##### August Comprehensive Exams

^{th}1pm - 5pm

Part II - August 20

^{th}1pm - 5pm

Math Tower P-131

August 4, 2021

We present a twistor correspondence for half-flat almost-Grassmannian structures on real manifolds. An almost-Grassmannian structure is (essentially) a factorization of the tangent bundle, which determines two preferred families of tangent subspaces, and this structure is said to be half-flat if one of these families is integrable. We provide global results when the underlying manifold is a Grassmannian of 2-planes, and show there exist nontrivial deformations of the standard almost-Grassmannian structure. Whereas twistor constructions typically involve moduli of closed curves in a complex manifold, we utilize and expand upon the more flexible approach pioneered by LeBrun and Mason using moduli of curves-with-boundary.

June 4, 2021

We describe the boundary of linear subvarieties in the moduli space of multi-scale differentials. Linear subvarieties are algebraic subvarieties of strata of (possibly) meromorphic differentials that in local period coordinates are given by linear equations. The main examples of such are affine invariant submanifolds, that is, closures of SL(2,R)-orbits. We prove that the boundary of any linear subvariety is again given by linear equations in generalized period coordinates of the boundary. Our main technical tool is an asymptotic analysis of periods near the boundary of the moduli space of multi-scale differentials which yields further techniques and results of independent interest.

May 20th, 2021

We propose a new approach to the question of prescribing Gaussian curvature on the 2-sphere with at least three conical singularities and angles less than $2\pi$, the main result being sufficient conditions for a positive function of class at least $C^2$ to be the Gaussian curvature of such a conformal conical metric on the round sphere. Our methods particularly differ from the variational approach in that they don’t rely on the Moser-Trudinger inequality. Along the way, we also prove a general precompactness theorem for compact Riemann surfaces with at least three conical singularities and angles less than $2\pi$.

May 18, 2021

Given a Lagrangian fibration, we provide a natural construction of a mirror Landau-Ginzburg model consisting of a rigid analytic space, a superpotential function, and a dual fibration based on Fukaya’s family Floer theory. The mirror in the B-side is constructed by the counts of holomorphic disks in the A-side together with the non-archimedean analysis and the homological algebra of the A infinity structures. It fits well with the SYZ dual fibration picture and explains the quantum/instanton corrections and the wall crossing phenomenon. Instead of a special Lagrangian fibration, we only need to assume a weaker semipositive Lagrangian fibration to carry out the non-archimedean SYZ mirror reconstruction

May 12, 2021

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a non-polynomial entire function, which we call a transcendental entire function. The Julia set of $f$ is defined to be the set of all points such that the iterates of $f$ do not form a normal family. In other words, a point is in the Julia set if and only if there exists points arbitrarily close by that have a different orbit under iteration by $f$.

Computer images of Julia sets show that they have a rich fractal structure. Through the work of Baker, McMullen, Stallard, Bishop, and many others, the Hausdorff dimension of Julia sets of transcendental entire functions must be between 1 and 2, and all values between 1 and 2 are attained. However, much less is known for other notions of dimension, such as the packing dimension. Bishop constructed an example where the Julia set has packing dimension and Hausdorff dimension equal to 1, and otherwise all other examples where the packing dimension has been computed, it has been equal to 2.

We will show how to construct transcendental entire functions whose Julia sets have packing dimension strictly between 1 and 2. In fact, we will show that the set of all values attained is dense in the interval (1,2), and we will show that the Hausdorff and packing dimension may be arranged to be arbitrarily close together.

May 10,2021

In his Annals of Mathematics paper (2009), Berndtsson proves an important result on the curvature of holomorphic infinite-rank vector bundles whose fibers are Hilbert spaces consisting of holomorphic $L^2$ functions with respect to a family of weight functions $e^{-{\varphi}^(t,\cdot)}$, varying in $t$, over a pseudoconvex domain. Using Hörmander's classical theorem on $L^2$-estimates for the $\bar{\partial}$-operator, he shows that such bundles are positively curved in the sense of Nakano provided that the function $\varphi$ is plurisubharmonic with respect to both variables. This result is at the center of a long-standing project of Berndtsson aiming at formulating a complex analogue of Brunn-Minkowski theory, which first started with his result on the log-plurisubharmonicity of Bergman kernels over pseudoconvex domains (2006). In addition to extending the Brunn-Minkowski theorem and its generalization -the Prékopa-Leindler theorem- to the complex setting, Berndtsson's result has deep applications in complex analysis and geometry. For example, his result leads to alternative proofs of existence and uniqueness theorems for Kähler-Einstein metrics, as well as optimal $L^2$-extension (or Ohsawa-Takegoshi type) theorems.

Berndtsson’s result can easily be extended to the geometric setting by taking the pseudoconvex domain to be a Stein manifold and the family of weights $e^{-{\varphi}^(t,\cdot)}$ to be a family of positively curved hermitian metrics for a line bundle over the manifold. Using a variant of Hörmander’s theorem due to Donnelly and Fefferman, we show that Berndtsson’s Nakano-positivity result holds under different (in fact, more general) curvature assumptions. This is of particular interest when the manifold admits a negative plurisubharmonic function, as these curvature assumptions then allow for some curvature negativity. We describe this setting as a "twisted" setting. In particular, we extend Berndtsson’s Nakano-positivity result to general trivial families of Stein manifolds. As immediate applications of this result, we prove $log$-plurisubharmonic variation theorems for Bergman kernels, as well as families of compactly supported measures and currents, over general trivial families of Stein manifolds. We then generalize these $log$-plurisubharmonic variation results to a certain class of non-trivial families of Stein manifolds. Finally, we also discuss Prékopa-Leindler type theorems showing, for instance, that the consequence of the Prékopa-Leindler theorem holds under weaker convexity assumptions.

May 5, 2021

This thesis is a demonstration of the applicability of Gromov-Witten Theory in the case of smooth complete intersections within a certain multidegree range. Gromov-Witten Theory provides a method for counting curves on smooth projective varieties. The invariants that are computed by Gromov-Witten Theory are frequently not enumerative and they cannot always be interpreted as actual curve counts. If the Kontsevich moduli space of genus-0 stable maps to the variety is irreducible of the expected dimension and contains an open dense subset parameterizing smooth embedded genus-0 curves in the variety, the genus-0 Gromov-Witten Invariants do provide us with actual curve counts.

This leads to the question: what are some classes of smooth varieties for which the Kontsevich space of genus-0 stable maps is irreducible of the expected dimension? The work of Harris, Roth and Starr shows the irreducibility of the Kontsevich space for smooth low degree hypersurfaces in projective space. We extend their work to study smooth complete intersections in projective space in instances where the dimension of the projective space is large compared to the multidegree of the complete intersection. Moreover, we use our results and methods of Starr and Tian to show the irreducibility of the space of Quasi-maps to every complete intersection within the same multidegree range.