PROGRAM SCHEDULE - TEXAS GEOMETRY AND TOPOLOGY CONFERNCE
SPRING 2024


The following is a tentative program schedule for the Texas Geometry and Topology Conference. Please check back here for the most up-to-date information.


Friday Schedule - Math Building Rm 11

Friday Hours Speaker Title Abstract
3:00 pm - 4:00 pm Banquet - Math Building Main Floor and Room 106 Tutoring Center
4:00 pm - 4:50 pm Catherine Searle When is an Alexandrov space smoothable? Alexandrov spaces of finite dimension $n\geq 1$ are locally compact, locally complete, length spaces with a lower curvature bound in the triangle comparison sense. They are a natural generalization of Riemannian manifolds with a lower sectional curvature bound. In this talk, I will discuss the problem of when an Alexandrov space is smoothable. We will review the history of this question and discuss a new result that partially answers it. This is joint work with Pedro Solórzano and Fred Wilhelm.

Saturday Schedule - Math Building Rm 11

Saturday Hours Speaker Title Abstract
9:00 am - 9:30 am Registration and Breakfast/Coffee
9.30 am - 10.20 am Brian Lawrence Rational points and computational complex analysis A classic problem in number theory asks to find all rational solutions to a polynomial equation f(x, y) = 0 -- geometrically, all rational points on an algebraic curve. I will briefly survey some attacks on this problem: Chabauty's method, quadratic Chabauty, and "Chabauty on covers". Motivated by this problem, I will suggest one or two problems in computational complex analysis. Computational number theory has traditionally been dominated by the algebraic approach; I will argue that analytic computations will be very useful for number theory.
10:30 am - 11:20 am Rui Loja Fernandes Invariant Kähler metrics for toric fibrations In this talk, I will discuss (extremal) invariant Kähler metrics for Lagrangian fibrations admitting only elliptic singularities. It turns out that such fibrations are precisely the Hamiltonian spaces of toric actions of symplectic torus bundles, which are a special type of symplectic groupoid. This allows us to extend the Abreu-Guillemin-Donaldson theory of invariant (extremal) Kähler metrics from toric manifolds to a much larger class of symplectic manifolds and to provide many more examples of invariant (extremal) Kähler metrics. In particular, this includes the case of complex ruled surfaces over elliptic curves, as studied by Apostolov et al via completely different methods. This presentation is based on ongoing joint work with Miguel Abreu (IST-Lisbon) and Maarten Mol (Max Planck-Bonn).
11:30 am - 12:20 pm Yi Lai Riemannian and Kahler flying wing steady Ricci solitons Steady Ricci solitons are fundamental objects in the study of Ricci flow, as they are self-similar solutions and often arise as singularity models. Classical examples of steady solitons are the most symmetric ones, such as the 2D cigar soliton, the O(n)-invariant Bryant solitons, and Cao’s U(n)-invariant Kahler steady solitons. Recently we constructed a family of flying wing steady solitons in any real dimension n≥3,which confirmed a conjecture by Hamilton in n=3. In dimension 3, we showed all steady gradient solitons are O(2)-symmetric. In the Kahler case, we also construct a family of Kahler flying wing steady gradient solitons with positive curvature for any complex dimension n≥2, which answers a conjecture by H.-D. Cao in the negative. This is partly collaborated with Pak-Yeung Chan and Ronan Conlon.
12:30 pm - 2:00 pm Lunch Break as group & Informal Discussions
2:00 pm - 2:50 pm Franz Pedit Higgs bundles, affine spheres, Monge-Ampere equations, and SYZ mirror symmetry In this talk I shall explain how to construct special Lagrangian 3-torus fibrations, singular over a trivalent discriminant locus, of Calabi-Yau 3-folds using parabolic non-Abelian Hodge theory. Along the way, we shall encounter Monge-Ampere equations and very classical differential geometric objects, the affine spheres introduced by the Blaschke school in the 1930s. The Strominger-Yau-Zaslow approach to mirror symmetry suggests existence of such fibrations in Calabi-Yau manifolds based on string/brane dualities. The results are joint work with Sebastian Heller (BIMSA, Beijing) and Charles Ouyang (Washington University, St. Louis).
3:00 pm - 3:50 pm Thomas A. Ivey mKdV Flows for Legendrian Curves in the 3-sphere As the unit sphere in C^2, the 3-sphere is acted on by U(2), preserving the standard contact structure generated by planes orthogonal to the Hopf fibers. For a given regular Legendrian curve in the sphere it is straightforward to construct a U(2)-invariant moving frame and curvature k. In terms of these, we define a natural symplectic structure on a suitable quotient of the space of periodic Legendrian curves, and identify an infinite hierarchy of commuting Hamiltonian flows which induce the modified Korteweg-DeVries hierarchy for the evolution of k. For example, the mKdV equation itself is induced by the Hamiltonian flow for Legendrian arclength. Moreover, initial curves which are the Legendrian analogue of elastica move by rigid motions under this flow, have periodic curvature functions, and realize a variety of Legendrian knot types. This is joint work with Annalisa Calini and Emilio Musso.
4:00 pm - 4:50 pm Joel Langer Quadratic Differentials and Plane Curves The notion of quadratic differential Q = q(z)dz2 was developed in the twentieth century as a tool in Riemann surface theory. The premise of this talk is that the concept also provides an ideal language for revisiting some special topics in the classical theory of real algebraic plane curves. Given such a curve f(x, y) = 0, I will consider two quadratic differentials. The clinant quadratic differential Qf extends the real arc length element ds2 = dx2 + dy2 along f to the corresponding complex projective curve. (More precisely, Qf is defined on the associated compact Riemann surface.) Arc length parameterization of f yields parameterizations of nearby horizontal geodesics of Qf—via analytic continuation—up to nearest zeros of Qf , which correspond to real foci of f. The focal quadratic differential Q{f} = dz2 δ(z) , on the other hand, is defined on the extended complex plane via a discriminant off. All curves in a confocal family {f} share the same Q{f}. The focal quadratic differential Q{f} has poles at the common real foci of {f}. The relationship between Qf and Q{f} is both unexpected and useful. In this talk I will use Qf and (two constructions for) Q{f} to reconsider some topics of nineteenth century origin. After a brief look at the Kiepert sextic, Story’s hyperbolic conics, and Siebeck’s bicircular quartics, I will consider a topic of more recent vintage—the Edwards theory of elliptic curves—which I will interpret geometrically via Cassinians and their confocal families.
5:30 pm - 8:00 pm Dinner & Discussions - Triple J Chophouse (Limited Reservations) - Rides must be requested in advance

