Michael Zhao Memorial Student Colloquium
The Michael Zhao Memorial Student Colloquium holds 45-minute talks by Columbia mathematics
faculty about their own research. The talks are intended for current PhD students in mathematics at Columbia.
If you are an undergraduate student or external graduate student and would like to come, please email
Ethan Hall or Shijie Dai .
When: Thursday 6:00 - 7:00 PM ET
Where: Mathematics Building, Room 528
Organizers: Ethan Hall, Shijie Dai
When: Thursday 6:00 - 7:00 PM ET
Where: Mathematics Building, Room 528
Organizers: Ethan Hall, Shijie Dai
| Date | Speaker | Title and Abstract |
|---|---|---|
| February 12 | Matthew Hase-Liu |
Abstract: One fruitful approach to studying the geometry of a variety X is through understanding the space Mor(C,X) of curves C on X. When X is a smooth hypersurface of low degree, I'll explain how the circle method from analytic number theory can be used to shed light on the dimension, number of components, size and quality of singularities, class in the Grothendieck ring of varieties, and cohomology of Mor(C,X).
|
| March 12 | Ovidiu Savin |
Abstract: I will discuss the classical obstacle problem and related problems, and provide an overview on the free boundary regularity theory for such problems.
|
| April 9 | Myungsin Cho |
Abstract: One of the central interests in algebraic topology is to study spaces that behave like abelian groups. The language developed to systematically study such homotopy-theoretic analogues of abelian groups is the category of spectra. In this talk, I will explain how this perspective provides a conceptual viewpoint on classical results such as Poincaré duality and the Universal Coefficient Theorem. I will also discuss how similar duality phenomena appear in algebraic K-theory.
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| April 16 | Mu-Tao Wang |
Abstract: We introduce a family of closed differential forms associated with minimal graphical submanifolds in Euclidean space in arbitrary codimension. For each minimal graph, we construct an explicit closed form whose restriction agrees with the induced volume form, and which admits a geometric interpretation as the pullback of a tautological form on the Grassmannian via the Gauss map. Unlike classical calibrations, these forms are generally non-parallel and do not arise from special holonomy.
The calibration problem is thus reduced to estimating their pointwise comass. We characterize this bound in terms of explicit inequalities involving the singular values of the defining map, expressed through its two-dilation, and identify sharp conditions ensuring the comass is at most one. It follows that any minimal graph satisfying these conditions is calibrated and therefore area-minimizing. This provides a broad new class of calibrated minimal graphs, extending the codimension-one theory, as well as a practical criterion for determining when a minimal graph is area-minimizing. As an application, we verify a conjecture of Lawson–Osserman under two-dilation conditions in arbitrary codimension. This is based on joint work with Chung-Jun Tsai.
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