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Aspects of enumerative and categorical algebraic geometry

 Author / Creator
 Chidambaram, Nitin Kumar

In this thesis, we study some aspects of algebraic geometry that have had a
significant influx of ideas from physics. The first part focuses on the Eynard
Orantin topological recursion and its variants as a theory of enumerative ge
ometry. We investigate the conjectural relationship between the topological
recursion and quantum curves in the case of elliptic curves. We show that the
perturbative wavefunction is not the solution to a quantum curve, while the
nonperturbative one is (up to certain order in ħ).
We define the formalism of Higher Airy Structures (HAS), which quantized
higher order Lagrangians in a symplectic vector space. By showing that the
BouchardEynard topological recursion is a HAS, we find a necessary and
sufficient condition on the spectral curve for producing symmetric correlators
ω g,n . We construct numerous other examples, some of which are (sometimes
conjecturally) related to FJRW theory or open (rspin) intersection theory.
In this thesis, we study some aspects of algebraic geometry that have had a significant influx of ideas from physics. The first part focuses on the EynardOrantin topological recursion and its variants as a theory of enumerative geometry. We investigate the conjectural relationship between the topological recursion and quantum curves in the case of elliptic curves. We show that the perturbative wavefunction is not the solution to a quantum curve, while the nonperturbative one is (up to certain order in $ \hbar $).We define the formalism of Higher Airy Structures (HAS), which quantized higher order Lagrangians in a symplectic vector space. By showing that the BouchardEynard topological recursion is a HAS, we find a necessary and sufficient condition on the spectral curve for producing symmetric correlators $ \omega_{g,n} $. We construct numerous other examples, some of which are (sometimes conjecturally) related to FJRW theory or open ($ r $spin) intersection theory.
In the second part of this thesis, we focus on the study of derived categories and its connections to birational geometry; in particular, we are interested in a conjecture of Bondal and Orlov about flops. Using the presentation of a flop as a variation of geometric invariant theory (VGIT) problem, Ballard, Diemer and Favero proposed a FourierMukai kernel and conjectured that it induces a derived equivalence. We verify this conjecture in the case of Grassmann flops first. Then, we tackle the case of singular VGIT problems using semifree commutative dgalgebra resolutions, and prove that we obtain derived equivalences of dgschemes, under some conditions.

 Subjects / Keywords

 Graduation date
 Fall 2020

 Type of Item
 Thesis

 Degree
 Doctor of Philosophy

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