Current Research Interests
1. Equivariant Commutative Orientations: The central notion in chromatic homotopy theory is the complex orientation of a ring spectrum which can be described as a map of homotopy commutative ring spectra MU ---> E. Hopkins-Lawson described an obstruction theory about lifting such maps to coherently multiplicative maps.
The current goal of my PhD project is to generalize this to a (global) equivariant setting.
See here for a poster.
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2. Real Equivariant Commutative Orientations (joint with Ryan Quinn): Hopkins and Lawson claim that their obstruction theory also works in the Real equivariant setting but say that the higher chromatic behaviour is unclear to them here. We hope to solve it with our methods from the equivariant story and hope to obtain further results specific to the Real equivariant setting.
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3. Universal Property of Equivariant Multiplicative Thom Spectra (joint with Ryan Quinn): If f is a multiplicative map, then its Thom spectrum Th(f) inherits multiplicative structures and Antolín-Camarena-Barthel proved a universal property to map out of such multiplicative Thom spectra Th(f). We establish the analogous statement in the equivariant setting by means of fibrous patterns and parametrized category theory.
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4. Fractured Structure on Condensed Anima (joint with Nima Rasekh): The categorical foundation of condensed mathematics lies in topos theory. This project has grown out of my master's thesis and our goal is to establish formal properties of Carchedi and Lurie's fractured topoi. One possible application is the study of condensed cohomology.
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See here for my master's thesis on this topic and here for a more recent talk.