Equivariant Cohomology: Positivity, Differential Equations

单变量函数的奇异点是函数导数消失的点。奇点和函数的全局方面之间的关系，比如极小值和极大值，只是冰山一角。Rimanyi研究高维空间之间函数的奇异性，以及它们与空间整体方面的关系。他研究的一个关键概念是等变上同调类，由一个几何上相关的变量表示。这类的自然无穷级数常常表现出非凡的性质。PI提出了一种新的方法来证明一组有趣的等变类的正结果，即所谓的颤振多项式。他还计划研究另一类的正性：Morin奇点的Thom多项式，与Green-Griffiths猜想有关的全纯映射到射影变体。他的新方法是基于等变类的迭代残余描述的灵活性。PI建议研究的另一个领域是等变上同调、微分方程和物理学之间非常有前途的相互作用。也就是说，等变类通常满足物理启发的微分方程。基于与Varchenko的早期工作，Rimanyi提出寻找量子Knizhnik-Zamolodchikov微分方程的拓扑方面，并研究为什么等变上同调类满足它们。作为一种应用，具有拓扑成分的量子Selberg型积分被寄予厚望。在物理科学中，一个非常具有挑战性的问题是如何理解由参数的平滑变化引起的突然变化。Rimanyi正在研究这些物理现象的数学方面。他的建议的一个重要方面是其教育成分。Rimanyi建议继续为北卡罗来纳大学的研究生举办几何学习研讨会，为教师和感兴趣的研究生提供一个重要的交流渠道。通过继续让一些聪明的教堂山高中学生参与数学研究，提议者正在不同机构之间创造新的机会和新的沟通渠道。他还提议，通过在学生开始写荣誉论文时提供建议，扩大本科生对系里数学生活的参与。他将继续为本科生举办每周一次的问题解决/竞赛准备研讨会——在那里，学生们不仅可以获得数学奥林匹克类型的训练，还可以亲身体验研究。他继续为硕士生和博士生提供建议。Rimanyi建议建立一个在线的Thom多项式门户网站，收集和整理分散的Thom多项式结果，以供科学家使用。

The singularities of a function in one variable are the points where the derivative of the function vanishes. The relationship between singularities and global aspects of the function, such as minima and maxima, is only the tip of the iceberg. Rimanyi studies singularities of functions between higher dimensional spaces, and their relations with global aspects of the spaces. A key notion of his research is the equivariant cohomology class represented by a geometrically relevant variety. Natural infinite series of these classes often display remarkable properties. The PI proposes a new approach to prove a positivity result for an interesting set of equivariant classes, the so-called quiver polynomials. He also plans to study the positivity of another class: the Thom polynomials of Morin singularities, in connection with the Green-Griffiths conjecture on the holomorphic maps to projective varieties. His new approach is based on the flexibility of Iterated Residue descriptions of equivariant classes. Another area the PI proposes to study is the very promising interplay between equivariant cohomology, differential equations, and physics. Namely, equivariant classes often satisfy physically inspired differential equations. Based on earlier works with Varchenko, Rimanyi proposes to find the topological aspects of quantum Knizhnik-Zamolodchikov differential equations, and study why equivariant cohomology classes satisfy them. As an application quantum Selberg type integrals with topological ingredients are expected.In the physical sciences, one very challenging problem is the understanding of sudden changes caused by smooth alterations of parameters. Rimanyi is working on the mathematical aspects of these physical phenomena. An essential aspect of his proposal is its educational component. Rimanyi proposes to continue a Geometry Learning Seminar for the graduate students at UNC, providing a critical source of communication between faculty and interested graduate students. By continuing to involve some bright Chapel Hill high school students in mathematics research, the proposer is creating new opportunities and new channels of communication between different institutions. He also proposes to broaden participation of undergraduates in the mathematical life of the department by advising students as they begin their honors theses. He will continue running a weekly problem solving/competition preparation seminar for undergraduates---where the students get not only math olympiad type training, but experience research firsthand. He continues to advice master and PhD student. Rimanyi proposes to set up an online Thom polynomial portal, where scattered results of Thom polynomials would be collected and organized for the benefit of scientists.