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Enumerative Invariants from Derived Categories. Here we review the definition of a quiver flag variety following Kalashnikov and recover Givental's small J function for CP^1 following the conventions of Cox-Katz by viewing CP^1 as a quiver flag variety.
Abstract: The study of enumerative invariants dates back at least as far as Euclid’s work circa 300 BC, who observed that through two distinct points in the plane there is a unique line. In 1849, Cayley-Salmon found that there are 27 lines on a nonsingular cubic surface. In 1879, Schubert found that there are 2875 lines on a generic non-singular quintic threefold; Katz correctly counted 609250 conics in a generic nonsingular quintic threefold in 1986. In 1991, physicists Candelas-de la Ossa-Green-Parkes gave a generating function for genus 0 Gromov-Witten invariants of a generic non-singular quintic threefold by studying the mirror space. This observation represented a change in our approach to enumerative problems by counting rational degree d curves inside of the quintic threefold “all at once;” other landmark achievements in modern enumerative geometry include Kontsevich-Manin’s recursive formula for the number of rational plane curves. From the perspective of homological mirror symmetry, enumerative invariants come from the Hochschild cohomology of the Fukaya category. I’m interested in a different question, which asks what enumerative data can be gleaned from the bounded derived category of coherent sheaves. I’ll share results on giving presentations of derived categories, and if time allows, will describe Kalashnikov’s method to recover Givental’s small J-function and the genus 0 Gromov-Witten potential for CP^1 by viewing it as a toric quiver variety associated to the Kronecker quiver; i.e., from a presentation of the bounded derived category of coherent sheaves.
Exposition on Gromov-Witten invariants and enumerating rational curves for \PP^1 and \PP^2. This includes some background on review of differential topology shared with undergraduate students at the Claremont Colleges. Abstract: Here we review background in differential topology related to the calculation of an euler characteristic, and background on localization in equivariant cohomology. We then outline Gromov-Witten invariants in algebraic geometry and give examples of the genus 0 Gromov-Witten potential for $\PP^1, \PP^2$, and a genus g>0 Riemann surface. Kontsevich-Manin's recursive formula for N_d, the number of degree d rational curves through 3d−1 points in general position on $\PP^2$ is recovered. Submitted.
"Localization in Equivariant Cohomology." Claremont Colleges ANTC Seminar. Estella building on Pomona campus. 9/10/2024.
Euler Characteristic of the Tangent Bundle. General Audience talk for Claremont McKenna College Summer Research Program (SRP). Summer 2024.
"Review of Differential Geometry." Claremont Consortium Analysis Seminar. Notes from Chapter 1 of Mirror Symmetry by Hori-Katz-Klemm-Pandharipande-Thomas-Vafa-Vakil-Zaslow.
"Homological Mirror Symmetry, Enumerative Geometry, and Curve Counting: 27 Lines on a Cubic Surface." Meet the New Faculty in Claremont McKenna College Department of Mathematical Sciences Seminar. Monday, 2/12/2024. Same talk given in ANTC on March 6
Macaulay2 output for 2875 lines on a quintic threefold
"Presentations of Derived Categories." UC Irvine Seminar 2/1/2024.
"Resolutions of the Diagonal and Fourier-Mukai Transforms." Claremont Consortium Colloquium. 9/27/2023
"A Positive Example for Modified King's Conjecture." Claremont Consortium Topology Seminar. 9/19/2023.
"Planar conics." Graduate student seminar at Kansas State University. 3/28/2023.
Beamer presentation from PhD dissertation defense. Kansas State University. March 9, 2023.
"Log Differential Forms." Junior Mirror Symmetry Seminar at Kansas State University. November 2022.
``Hilbert Polynomials." Graduate Student Seminar (GSS) at Kansas State University. 9/1/2022.
``What is: an Infinitesimal?" Differential Geometry Seminar at UIowa. March 2019. A version of these notes was published in the AMS Graduate Student Blog on August 28, 2019 at https://blogs.ams.org/mathgradblog/2019/08/28/32749/.
Title: Calculus with Schemes: Using the Dual Numbers to Understand Infinitesimals
Abstract: The dual numbers over the real line (R) offer a framework to understand algebraic constructions in undergraduate calculus by considering infinitesimal lengths to be nilpotent elements in the coordinate ring of the origin, considered as a subscheme of R. The main result here is that the dual numbers are isomorphic to the first infinitesimal neighborhood of the origin. Though this is not a new result, this construction invites questions of how to interpret additional scheme-theoretic infinitesimal neighborhoods of the origin, as well as infinitesimal deformations of varieties and schemes more generally. I will describe features of the dual numbers which suggest that there is an imaginary or non-real quality to infinitesimals over the real line. If time permits, I will also mention a connection to Lie algebras and tangent bundles. This talk references ResearchGate pre-print DOI: 10.13140/RG.2.2.17102.10563.
``DG Categories and Homotopy Category of a DG Category" 7/9/18. Categories of Modules Mini-Course Lecture at Kansas State University.
``Survey of Results in K-Theory", May 2018. Final presentation for Homological Algebra course at Kansas State University.
A Sketch of Mirror Symmetry as Batyrev Duality for Toric Varieties. Notes for a previous blog post. 2/13/2021
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