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:
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10/3:
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10/10: Spencer Cattalani (Stony Brook University)
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10/17: Prof. Marco Castronovo (Barnard College, Columbia University)
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10/24: no meeting (Monday schedule)
10/31: Prof. Chung-Ming Pan (Université de Quebec à Montréal)
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11/7: Prof. Joshua Sabloff (Haverford College)
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11/14:
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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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