Difference between revisions of "Geometry and Topology Seminar 20192020"
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Max Hallgren (Cornell)  Max Hallgren (Cornell)  
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Yi Lai (Berkeley)  Yi Lai (Berkeley)  
TBA  TBA  
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(Huang)  (Huang)  
Revision as of 12:14, 22 July 2020
The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm  2:10pm.
For more information, contact Shaosai Huang.
Contents
Fall 2020
date  speaker  title  host(s)  

TBA  Max Hallgren (Cornell)  TBA  (Huang)  
TBA  Yi Lai (Berkeley)  TBA  (Huang)
Spring 2020
Fall 2019
Spring AbstractsXiangdong XieThe quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played an important role in various rigidity questions in geometry and group theory. In these talks I shall give an introduction to this topic. In the first talk I will introduce Gromov hyperbolic spaces, define their ideal boundary, and discuss their basic properties. In the second and third talks I will define the visual metrics on the ideal boundary, explain the connection between quasiisometries of Gromov hyperbolic space and quasiconformal maps on their ideal boundary, and indicate how the quasiconformal structure on the ideal boundary can be used to deduce rigidity. KuangRu WuFollowing Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge– Ampere type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kahler geometry, related to the construction of flat maps for the Mabuchi metric. This is joint work with Tamas Darvas. Yuanqi Wang$G_{2}$instantons are 7dimensional analogues of flat connections in dimension 3. It is part of DonaldsonThomas’ program to generalize the fruitful gauge theory in dimensions 2,3,4 to dimensions 6,7,8. The moduli space of $G_{2}$instantons, with virtual dimension $0$, is expected to have interesting geometric structure and yield enumerative invariant for the underlying $7$dimensional manifold. In this talk, in some reasonable special cases and a fairly complete manner, we will describe the relation between the moduli space of $G_{2}$instantons and an algebraic geometry moduli on a CalabiYau 3fold. Karin MelnickD'Ambra proved in 1988 that the isometry group of a compact, simply connected, realanalytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture. Joerg SchuermannWe give an introduction to PoincareHopf theorems for singular spaces via characteristic cycles, based on stratified Morse theory for constructible functions. The corresponding local index of an isolated critical point (in a stratified sense) of a oneform depends on the constructible function, specializing for different choices to well known indices like the radial, GSV or Euler obstruction index. David MasseyGiven a complex analytic function on an open subset U of C^{n+1}, one may consider the complex of sheaves of vanishing cycles along f of the constant sheaf Z_{U}. This complex encodes on the cohomological level the reduced cohomology of the Milnor fibers of f at each of f^{1}(0). The question is: how does one calculate (ideally, by hand) any useful numbers about this vanishing cycle complex? One answer is to look at the Lê numbers of f. We will discuss the precise relationship between these objects/numbers. Antoine SongTBA Fall AbstractsRuobing ZhangThis talk centers on the degenerations of CalabiYau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing CalabiYau metrics may exhibit various wild geometric properties with highly nonalgebraic features. First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating CalabiYau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner. Emily StarkThe relationship between the largescale geometry of a group and its algebraic structure can be studied via three notions: a group's quasiisometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic nmanifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasiisometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse. Max ForesterI will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of rightangled Artin groups and certain rightangled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao. Yu LiWe develop a structure theory for noncollapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere. Archive of past Geometry seminars20182019 Geometry_and_Topology_Seminar_20182019
