Abstract: Coercivity thresholds are a central theme in geometry. They appear classically in the Yamabe problem (constant scalar curvature in a conformal class), in the Nirenberg problem (prescribed curvature on the 2-sphere), and in numerous problems on determining best constants in Sobolev embeddings and related functionals inequalities. In 1980's Aubin and Tian introduced the first such thresholds in the Kahler-Einstein problem and their study has been a central and still very active field. In 1988 Tian observed that these thresholds have quantum versions and he posed the so-called Stabilization Problem: do the equivariant quantum thresholds become constant (and hence equal to the classical thresholds)? Cheltsov conjectured that these invariants coincide with the algbero-geometric log canonical thresholds (lct), and this was verified by Demailly (2008). The best result so far has been Birkar's theorem (2019) that shows that the quantum lcts are constant along a subsequence in the absence of group actions. In joint work with Chenzi Jin (PhD student at UMD) we offer a new approach and solve Tian's problem in the toric case. Surprisingly, the equivariant lcts are constant already from the first quantum level. For more general Grassmannian lcts we offer counterexamples to stabilization and determine when it holds. The key new ideas are understanding the effect of finite group actions on these invariants, and relating these thresholds to support and gauge functions from convex geometry. Time permitting I will discuss extensions and generalizations to other invariants, e.g., the Fujita-Odaka stability thresholds. 

Biography: Rubinstein received his doctorate from M.I.T. in 2008. He then held positions at Johns Hopkins University and Stanford University before accepting a tenured position at the University of Maryland in 2012 where he has been since. His research has spanned problems originating from geometric analysis, algebraic geometry, microlocal analysis, complex and convex analysis, and numerical analysis. He has led initiatives for the development and expansion of Research Experiences for Undergraduates (REUs) and has edited a volume on ``Directions for Mathematics REUs" in 2016. In 2023 he received the University of Maryland Grand Challenges award and in 2024 the Do Good award for increasing the participation of underrepresented minorities and women in STEM. He is a Sloan Fellow (2013) and a Simons Fellow (2024).

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