Equivariant Cohomology: Positivity, Differential Equations

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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.