About

Daniel Berwick-Evans is an Associate Professor in the Department of Mathematics at the University of Illinois at Urbana-Champaign. He earned his PhD in 2013 from UC Berkeley, working with Peter Teichner. His research studies connections between supersymmetric (quantum) field theories, differential geometry, and algebraic topology. Prior to his current position, he was a Szego Assistant Professor at Stanford University from 2013 to 2015. His work involves exploring the interplay between advanced mathematical structures and quantum field theories, contributing to the understanding of elliptic cohomology, modularity, and topological quantum field theories.

Research topics

• Mathematics
• Geometry
• Pure mathematics
• Computer Science
• Mathematical analysis
• Combinatorics

Selected publications

• Flat principal 2-group bundles and flat string structures

  Contemporary mathematics - American Mathematical Society · 2025-01-01

  other1st authorCorresponding

  For a weak 2-group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we construct a bicategory of flat <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bundles over differentiable stacks as a localization of a functor bicategory. This description is amenable to explicit geometric constructions. For example, we show that flat <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bundles can be described in terms of ordinary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bundles together with a trivialization of a certain 2-gerbe. This specializes to a characterization of flat string structures on vector bundles over differentiable stacks.

  PublisherDOI

• The Freed--Quinn line bundle from higher geometry

  ArXiv.org · 2025-10-12

  preprintOpen access1st authorCorresponding

  For a finite group $G$, and level $α\in Z^3(BG;{\rm U}(1))$, Freed and Quinn construct a line bundle over the moduli space of $G$-bundles on surfaces. Global sections determine the values of Chern--Simons theory at level $α$ on surfaces. In this paper, we provide an alternate construction using tools from higher geometry: the pair $(G,α)$ determines a 2-group group, and the Freed--Quinn line arises as a categorical truncation of the bicategory of 2-group bundles.

  PublisherOA PDFDOI

• Classifying spaces of infinity-sheaves

  Algebraic & Geometric Topology · 2024-12-27 · 2 citations

  articleOpen access1st authorCorresponding

  We prove that the set of concordance classes of sections of an 1-sheaf on a manifold is representable, extending a theorem of Madsen and Weiss for sheaves of sets.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?18F20, 18N60, 55N30 1. Introduction 4891 2. The concordance resolution is concordance-invariant 4896 3. Weak Kan fibrations 4901 4. Weak Kan fibrancy of the concordance resolution 4908 5.The shape functor preserves the 1-sheaf property 4920 6.What does the classifying space of an 1-category classify?4927 Appendix.Technical lemmas on simplicial sets and spaces 4930

  PublisherOA PDFDOI

• Elliptic Cohomology and Quantum Field Theory

  Encyclopedia of Mathematical Physics · 2024-10-03

  book-chapter1st authorCorresponding
  PublisherDOI

• Supersymmetric localization, modularity and the Witten genus

  Journal of Differential Geometry · 2024-02-01 · 3 citations

  article1st authorCorresponding

  We use equivariant localization techniques to give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In particular, we identify a candidate target for the elliptic Bismut–Chern character.

  PublisherDOI

• Elliptic cohomology and quantum field theory

  arXiv (Cornell University) · 2024-08-14

  preprintOpen access1st authorCorresponding

  This survey provides an introduction to the Stolz-Teichner program on elliptic cohomology and quantum field theory.

  PublisherOA PDFDOI

• Power operations preserve Thom classes in twisted equivariant Real K-theory

  arXiv (Cornell University) · 2024-07-17

  preprintOpen access1st authorCorresponding

  We construct power operations for twisted KR-theory of topological stacks. Standard algebraic properties of Clifford algebras imply that these power operations preserve universal Thom classes. As a consequence, we show that the twisted Atiyah-Bott-Shapiro orientation commutes with power operations.

  PublisherOA PDFDOI

• Averaging property of wedge product and naturality in discrete exterior calculus

  Advances in Computational Mathematics · 2024-07-31

  article
  PublisherDOI

• The families Clifford index and differential KO-theory

  arXiv (Cornell University) · 2023-03-16

  preprintOpen access1st authorCorresponding

  Extending ideas of Atiyah--Bott--Shapiro and Quillen, we construct a model for differential $\rm KO$-theory whose cocycles are families of Clifford modules with superconnection. The model is built to accommodate an analytic pushforward for bundles of spin manifolds, affording a differential refinement of Atiyah and Singer's families index.

  PublisherOA PDFDOI

• Averaging Property of Wedge Product and Naturality in Discrete Exterior Calculus

  arXiv (Cornell University) · 2023-10-01

  preprintOpen access

  In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and the antisymmetrized cup product provides a discrete wedge product. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson's cochain product defined using Whitney and de Rham maps.

  PublisherOA PDFDOI

Frequent coauthors

• Mark D. Schubel

  University of Illinois Urbana-Champaign

  4 shared

• Anil N. Hirani

  University of Illinois Urbana-Champaign

  4 shared

• Arnav Tripathy

  Beijing Institute of Mathematical Sciences and Applications

  4 shared

• Dmitri Pavlov

  Texas Tech University

  3 shared

• Nathaniel Stapleton

  2 shared

• Tobias Barthel

  Max Planck Institute for Mathematics

  2 shared

• Pedro Boavida de Brito

  2 shared

• Д. А. Павлов

  N. I. Lobachevsky State University of Nizhny Novgorod

  1 shared