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## UT Austin Geometry Seminar 2017/2018The Geometry Seminar meets every Thursday at 3:30 pm in RLM 9.166.
On the one hand, there is a 3d Calabi Yau category with stability conditions associated to a quiver without loops or 2-cycles with generic potential, and one can study its Donaldson-Thomas invariants. On the other hand, such a quiver also defines a cluster Poisson variety, which often has geometric realizations. In certain cases, the Donaldson-Thomas invariants of the former can be captured by a birational automorphism of the latter. In this talk, I will describe the cluster Poisson structure on the moduli space of configurations of points in a projective space, and state my result on the geometric realization of the corresponding cluster Donaldson-Thomas transformation.
It is an interesting problem to attach moduli meanings to locally symmetric domains via period maps. Besides the classical cases like polarized abelian varieties and lattice polarized K3 surfaces, such examples include quartic curves (by Kondo), cubic surfaces and cubic threefolds (by Allcock, Carlson and Toledo), and some Calabi-Yau varieties (by Borcea, Voisin, and van Geemen). In the talk we will discuss two examples along these lines: (1) certain surfaces of general type with p_g=2 and K^2=1; (2) pairs consisting of a cubic threefold and a hyperplane section. This is joint work with R. Laza and G. Pearlstein.
In symplectic geometry, most Floer homologies theories come with products. Sometimes these products are obviously commutative, and sometimes this is less obvious, but there are still geometric reasons for it. On the other hand, when we do have a commutative product, we can ask for its noncommutative deformations, and the geometric basis for those. I will describe several instances of these phenomena, with an emphasis torus fibrations and the cases of interest in homological mirror symmetry.
I will present a new tool for the calculation of motivic invariants appearing in Donaldson-Thomas theory, such as the motivic Milnor fiber, starting from a theory of volumes of semi-algebraic sets introduced a decade ago by Hrushovski and Kazhdan. They key new result for applications is a tropical Fubini theorem - the invariants of interest can be computed by integrating the volumes of fibers of the tropicalization map with respect to Euler characteristic on the tropicalization. Based on joint work with J. Nicaise.
In 2007, Baird-Danielo and independently John Lott discovered non-gradient Ricci Soliton structures on the 3-dimensional geometries Nil and Sol. A soliton is a fixed point of the Ricci flow. These were analyzed by Guenther-Isenberg-Knopf, who studied the linearization of Ricci flow about these fixed points, and Williams-Wu, who studied its dynamic stability. The examples of Baird-Danielo were constructed by exploiting semi-conformal mappings from these geometries to the plane. In essence, the existence of such a map allows for a representation of the Ricci tensor in terms of geometric quantities such as the mean curvature of the fibres of the map and the integrability of the horizontal distribution. It is my aim to take this work further by studying the behaviour of solitons under biconformal deformations of the metric.
The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods. This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.
Since the introduction of generalized Kahler geometry in 1984 by Gates, Hull, and Rocek in the context of two-dimensional supersymmetric sigma models, we have lacked an understanding of the degrees of freedom inherent in the geometry. In particular, the description of a usual Kahler structure in terms of a complex manifold together with a local Kahler potential function is not available for generalized Kahler structures, despite many positive indications in the literature over the last decade. I will explain recent work showing that a generalized Kahler structure may be viewed in terms of a Morita equivalence between holomorphic Poisson manifolds; this allows us to solve the problem of existence of a generalized Kahler potential.
I will describe some results about the spaces of rational curves on a general hypersurface. In particular, I discuss results with David Yang where we compute the dimensions of the space of degree e curves on a very general hypersurface of degree d in P^n for d < n-1 and d > 3n/2, and show that they have the expected dimension.
A Bi-Lagrangian structure in a smooth manifold consists of a symplectic form and a pair of transverse Lagrangian foliations. Equivalently, it can be defined as a para-Kähler structure, that is the para-complex equivalent of a Kähler structure. Bi-Lagrangian manifolds have interesting features that I will discuss in both the real and complex settings. I will proceed to show that the complexification of a real-analytic Kähler manifold has a natural complex bi-Lagrangianstructure, and showcase its properties. I will then specialize this discussion to moduli spaces of geometric structures on surfaces, which have a rich symplectic geometry. I will show that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory, while revealing other new geometric features; as well as deriving several well-known results of Teichmüller theory by pure differential geometric machinery. Time permits, I will also mention the construction of an almost hyper-Hermitian structure in the complexification of any real-analytic Kähler manifold, and compare it to the Feix-Kaledin hyper-Kähler structure. This is joint work with Andy Sanders.
Donaldson-Thomas theory is branch of (virtual) enumerative geometry, with a close relationship to the older Gromov-Witten invariants. I will discuss how these invariants behave under birational transformations and also a formula (known as the crepant resolution conjecture) in the setting of the McKay correspondence. The latter is work in progress with Beentjes and Rennemo.
Many sporadic groups G admit distinguished "moonshine" representations on conformal field theories --- the original and most famous example being the "defining" representation of the Monster group. Any such action produces an accompanying gauge anomaly living in H^4(G;Z). I will discuss the values of these anomalies, and suggest that often the anomaly of the distinguished "moonshine" action generates the corresponding cohomology group. In particular, I will report on my calculations, joint with David Truemann, of the fourth cohomology of the largest Conway group and of the O'Nan group, and on my calculation of the order of the Monster's moonshine anomaly. The latter calculation uses a construction I call "finite group T-duality" which may be of independent interest.
We explain how a doubled version of the Beilinson-Bernstein localization functor can be understood using the geometry of the wonderful compactification of a group. Specifically, bimodules for the Lie algebra give rise to monodromic D-modules on the horocycle space, and to filtered D-modules on the group that respect a certain matrix coefficients filtration. These two categories of D-modules are related via an associated graded construction in a way compatible with localization, Verdier specialization, and additional structures. This is joint work with David Ben-Zvi and David Nadler.
A symmetric quiver with g nodes is described by a symmetric adjacency matrix of size g. The same data defines a "framing" of a certain genus-g Legendrian surface in the five-sphere, and the invariants of the quiver conjecturally relate to the open Gromov-Witten (GW) invariants of a non-exact Lagrangian filling of the surface. (Physically, both data count the same BPS states but from different perspectives.) Further, cluster theory can be exploited to conjecturally obtain all open GW invariants of Lagrangian fillings of a wider class of Legendrian surfaces described by cubic planar graphs. In this talk, I will describe these observations, which build on prior work of others and are explored in joint works with David Treumann and Linhui Shen.
Chern conjectured long ago that the Euler number of a closed affine manifold vanishes. This was proved in special cases by Benzecri, Milnor, Kostant, and Sullivan. Recently Klingler proved another special case, the conjecture for special affine manifolds. Still, the general Chern conjecture remains open. I will give an exposition of these developments.
A degree one del Pezzo surface is the blowup of P^2 at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we can construct a stable pair compactification of the moduli space of anti-canonically polarized degree one del Pezzo surfaces. The KSBA theory of stable pairs (X,D) is the natural extension to dimension 2 of the Deligne-Mumford-Knudsen theory of stable curves. I will discuss the construction of the space of interest as a limit of a space of weighted stable elliptic surface pairs and explain how it relates to some previous compactifications of the space of degree one del Pezzo surfaces. This is joint work with Kenny Ascher. |