Differential Geometry, Topology,
and special structures

Department of Mathematics
The Graduate Center of CUNY

Fridays 11:45noon - 1:45pm
Location: Room 5417 (NOTE: room change!)
Organizers: Luis Fernandez, Mehdi Lejmi, Scott Wilson

Scope: this seminar is devoted to geometric structures on manifolds, and topological properties of them. We are particularly interested in (almost) complex manifolds, symplectic topology and geometry, (almost) Kahler geometry, and spin^c geometry, as well the tools of algebraic topology and geometric analysis that have proven useful in studying such structures. The goal is expose faculty and students to present day research and foundational material in a relaxed format that includes traditional lectures with an actively engaged audience.
Here is a link to previous semesters: DGT Seminar Spring 2022, DGT Seminar Fall 2022
Here is a link to the formerly named seminar of similar scope: ACG seminar

Spring 2023: (all meetings in person)

1/27: no meeting (Semester Kickoff & Chalk Talk this day, 11am-3:30pm in the Math Lounge.)

2/3: Jiahao Hu (Stony Brook University)
Title: Differential K-theory and its topology

Abstract: I will define differential K-theory of a smooth manifold using complex vector bundles with connections and survey some of its properties. Then I will show (1) differential K-theory carries a natural topology induced from the smooth structure of the manifold; and (2) from this topology one can recover topological K-theory and some information of the manifold concerning differential forms. If time permits, I will also discuss how this topology helps axiomatically characterize the differential K-theory functor.

2/10: no talk scheduled this week (see ENYGMMa III)

2/17: Prof. Thomas Tradler (City Tech, CUNY)
Title: Constructing Poincare duality on the chain level

Abstract: We will present a construction of a Poincare duality structure on the (co-)chain level of a manifold. When building these structures, higher homotopies will naturally appear to adjust for algebraic properties that may not hold strictly on the chain level. A goal for such a construction is to define a "minimal Poincare duality structure" on the (co-)homology level, which will include the cohomological Poincare duality as well as higher homotopies that encode local information of the space. For example, using a result by Menichi, it can be shown that the minimal Poincare duality for the 2-sphere with Z_2 coefficients differs from the purely cohomological Poincare duality. This is joint work with Kate Poirier.

2/24: Prof. Kate Poirier (City Tech, CUNY)
Title: Applying chain-level Poincare duality to string topology

Abstract: String topology studies algebraic structures that arise by intersecting loops, where a "loop" can mean something topological or algebraic. For example (on the topological side) on the homology of the free loop space of a closed, oriented manifold, there is a binary operation called the "loop product" and a unary operation called the "BV operator". These two operations together give the homology of the free loop space the structure of a "BV algebra". Separately (on the algebraic side) in the presence of an algebraic version of Poincare duality, there is product and BV operator on the Hochschild cohomology of this algebra. These operations give the Hochschild cohomology of the algebra the structure of a BV algebra as well. Further, when the algebra is the cochain algebra of a closed, oriented, simply connected manifold there is an isomorphism between its Hochschild cohomology and the homology of the free loop space of the manifold. While Cohen and Jones showed that this isomorphism respects the product structure, subsequent work of Menichi suggested that, in the case of the 2-sphere with mod 2 coefficients, it did not respect the BV operator. In this talk, we describe these operations and show that with an appropriate updated algebraic version of Poincare duality for algebras-one involving higher homotopies-Hochschild cohomology can be given a BV operator that is, in fact, preserved by the isomorphism from the homology of the free loop space of the 2-sphere with mod 2 coefficients. This is joint work with Thomas Tradler.

3/3: Prof. Valentino Tosatti (Courant Institute)
Title: Collapsing Calabi-Yau manifolds

Abstract: I will give an introduction to compact Calabi-Yau manifolds and their Ricci-flat Kahler metrics. I will then study how these Ricci-flat metrics behave when the cohomology class of their Kahler forms degenerates. While the non-collapsing case is by now well-understood, the collapsing case presents a formidable challenge. I will report on recent progress in this direction, and discuss open questions.

