Conference on Categorical Methods in Topology and Quantum Geometry; Fall 2009, Dallas, Texas

题为“拓扑学和量子几何中的绝对方法”的科学研讨会将于2009年10月15日至18日在德克萨斯州达拉斯举行的奇卡诺和美洲原住民科学促进会(SACNAS)全国会议上举行。低维拓扑学和量子几何这两个先验的完全不同的领域最近通过范畴方法联系在一起。最近，这些主题中的每一个都独立地取得了令人兴奋的进展，包括三维空间中的温斯坦猜想和所有环面三重空间的Donaldson-Thomas/Gromov-Witten二元性。重要的悬而未决的问题推动了继续研究；这些问题包括一般的温斯坦猜想，以及格罗莫夫-维腾理论、唐纳森-托马斯理论和Pandharipande-Thomas理论之间的等价性三部曲。作为我们研讨会的中心，这些领域最近通过它们对派生范畴和范畴的使用而相互关联。这包括霍万诺夫的范畴及其许多推广，Behrend和Joyce-Song对Donaldson-Thomas理论的范畴，以及Pandharipande-Thomas理论的派生范畴。此外，CAUTIS和Kamnitzer的程序表明，拓扑学中的范畴结果在某些情况下可以被重塑为代数几何中派生范畴的同构。本次研讨会的发言者将宣布取得的成就，并讨论新的研究方向。具体地说，Renzo Cavalieri将讨论GW和DT理论中的派生范畴，Emily le Davie将讨论低维拓扑学的应用，Juan Ortiz-Navaro将讨论范畴化、Khovanov同调及其推广。这次研讨会将使学生和其他科学家能够并鼓励他们在与量子几何和拓扑学相互作用的领域进行研究，为科学家提供互动的机会，促进合作和新的研究，并向广泛和异常不同的受众传播知识。尽管组织一次关于拓扑学和量子几何的科学研讨会的原因很多，但对于代表不足的少数群体来说，还有一个迫切的需要这样做。目前，在数学科学领域，少数族裔的代表性不足是不可能的；这种代表性不足在拓扑学和量子几何学中显然都很严重，当然，在这些学科的交叉领域也是如此。在这些主题中有一些非常重要的问题需要解决，有必要吸引广泛和不同的受众来解决这些问题。格罗莫夫-维腾理论和相关领域在解决数学和物理的几个分支中的突出问题方面取得了极大的成功，其中一些问题已有100多年的历史。低维拓扑学不仅在几何学和拓扑学中具有基本的重要性，而且在应用数学的几个领域也是如此，正如本次研讨会所强调的那样。据预测，在不远的将来，代表人数不足的少数族裔将成为美国公民的多数；因此，增加这些群体成员的参与符合拓扑学和量子几何学的长期利益。此外，鉴于这些学科对一般数学和科学的重要性，努力解决在这些领域进行研究的少数族裔代表性不足的问题符合我们的国家利益。

A scientific symposium titled "Categorical Methods in Topology and Quantum Geometry" will take place at the National Meeting of the Society for Advancement of Chicano and Native Americans in Science(SACNAS) in Dallas, Texas, October 15-18, 2009. The two a priori disparate fields of low dimensional topology and quantum geometry have recently been related through the involvement of categorical methods. Exciting progress has been recently made in each of these subjects independently, including the Weinstein conjecture in dimension three and Donaldson-Thomas/Gromov-Witten duality for all toric threefolds. Important open questions motivate continued study; these include the general Weinstein conjecture, and the trilogy of equivalences between Gromov-Witten theory, Donaldson-Thomas theory and Pandharipande-Thomas theory. Central to our symposium, these fields have recently been related through their use of derived categories and categorification. This includes the categorification of Khovanov and its many generalizations, the categorification of Donaldson-Thomas theory by Behrend and Joyce-Song and the derived categories of Pandharipande- Thomas theory. In addition, the program of Cautis and Kamnitzer shows categorification results in topology may be recast in some instances as isomorphisms of derived categories in algebraic geometry. The speakers at this symposium will announce accomplishments and also discuss new directions of research. Specifically, Renzo Cavalieri will discuss derived categories in GW and DT theories, Emille Davie will discuss applications to low dimensional topology, and Juan Ortiz- Navaro will discuss categorification, Khovanov homology and its generalizations.This symposium will enable and encourage students and other scientists to pursue research in areas related to the interaction of quantum geometry and topology, provide the opportunity for scientists to interact and foster collaboration and new research, and disseminate knowledge to a wide and extraordinarily diverse audience. While the reasons for organizing a scientific symposium on topology and quantum geometry are many, there is additionally an acute need to do so for an audience of underrepresented minorities. There is at this time significant underrepresentation of minorities in the mathematical sciences; this underrepresentation is evidently severe in both topology and quantum geometry and certainly the intersection of these subjects. There are very important questions that need to be addressed in these subjects, and it is necessary to attract a broad and diverse audience to work on these problems. Gromov-Witten theory and related fields have been extremely successful in solving outstanding problems, some over 100 years old, in several branches of mathematics and physics. Low dimensional topology is not only of basic importance in geometry and topology, but in several areas of applied mathematics as well, as highlighted in this symposium. It is predicted that underrepresented minorities will become the majority of United States Citizens in the not so distant future; as such, it is in the long term interest of topology and quantum geometry to have increased participation from members of these groups. Moreover, given the importance of these subjects to mathematics and science in general, it is in our national interest to work against the underrepresentation of minorities conducting research in these fields.