Papers
Dmitri Pavlov's articles on arXiv.
Functorial field theory
(with Daniel Grady)
The geometric cobordism hypothesis.
Submitted.
We prove a generalization of the cobordism hypothesis of Baez--Dolan and Hopkins--Lurie for bordisms with arbitrary geometric structures, such as Riemannian metrics, complex and symplectic structures, principal bundles with connections, or geometric string structures. Our methods rely on the locality property for fully extended functorial field theories established in arXiv:2011.01208, reducing the problem to the special case of geometrically framed bordism categories. As an application, we upgrade the classification of invertible fully extended topological field theories by Bökstedt--Madsen and Schommer-Pries to nontopological field theories, generalizing the work of Galatius--Madsen--Tillmann--Weiss to arbitrary geometric structures.
(with Daniel Grady)
Extended field theories are local and have classifying spaces.
Submitted.
We show that all extended functorial field theories, both topological and nontopological, are local. We define the smooth (infinity,d)-category of bordisms with geometric data, such as Riemannian metrics or geometric string structures, and prove that it satisfies codescent with respect to the target S, which implies the locality property. We apply this result to construct a classifying space for concordance classes of functorial field theories with geometric data, solving a conjecture of Stolz and Teichner about the existence of such a space. We use our classifying space construction to develop a geometric theory of power operations, following the recent work of Barthel, Berwick-Evans, and Stapleton.
(with Daniel Berwick-Evans)
Smooth one-dimensional topological field theories are vector bundles with connection.
Algebraic & Geometric Topology 23:8 (2023), 3707–3743.
doi:10.2140/agt.2023.23.3707
We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth version of Rezk's complete Segal spaces. With such a definition in hand, we analyze the category of field theories using a combination of descent, a smooth version of the 1-dimensional cobordism hypothesis, and standard differential-geometric arguments.
Sheaf theory
Projective model structures on diffeological spaces and smooth sets and the smooth Oka principle.
Homology, Homotopy, and Applications 26:2 (2024), 375–408.
doi:10.4310/HHA.2024.v26.n2.a18
In the first part of the paper, we prove that the category of diffeological spaces does not admit a model structure transferred via the smooth singular complex functor from simplicial sets, resolving in the negative a conjecture of Christensen and Wu, in contrast to Kihara's model structure on diffeological spaces constructed using a different singular complex functor. Next, motivated by applications in quantum field theory and topology, we embed diffeological spaces into sheaves of sets (not necessarily concrete) on the site of smooth manifolds and study the proper combinatorial model structure on such sheaves transferred via the smooth singular complex functor from simplicial sets. We show the resulting model category to be Quillen equivalent to the model category of simplicial sets. We then show that this model structure is cartesian, all smooth manifolds are cofibrant, and establish the existence of model structures on categories of algebras over operads. Finally, we use these results to establish analogous properties for model structures on simplicial presheaves on smooth manifolds, as well as presheaves valued in left proper combinatorial model categories, and prove a generalization of the smooth Oka principle established in arXiv:1912.10544. We apply these results to establish classification theorems for differential-geometric objects like closed differential forms, principal bundles with connection, and higher bundle gerbes with connection on arbitrary cofibrant diffeological spaces.
Numerable open covers and representability of topological stacks.
Topology and its Applications 318 (2022), 108203, 1–28.
doi:10.1016/j.topol.2022.108203
We prove that the class of numerable open covers of topological spaces is the smallest class that contains covers with pairwise disjoint elements and numerable covers with two elements, closed under composition and coarsening of covers. We apply this result to establish an analogue of the Brown--Gersten property for numerable open covers of topological spaces: a simplicial presheaf on the site of topological spaces satisfies the homotopy descent property for all numerable open covers if and only if it satisfies it for numerable covers with two elements and covers with pairwise disjoint elements. We also prove a strengthening of these results for manifolds, ensuring that covers with two elements can be taken to have a specific simple form. We apply these results to deduce a representability criterion for stacks on topological spaces similar to arXiv:1912.10544. We also use these results to establish new simple criteria for chain complexes of sheaves of abelian groups to satisfy the homotopy descent property.
(with Daniel Berwick-Evans and Pedro Boavida de Brito)
Classifying spaces of infinity-sheaves.
Algebraic & Geometric Topology 24:9 (2024), 4891–4937.
doi:10.2140/agt.2024.24.4891
We prove that the set of concordance classes of sections of an infinity-sheaf on a manifold is representable, extending a theorem of Madsen and Weiss. This is reminiscent of an h-principle in which the role of isotopy is played by concordance. As an application, we offer an answer to the question: what does the classifying space of a Segal space classify?