3/10: Giuseppe Barbaro (Sapienza U., Rome)
Title: Non-flat pluriclosed Calabi-Yau with torsion metrics

Abstract: We describe the CYT condition on principal bundles over Hermitian manifolds with complex tori as fibers. Thus we show explicit examples of Calabi-Yau with torsion Hermitian structures and a uniqueness result for them. We then analyze the existence of pluriclosed CYT metrics on homogeneous spaces and their relation with a stability condition for the pluriclosed flow.

3/17:Nicholas McCleerey (U. Michigan)
Title: Geodesic Rays in the Donaldson-Uhlenbeck-Yau Theorem

Abstract: The theorem of Donaldson-Uhlenbeck-Yau says that a holomorphic vector bundle E over a compact Kahler manifold admits a Hermite-Einstein (HE) metric iff E is stable. Historically, this was the first example of a general program linking solvability of certain geometric PDE (the HE metric) with a stability condition, and is something of a spiritual predecessor to the Yau-Tian-Donaldson conjecture. Work on this subsequent conjecture has revealed an important link with a third object, namely, geodesic rays of ``weak" metrics. In joint work with Jonsson, Shivaprasad, we return to the Donaldson-Uhlenbeck-Yau theorem and, by focusing on the analogous geodesic rays in this setup, find a new proof of this celebrated result.

3/24: Prof. Sisi Shen (Columbia U.)
Title: The Kahler-Ricci flow and some applications

Abstract: We give an introduction to the Kahler-Ricci flow, which is a special instance of the Ricci flow starting from a Kahler metric. This flow preserves the complex structure and the Kahler condition and can be used to prove the existence of Kahler-Einstein metrics for manifolds with negative or vanishing first Chern class. In addition, we discuss how the flow displays the behavior of contracting exceptional divisors on a Kahler manifold as well as collapsing the fibers of a holomorphic fiber bundle.

3/31: Yueqiao Wu, (U. Michigan)
Title: A non-Archimedean characterization of local K-stability

Abstract: K-stability of Fano manifolds serves as the obstruction to the existence of a Kahler-Einstein metric. While K-stability is defined purely algebraically, it admits a non-Archimedean analytic characterization which connects well back to the differential geometric side. In this talk, we will give an introduction to an analogous local K-stability, which generalizes the K-stability of Fano manifolds and serves as the obstruction to the existence of a Ricci-flat Kahler cone metric. We will then give a non-Archimedean characterization of local K-stability, extending the one for Fano manifolds.

4/7: no meeting (Spring break)

4/14: no meeting


4/21: Prof. Chi Li (Rutgers)
Title: Kahler structures for holomorphic submersions

Abstract: We prove a criterion for the existence of Kahler structure for any holomorphic submersion. This criterion generalizes Blanchard's criterion for isotrivial holomorphic submersions. We then discuss its application to a question of Harvey-Lawson in the case of fiber dimension one and to the existence of Hermitian-Symplectic structures.

4/28: Michael Albanese (U. Waterloo)
Title: Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics

Abstract: Using results from the theory of harmonic maps, Kotschick proved that a closed hyperbolic four-manifold cannot admit a complex structure. We give a new proof which instead relies on properties of Einstein metrics in dimension four. The benefit of this new approach is that it generalizes to prove that another class of aspherical four-manifolds (graph manifolds with positive Euler characteristic) also fail to admit complex structures. This is joint work with Luca Di Cerbo.

5/5: Jiahao Hu (Stony Brook University)
Title: Characters for generalized cohomology theories

Abstract: Similar to that a de Rham cohomology class is determined by its periods over singular cycles, I will explain how a generalized cohomology class is determined by its periods over a suitable class of cycles. Moreover, I will discuss an enhanced version of this result which identifies differential generalized cohomology classes with differential characters over suitable cycles. Concrete examples such as singular cohomology and K-theories will be emphasized. For example, in the case of K-theory, I will show that a vector bundle up to stable equivalence is completely determined by the corresponding twisted Dirac indices.

5/12: no meeting