Qi Zhu

Current Projects

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1. Monoidal Parametrized Unstraightening (joint with Siddharth Gurumurthy and Ryan Quinn)

Straightening-Unstraightening compares a slice construction to a functor category construction. Both of these acquire a natural monoidal structure through the slice monoidal structure and Day convolution. Classically, these are known to be equivalent as (symmetric) monoidal ∞-categories due to Ramzi. We wish to spell out a generalization to parametrized higher category theory.

2. Norms of Real Algebraic K-Theory (joint with Kaif Hilman and Ryan Quinn)

The project is to endow Poincaré categories with a C_2-symmetric monoidal structure and realize Real algebraic K-theory as a normed algebra. This in particular yields a natural map K ---> L as well as parts of red- and whiteshift results.

3. Hermitian K-Theory through Real Homotopy Theory (joint with Christian Carrick and Ryan Quinn)

We start from Real analogs of Lee-Levy's THH computations leading to THR computations of Real chromatic spectra. The goal is to use this to make computations in Hermitian K-theory.

4. Equivariant Commutative Orientations [poster]

Hopkins-Lawson describe an obstruction theory about lifting complex orientations MU ---> E to coherently multiplicative maps. I'm hoping to generalize this to an equivariant setting.​

5. Real Equivariant Commutative Orientations (joint with Ryan Quinn) ​​​​

Hopkins-Lawson describe an obstruction theory about lifting complex orientations MU ---> E to coherently multiplicative maps. We hope to lift it to the Real equivariant setting and obtain highly structured Real orientations from it.

6. Multiplicative Equivariant Morita Theory

In the language of parametrized higher algebra, we wish to prove a multiplicative equivariant Schwede-Shipley theorem. The aim is to apply this to a good theory of equivariantly inverting elements in a multiplicative (but not fully multiplicative) setting.

7. More Multiplication on BP (joint with Ryan Quinn)

The best known multiplicative structure on BP is E_4 obtained by Basterra-Mandell. We have ongoing work to improve on this structure.