Homotopy theory
The enriched Thomason model structure on 2-categories.
Journal of Pure and Applied Algebra 228:5 (2023), 107496, 1–23.
doi:10.1016/j.jpaa.2023.107496
We prove that categories enriched in the Thomason model structure admit a model structure that is Quillen equivalent to the Bergner model structure on simplicial categories, providing a new model for (infinity,1)-categories. Along the way, we construct model structures on modules and monoids in the Thomason model structure and prove that any model structure on the category of small categories that has the same weak equivalences as the Thomason model structure is not a cartesian model structure.
Combinatorial model categories are equivalent to presentable quasicategories.
Journal of Pure and Applied Algebra 229:2 (2025), 107860, 1–39.
doi:10.1016/j.jpaa.2024.107860
We establish a Dwyer-Kan equivalence of relative categories of combinatorial model categories, presentable quasicategories, and other models for locally presentable (infinity,1)-categories. This implies that the underlying quasicategories of these relative categories are also equivalent.
(with Owen Gwilliam)
Enhancing the filtered derived category.
Journal of Pure and Applied Algebra 222:11 (2018), 3621–3674.
doi:10.1016/j.jpaa.2018.01.004
The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods of homotopical algebra, in the formalisms of stable infinity-categories, stable model categories, and pretriangulated, idempotent-complete dg categories. We characterize the filtered stable infinity-category Fil(C) of a stable infinity-category C as the left exact localization of sequences in C along the infinity-categorical version of completion (and prove analogous model and dg category statements). We also spell out how these constructions interact with spectral sequences and monoidal structures. As examples of this machinery, we construct a stable model category of filtered D-modules and develop the rudiments of a theory of filtered operads and filtered algebras over operads.
(with Jakob Scholbach)
Homotopy theory of symmetric powers.
Homology, Homotopy, and Applications 20:1 (2018), 359–397.
doi:10.4310/HHA.2018.v20.n1.a20
We introduce the symmetricity notions of symmetric h-monoidality, symmetroidality, and symmetric flatness. As shown in our paper arXiv:1410.5675, these properties lie at the heart of the homotopy theory of colored symmetric operads and their algebras. In particular, the former property can be seen as the analog of Schwede and Shipley's monoid axiom for algebras over symmetric operads and allows one to equip categories of such algebras with model structures, whereas the latter ensures that weak equivalences of operads induce Quillen equivalences of categories of algebras. We discuss these properties for elementary model categories such as simplicial sets, simplicial presheaves, and chain complexes. Moreover, we provide powerful tools to promote these properties from such basic model categories to more involved ones, such as the stable model structure on symmetric spectra.
(with Jakob Scholbach)
Admissibility and rectification of colored symmetric operads.
Journal of Topology 11:3 (2018), 559–601.
doi:10.1112/topo.12008
We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, i.e., the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, i.e., the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categories yield Quillen equivalences of model categories of algebras over operads. Applications of these results include enriched categories, colored operads, prefactorization algebras, and commutative symmetric ring spectra.
(with Jakob Scholbach)
Symmetric operads in abstract symmetric spectra.
Journal of the Institute of Mathematics of Jussieu 18:4 (2019), 707–758.
doi:10.1017/S1474748017000202
This paper sets up the foundations for derived algebraic geometry, Goerss--Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category.
We show that one can do derived algebraic geometry a la Toën--Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology.
These results are obtained by first establishing Smith's stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and E-infinity ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of E-infinity rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent.
von Neumann algebras
Gelfand-type duality for commutative von Neumann algebras.
Journal of Pure and Applied Algebra 226:4 (2022), 106884, 1–53.
doi:10.1016/j.jpaa.2021.106884
We show that the following five categories are equivalent: (1) the opposite category of commutative von Neumann algebras; (2) compact strictly localizable enhanced measurable spaces; (3) measurable locales; (4) hyperstonean locales; (5) hyperstonean spaces. This result can be seen as a measure-theoretic counterpart of the Gelfand duality between commutative unital C*-algebras and compact Hausdorff topological spaces.
Algebraic tensor products and internal homs of noncommutative -spaces.
