Student Enumerative Geometry Seminar

Fall 2017

In the Fall 2017 semester, Noah Arbesfeld, Henry Liu, Petr Pushkar and Shuai Wang organized a seminar on Enumerative Geometry. The main reference is Professor Andrei Okounkov's Lectures on K-theoretic computations in enumerative geometry.

If you're interested in counting, please check out the seminar webpage and join us!
The seminar meets at 2:30pm, Wednesdays in Mathematics Building Room 622.
Every talk lasts around one and half hours and will be followed by dinner.

In his lectures, Vanya explained how to construct a geometric R-matrix using stable envelopes. I'll give another construction in the special case of instanton moduli space that uses the action of a certain Virasoro algebra on the cohomology of the Hilbert scheme of points on C^2 and explain why this construction coincides with the one we've already seen

Date Speaker Title
September 13 Henry Liu  13/2 ways of counting curves.
September 20 Dimitry Korb  Donaldson-Thomas theory.
September 27 Shuai Wang
Curve counting via stable pairs in the derived category by R. Pandharipande, R. P. Thomas. For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $C\subset X$ is an embedded curve and $D\subset C$ is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of $X$. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of $X$. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category.
Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described.
The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.
October 4 Shuai Wang  Wall crossing, curve counting and Jacobi forms
October 11 (Tuesday 7pm) Iakov Kononov.  Equivariant vertex and topological strings
October 18 Ivan Danilenko  Equivariant quantum cohomology of cotangent bundle of G/P I
October 25 Ivan Danilenko  Equivariant quantum cohomology of cotangent bundle of G/P II
November 1 Noah Arbesfeld

In his lectures, Vanya explained how to construct a geometric R-matrix using stable envelopes. I'll give another construction in the special case of instanton moduli space that uses the action of a certain Virasoro algebra on the cohomology of the Hilbert scheme of points on C^2 and explain why this construction coincides with the one we've already seen

November 8 Noah Arbesfeld  The R-matrix from the Virasoro algebras II
November 14 Postponed (no talk)  Postponed (no talk)
November 21 Thanksgiving (no talk)  Thanksgiving (no talk)
November 29 Dimitry Korb  Quasimaps I
December 6 Petr Pushkar  Quasimaps II
December 13 Renata Picciotto  Landau-Ginzburg/Calabi-Yau correspondence for Fermat quintic