Sunday Schedule - Math Building Rm 11

Sunday Hours Speaker Title Abstract
10.00 am - 11.00 am Igor Zelenko On projective and affine equivalence of sub-Riemannian metrics: generalized Eisenhart and Levi-Civita theorems. Sub-Riemannian metrics on a manifold are defined by a distribution (a subbundle of the tangent bundle) together with a Euclidean structure on each fiber. The Riemannian metrics correspond to the case when the distribution is the whole tangent bundle. Two sub-Riemannian metrics are called projectively equivalent if they have the same geodesics up to a reparameterization and affinely equivalent if they have the same geodesics up to affine reparameterization. In the Riemannian case, both equivalence problems are classical: local classifications of projectively and affinely equivalent Riemannian metrics were established by Levi-Civita in 1898 and Eisenhart in 1923, respectively. In particular, a Riemannian metric admitting a nontrivial (i.e. non-constant proportional) affinely equivalent metric must be a product of two Riemannian metrics i.e. separation of variables (the de Rham decomposition) occurs, while for the analogous property in the projective equivalence case, a more involved (``twisted) product structure is necessary. The latter is also related to the existence of commuting nontrivial integrals quadratic with respect to velocities for the corresponding geodesic flow. We will describe the recent progress toward the generalization of these classical results to sub-Riemannian metrics. The talk is based on the joint works with Frederic Jean , Sofya Maslovskaya, Zaifeng Lin, and Christopher Sinkule.
11:00 pm - 12:00 pm Conclusion



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Program Schedule - Location


The conference will be held in the Mathematics and Statistcis Building Room 11 at Texas Tech University.