Combinatorics with Applications

研究人员正在继续他们的核心研究计划，攻击组合学的重要经典问题，包括（1）与离散几何连接的图的交叉数的理论;（2）寻找关于子集的反链和交叉族的新见解，以及一组子集的链分解与对称维恩图之间的联系。 具有更广泛影响的研究人员的研究包括他们对（3）计算生物学：系统发育树重建，生物分子序列聚类，（4）理论计算机科学：图形绘制，药物数据库的安全性;以及（5）通信：实数顶点标号与距离条件的新模型，以找到最佳或接近，大型发射机网络的最佳信道分配。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层次的组合结果，这些结果显示出对应用程序至关重要的结构，或者提供任何算法可以实现的范围。 这些领域的进展有助于解决现实生活中的重要问题，包括开发更有效的生物信息学研究工具;设计更好的计算机屏幕可视化工具;提高计算机和数据库安全性;以及更有效的手机通信。 调查人员正在增加他们招募和培训研究生并让他们参与研究的令人印象深刻的记录。 他们正在扩大努力，以吸引更多的少数民族和女性学生进入博士学位。数学方面的课程。 他们继续在管理研究期刊，组织会议，指导高中学生的加拿大-美国数学夏令营，并为本科生建模国际数学竞赛做出贡献。

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.