样。 这项研究预计将在多个领域产生影响，因为研究人员将开发通用工具来了解这些系统随时间的演变。

The majority of this research is in the area of probabilistic combinatorics, which is comprised of the study of random combinatorial structures, randomized algorithms and probabilistic existence proofs for combinatorics (i.e. the probabilistic method). In recent years, this field has seen the development of a technique for proving that key statistics of a discrete stochastic process are likely to remain close to their expected trajectories as the process evolves. This method is known as the differential equations method for greedy algorithms and random graph processes. The investigator recently extended this technique to analyze the triangle-free process, a `controlled' random graph process that produces an interesting distribution on the space of triangle-free graphs. The investigator showed that a graph drawn from this distribution is likely to have independence number within a constant multiplicative factor of the smallest possible. In other words, the triangle-free process produces a Ramsey R(3,t) graph. The investigator shall continue the development of this method for the study of related `controlled' random processes, with an emphasis on processes that are interesting from the perspective of extremal combinatorics. Related questions are motivated by computer science. Here the investigator attempts to develop applications of the differential equations method to prove good performance of randomized algorithms without an explicit solution, or numerical approximation of the solution, of the associated differential equation (the need for such information often significantly complicates application of this method).This research project is an investigation of discrete mathematical objects, like networks or codes, whose evolution over time is determined by a sequence of random choices. We are particularly interested in processes where there is dependence between the choices made in different rounds. While such processes can be good models for the dynamics of real-world phenomenon, like the spread of a disease in a network or phase transitions in materials, the focus of this research is on processes that generate interesting mathematical objects. Randomness has long played an important role in the construction of sophisticated combinatorial objects, just as randomness plays an a central role in many algorithms in computer science. This research is expected to have an impact in multiple domains as the investigator shall develop general tools for understanding the evolution of these systems over time.