Combinatorics with Applications

研究人员正在继续他们的核心研究项目，研究组合学的重要经典问题，包括(1)与离散几何有联系的图的交叉数理论；(2)寻找关于子集的反链和相交族的新见解，以及子集的链分解与对称维恩图之间的联系。具有广泛影响的研究者的研究包括他们在以下方面的贡献：(3)计算生物学：系统发育树重建，聚类生物分子序列；(4)理论计算机科学：图形绘制，统计数据库的安全性；(5)通信：带有距离条件的实数顶点标记的新模型，用于为大型发射机网络找到最优或接近最优的信道分配。组合学的研究是在我们理解离散结构如何工作以及如何最佳地使用它们的核心问题上进行的。研究人员还将组合理论应用于跨学科领域（包括计算生物学、理论计算机科学和通信），这些领域的进展通常需要使用或发明更深入的组合结果，这些结果展示了对应用至关重要的结构，或者提供了任何算法可以实现的界限。这些领域的进展有助于解决重要的现实问题，包括开发更有效的生物信息学研究工具；为电脑屏幕设计更好的可视化工具；提高计算机和数据库的安全性；以及更高效的手机通讯。研究人员在招募和培训研究生并让他们参与研究方面取得了令人印象深刻的成绩。他们正在加大努力，吸引更多的少数族裔和女性学生攻读数学博士学位。他们在管理研究期刊、组织会议、指导加拿大-美国高中数学夏令营以及为本科生国际数学建模竞赛做出贡献等方面继续开展广泛的活动。

The investigators are continuing their core research programs, attackingimportant classic problems of combinatorics, including (1) The theoryof crossing numbers of graphs with connections to discrete geometry; and(2) Searching for new insights on antichains and intersecting familiesof subsets, and for connections between chain decompositions of the subsetsof a set and symmetric Venn diagrams. Research by the investigators with abroader impact includes their contributions to (3) Computationalbiology: phylogenetic tree reconstruction, clustering biomolecular sequences,(4) Theoretical computer science: graph drawing, security of statisticaldatabases; and (5) Communications: New models of real-number vertex labellings with distance conditions to find optimal or near-optimal channel assignments for large networks of transmitters.Research in combinatorics is carried out on problems that lie at the very core of our understanding of how discrete structures work and how to use them optimally. The investigators also apply combinatorial theory to interdisciplinary areas (including computational biology, theoretical computer science, and communications), where progress often requires using or inventing deeper combinatorial results which exhibit structures that are key to applications, or which provide bounds on what can be achieved by any algorithm. Progress in these areas contributes to important real-life problems, including the development of more efficient research tools in bioinformatics; the design of better visualisation tools for computer screens; increased computer and database security; and more efficient cell phone communication. The investigators are adding to their impressive record of recruiting and training graduate students and involving them in their research. They are expanding their efforts to attract more minority and women students into Ph.D. programs in mathematics. They are continuing their extensive activities in managing research journals, organizing conferences, guiding the Canada-USA Mathcamp for high school students, and contributing to the international Mathematical Competition in Modeling for undergraduates.