Spring 2023: (all meetings in person)

1/27: no meeting (Semester Kickoff & Chalk Talk this day, 11am3:30pm in the Math Lounge.)

2/3: Jiahao Hu (Stony Brook University)
Title: Differential Ktheory and its topology
Abstract: I will define differential Ktheory of a smooth manifold using complex vector bundles with connections and survey some of its properties. Then I will show (1) differential Ktheory carries a natural topology induced from the smooth structure of the manifold; and (2) from this topology one can recover topological Ktheory and some information of the manifold concerning differential forms. If time permits, I will also discuss how this topology helps axiomatically characterize the differential Ktheory 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 2sphere 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 chainlevel 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 2sphere
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
algebrasone involving higher homotopiesHochschild 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 2sphere with mod 2 coefficients. This is joint work with Thomas Tradler.

3/3: Prof. Valentino Tosatti (Courant Institute)
Title: Collapsing CalabiYau manifolds
Abstract: I will give an introduction to compact CalabiYau manifolds and their Ricciflat Kahler metrics. I will then study how these Ricciflat metrics behave when the cohomology class of their Kahler forms degenerates. While the noncollapsing case is by now wellunderstood, 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: Nonflat pluriclosed CalabiYau 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 CalabiYau 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 DonaldsonUhlenbeckYau Theorem
Abstract: The theorem of DonaldsonUhlenbeckYau says that a holomorphic vector bundle E over a compact Kahler manifold admits a HermiteEinstein (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 YauTianDonaldson 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 DonaldsonUhlenbeckYau 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 KahlerRicci flow and some applications
Abstract: We give an introduction to the KahlerRicci 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 KahlerEinstein 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 nonArchimedean characterization of local Kstability
Abstract: Kstability of Fano manifolds serves as the obstruction to the existence of a KahlerEinstein metric. While Kstability is defined purely algebraically, it admits a nonArchimedean analytic characterization which connects well back to the differential geometric side. In this talk, we will give an introduction to an analogous local Kstability, which generalizes the Kstability of Fano manifolds and serves as the obstruction to the existence of a Ricciflat Kahler cone metric. We will then give a nonArchimedean characterization of local Kstability, extending the one for Fano manifolds.

4/7: no meeting (Spring break)

4/14: no meeting
Title:
Abstract:

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 HarveyLawson in the case of fiber dimension one and to the existence of HermitianSymplectic structures.

4/28: Michael Albanese (U. Waterloo)
Title: Aspherical 4Manifolds, Complex Structures, and Einstein Metrics
Abstract: Using results from the theory of harmonic maps, Kotschick proved that a closed hyperbolic fourmanifold 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 fourmanifolds (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 Ktheories will be emphasized. For example, in the case of Ktheory, I will show that a vector bundle up to stable equivalence is completely determined by the corresponding twisted Dirac indices.

5/12: no meeting
