Fall 2025: (all meetings in person)
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9/5: Prof. David Pham (CUNY)
Title: Hypercomplex Geometry and Compact Lie groups
Abstract: In the early 1950s, Samelson and Wang (working independently) both showed that every even dimensional compact Lie group admits a left-invariant complex structure. Hypercomplex geometry necessarily requires the dimension of the manifold to be a multiple of 4. Motivated by the work of Samelson and Wang, it is natural to conjecture that every compact Lie group of dimension 4n admits a left-invariant hypercomplex structure. Dominic Joyce, in the early 1990s, showed that (at the very least) if G is any compact Lie group, then GxT always admits a left-invariant hypercomplex structure where T is a torus of sufficient size. Interestingly, there are examples in the recent literature that assert that Joyce’ s theorem implies that the aforementioned conjecture is true. In this talk, we show that the conjecture is actually false. More precisely, we show that the Lie group SU(2)^m never admits a left-invariant hypercomplex structure for all m>0. (In particular, this is true for the non-trivial case m=4n.)
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9/12: Prof. Izar Alonso Lorenzo (Rutgers)
Title: Gauge theory on hyperkähler manifolds
Abstract: H-instantons are a distinguished type of connections on Riemannian n-manifolds, as they are generalizations of anti-self-dual connections to dimensions \(n \geq 4\). Examples of H-instantons include primitive Hermitian Yang-Mills (pHYM) connections, G_2- and Spin(7)-instantons, which have been of great interest in the recent years, and the less studied Sp(n)-instantons.
In this talk, we describe Sp(2)-instantons on hyperkähler eight-manifolds and their relations with other gauge-theoretical objects.
We then construct examples of Sp(2)-instantons, pHYM connections and Spin(7) instantons with symmetry on the manifold \(T^*CP^2\) with the Calabi hyperkähler structure.
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9/19: Prof. Nick McCleerey (Purdue)
Title: Lines in the space of Kähler metrics
Abstract: We report on joint work with Tamás Darvas, in which we establish a Ross-Witt Nyström correspondence for weak geodesic lines in the (completed) space of Kähler metrics. Using this, we construct a wide range of weak geodesic lines on any projective Kähler manifold which are not generated by holomorphic vector fields, thus disproving a folklore conjecture popularized by Berndtsson. Remarkably, some of these weak geodesic lines turn out to be smooth. In the case of Riemann surfaces, our results can be significantly sharpened. Finally, we investigate the validity of Euclid's fifth postulate for the space of Kähler metrics.
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9/26: Prof. Scott Wilson (CUNY)
Title: Bicomplexes, both linear and multiplicative, and their homotopy theory
Abstract: I will survey recent work (mostly by Jonas Stelzig) on the homotopy theory of bicomplexes and commutative bigraded bi-differential algabras (cbba's), with a view towards complex amanifolds and applications. I will describe the Appeli-Bott-Chern (ABC) Massey products in this context (due to Angella and Tomassini) with the hopes of generalizing this to the case of cbba's with compatible (homotopy) inner products.
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10/3: Prof. Mehdi Lejmi (CUNY)
Title: 4-dimensional Almost-Hermitian manifolds with Kahler-Type curvature.
Abstract: In this talk, we will characterize 4-dimensional almost-Hermitian manifolds that satisfy the first Gray curvature condition. The goal is to obtain a classification of such manifolds. This is a joint work with Ethan Addison and Tedi Draghici.
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10/10: Spencer Cattalani (Stony Brook University)
Title: Ahlfors currents in symplectic geometry
Abstract: Rational curves are one of the main tools in symplectic geometry and provide a bridge to algebraic geometry. Complex lines are a more general class of curve that has the potential to connect symplectic to complex analytic geometry. These curves are non-compact, which presents a serious difficulty in understanding their symplectic aspects. In this talk, I will explain how Ahlfors currents can be used to resolve this difficulty and produce a theory parallel to that of rational curves. In particular, Ahlfors currents can be constructed via a continuity method, they control bubbling of holomorphic curves, and they form a convex set.
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10/17: Prof. Marco Castronovo (Barnard College, Columbia University)
Title: Cluster gluings and open-string Schubert calculus
Abstract: When is a gluing of affine schemes affine? I will state a complete but conjectural answer when the pieces are tori, and the gluing maps are cluster mutations in the sense of Fomin-Zelevinsky. The combinatorics of Demazure weaves allows to prove the conjecture in interesting cases, notably those in which there are finitely many tori, as well as the simplest ones with infinitely many. This has applications to open-string Schubert calculus, i.e. computations of the Fukaya category of G/P. Joint work with M. Gorsky, J. Simental, D. Speyer.
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10/24: no meeting (Monday schedule)
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10/31: Note: We have two talks this day.
10/31: Prof. Gueo Grantcharov (Florida International University)
Note special location: Thesis Room, 4214.
Title: Some geometric applications of twistor spaces
Abstract: I'll review the definition of twistor space over 4-dimensional Riemannian manifold and report on some of its properties. Then I'll explain its relation to generalized Kähler structures and some further extensions to the higher dimensions.
10/31: Prof. Chung-Ming Pan (Université de Quebec à Montréal)
Title: Singular canonical Kähler metrics and orbifold regularity problem
Abstract: The talk aims to explain the orbifold regularity problem for canonical Kähler metrics on singular spaces. I will begin with a brief overview of canonical Kähler metrics and the Calabi problem, focusing more on the Kähler-Einstein setting. I will then discuss how singular Kähler-Einstein metrics are defined and what kind of singular spaces interest us. Finally, I will present the orbifold regularity problem of canonical Kähler metrics and explain the idea and strategy for solving the Kähler Calabi-Yau case.
This talk is based on joint work with Henri Guenancia and Mihai Păun.
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11/7: Prof. Joshua Sabloff (Haverford College)
Title: Weak Relative Calabi-Yau Structures for Legendrian Contact Homology
Abstract: The Legendrian Contact Homology (LCH) differential graded algebra (DGA) was among the first, and is still among the most important, non-classical invariants of Legendrian knots. In this talk, I will tell a story that builds up ever more sophisticated analogues of Poincare Duality as we delve more deeply into the structure of the DGA. Despite the algebraic nature of the results, I promise pictures, examples, and analogies with Morse Theory. This is joint work with Jason Ma (Duke).
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11/14: Caleb Jonker (U. Toronto)
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11/21:
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11/28: no meeting (Thanksgiving)
12/5:
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12/12:
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