Journal of Mathematical Analysis and Applications 456:1 (2017), 229–244.
doi:10.1016/j.jmaa.2016.11.060
We prove that the multiplication map L^a(M)\otimes_M L^b(M)\to L^{a+b}(M) is an isometric isomorphism of (quasi)Banach M-M-bimodules. Here L^a(M)=L_{1/a}(M) is the noncommutative L_p-space of an arbitrary von Neumann algebra M and \otimes_M denotes the algebraic tensor product over M equipped with the (quasi)projective tensor norm, but without any kind of completion. Similarly, the left multiplication map L^a(M)\to Hom_M(L^b(M),L^{a+b}(M)) is an isometric isomorphism of (quasi)Banach M-M-bimodules, where Hom_M denotes the algebraic internal hom. In particular, we establish an automatic continuity result for such maps. Applications of these results include establishing explicit algebraic equivalences between the categories of L_p(M)-modules of Junge and Sherman for all p≥0, as well as identifying subspaces of the space of bilinear forms on L^p-spaces.
Teaching
Topology and Geometry Seminar.
Quantum Homotopy Seminar.
Fall 2025: Mathematics 6323 (Algebraic Geometry I), TuTh 2–3:30.
Spring 2025: Mathematics 6333 (Lie Groups), TuTh 2–3:30 in MA 10.
Syllabus.
Fall 2024: Mathematics 6330 (Manifold Theory), TuTh 11–12:20 in MA 10.
Syllabus,
handwritten notes by Stone Fields.
Spring 2024: Mathematics 6321 (Homological Algebra), TuTh 11–12:20.
Fall 2023: Mathematics 4350 (Advanced Calculus I), TuTh 9:30–11.
Fall 2023: Mathematics 1451 (Calculus I), TuTh 3–5.
Spring 2023: Mathematics 6332 (Geometric Mechanics), TuTh 11–12:30.
Fall 2022: Mathematics 6325 (Category Theory), TuTh 11–12:30.
Homework:
1, 2.
Fall 2022: Mathematics 4362 (Theory of Numbers), TuTh 2–3:30 in MA 12.
Spring 2022: Mathematics 4351 (Advanced Calculus II), TuTh 2–3:20 in MA 109.
Fall 2021: Mathematics 2450.021 (Calculus III), TuTh 9:30–11:30 in MA 16.
Syllabus.
Fall 2021: Mathematics 6330 (Manifold Theory), TuTh 2–3:30 in MA 114.
Syllabus,
notes,
homework.
Spring 2021: Mathematics 5399.002 (Introduction to Modern Algebra II), TuTh 9:30–11 in MA 17.
Syllabus,
handwritten notes by Jack Weiland,
homework 1, 2.
Spring 2021: Mathematics 2360.022 (Linear Algebra), TuTh 11–12:30 in HUMSCI 63.
Syllabus,
digest.
Fall 2020: Mathematics 5317 (Introduction to Modern Algebra), TuTh 9:30–11 in MA 109.
Syllabus,
handwritten notes by Jack Weiland,
homework 1, 2, 3, 4, 5, 6, 7, 8, 9.
Fall 2020: Mathematics 2360.121 (Linear Algebra), online.
Syllabus,
study guide,
digest,
review 1, 2, 3.
Spring 2020: Mathematics 6322 (Homological Algebra II), TuTh 9:30–11 in MA 113 and online.
Syllabus,
handwritten notes by Rachel Harris,
midterm,
projects.
Spring 2020: Mathematics 6332 (Geometric Mechanics), TuTh 11–12:30 in MA 17 and online.
Syllabus,
projects.
Fall 2019: Mathematics 6321 (Homological Algebra I), TuTh 9:30–11 in MA 109.
Syllabus,
notes,
handwritten notes by Rachel Harris,
homework,
projects.
Fall 2019: Mathematics 6333 (Lie Groups), TuTh 11–12:30 in MA 113.
Syllabus,
handwritten notes by Rachel Harris,
homework,
projects.
Summer 2019: Topology Doctoral Preliminary Examination.
Practice problems,
supplement,
May 2019,
August 2019,
August 2020.
Spring 2019: Mathematics 5325 (Topology II), TuTh 11–12:30 in MA 113.
Syllabus,
notes,
handwritten notes by Rachel Harris,
midterm 1, 2,
final.
Fall 2018: Mathematics 5324 (Topology I), TuTh 11–12:30 in MA 113.
Syllabus,
notes,
midterm 1, 2, final.
Fall 2018: Mathematics 5365 (Analysis of Algorithms), TuTh 12:30–2 in MA 113.
Syllabus,
homework: 1, 2, 3,
midterm 1, 2, final.
Spring 2018: Mathematics 6325 (Category Theory), MWF 9–10 in MA 10.
Syllabus,
notes,
handwritten notes by Nilan Manoj Chathuranga,
homework: 1, 2, 3.
Fall 2017: Mathematics 2360.002 (Linear Algebra), MWF 2–3 in MA 110.
Syllabus.