David Carchedi
George Mason University, VA·Mathematics & Statistics
Calculus was first built for flat spaces like lines and planes. Differential geometry extends it to smoothly curved spaces called manifolds, such as spheres, donuts and their higher-dimensional versions.
- Fields
Mathematics & Statistics
Physics
- Topics
derived differential geometry
derived stacks
higher stacks
- Grants
- 1 known grant, through Aug 2028
What they work on
- Build a theory of integration on quasi-smooth derived higher stacks, which are spaces that can handle non-smooth intersections, by developing a six-functor formalism within derived differential supergeometry.
- Identify dualizing complexes, which act as volume forms, with Berezinians of cotangent complexes, and define integration through the co-unit of exceptional inverse and direct image functors, with relevance to Feynman path integrals.
- Develop geometric quantization for shifted symplectic derived smooth stacks that outputs fully extended topological field theories, refining Safronov's shifted quantization program for the smooth setting.
- Prove that the fully extended framed TQFT of Chern-Simons theory is determined by a linear 2-category of representations of the string 2-group, using the cobordism hypothesis.
More about the research
This framework underlies much of modern physics, from Einstein's picture of gravity as curved spacetime to the Standard Model of particle physics. Solving equations corresponds geometrically to intersecting shapes. However, the intersection of two manifolds is often not a smooth manifold, and ordinary differential geometry cannot handle it. Derived differential geometry (DDG), a framework to which the PI has made significant contributions, is designed to deal with such non-smooth intersections. Integration, a cornerstone of calculus, has not yet been fully developed in this setting.
Why it matters
This work is pure mathematics with direct ties to physics. New formalisms for integration and quantization over derived stacks could provide rigorous tools for calculations in quantum field theory, such as path integrals. They could also yield new quantum invariants from topological field theories. Both mathematicians and theoretical physicists working where geometry, topology and quantum theory meet could benefit.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Chelsea Walton
Rice University, TX·Mathematics & Statistics
Extend classical properties of algebras over a field to algebras in monoidal categories, and identify properties that carry significant meaning only in the categorical setting.
Cris Negron
University of Southern California, CA·Mathematics & Statistics·Early-career
Build rigorous mathematical realizations of topological quantum field theories whose spaces of lowest-energy states (moduli of vacua) are non-compact, using homotopical methods in representation theory.
Ron Donagi
University of Pennsylvania, PA·Mathematics & Statistics
Attack the geometric Langlands conjecture, a proposed deep link between areas of mathematics, using non-abelian Hodge theory and ideas from quantum field theory.
The school
George Mason University: 15 labs on PI Pages, $282M a year on research, 297 research doctorates a year.
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