A valuable collaboration

Chris Elliott

Address:
Department of Mathematics and Statistics
Amherst College
Seeley Mudd Building
31 Quadrangle Dr
Amherst, MA 01002

Office: Chapin Hall 018
E-mail: celliott AT amherst DOT edu

A pdf copy of my CV is available here.

I am a visiting assistant professor in mathematics at Amherst College. I'm interested in mathematical constructions in classical and quantum field theories using derived algebraic geometry. For example, I think about:

The construction and classification of (not necessarily topological) twists of classical and quantum field theories, especially using techniques of derived algebraic geometry and homotopical algebra.
The connection between structures appearing in various versions of the geometric Langlands correspondence and twists of four- and five-dimensional supersymmetric gauge theories.
The theory of factorization algebras as a model for perturbative field theory.

A more detailed description of some of my interests can be found in my research statement. I also have a (somewhat older) non-technical description of my research.

Teaching

Spring 2024: : Math 121 -- Intermediate Calculus, Math 350 -- Groups, Rings and Fields, and Math 490 -- Independent study on Lie Algebras
Fall 2023: Math 211 -- Multivariate Calculus, Math 350 -- Groups, Rings and Fields, and Math 490 -- Independent study on Algebraic Topology
Spring 2023: Math 250 -- Number Theory and Math 455 -- Topology
Fall 2022: Math 211 -- Multivariate Calculus and Math 271 -- Linear Algebra
Spring 2022: Ordinary Differential Equations for Scientists and Engineers

Course Webpage
Fall 2021: Math 411: Introduction to Abstract Algebra

Syllabus
Spring 2021: Math 797RM -- Moduli Spaces in Representation Theory.

Course Webpage
Spring 2021: Math 132 -- Calculus II.
Fall 2020: Math 132H -- Calculus II Honors

Syllabus
Spring 2020: Math 132 -- Calculus II.

Course Webpage

Syllabus
Fall 2019: Math 131H -- Calculus I Honors

Papers and Preprints

Twists of Superconformal Algebras (joint with Owen Gwilliam and Matteo Lotito): https://arxiv.org/abs/2403.19753
Defects via factorization algebras (joint with Ivan Contreras and Owen Gwilliam), https://arxiv.org/abs/2208.01730
The derived pure spinor formalism as an equivalence of categories (joint with Fabian Hahner and Ingmar Saberi), http://arxiv.org/abs/2205.14133
Framed E_n-Algebras from Quantum Field Theory (joint with Owen Gwilliam), arxiv.org/abs/2204.03702
Higher Deformation Quantization for Kapustin-Witten Theories (joint with Owen Gwilliam and Brian Williams), arxiv.org/abs/2108.13392
Quantum Geometric Langlands Categories from N = 4 Super Yang-Mills Theory (joint with Philsang Yoo), arxiv.org/abs/2008.10988
Spontaneous Symmetry Breaking: a view from derived geometry (joint with Owen Gwilliam), Journal of Geometry and Physics, Vol 162, 2021, arxiv.org/abs/2008.03599
Holomorphic Poisson Field Theories (joint with Brian Williams), Higher Structures, Vol 5, Issue 1, 2021, arxiv.org/abs/2008.02302
A Taxonomy of Twists of Supersymmetric Yang-Mills Theory (joint with Pavel Safronov and Brian Williams), accepted for publication in Selecta Mathematica, arxiv.org/abs/2002.10517
Multiplicative Hitchin Systems and Supersymmetric Gauge Theory (joint with Vasily Pestun), Selecta Mathematica, Vol 25, Issue 64, 2019, arxiv.org/abs/1812.05516
Topological Twists of Supersymmetric Algebras of Observables (joint with Pavel Safronov), Communications in Mathematical Physics, Vol 371, pages 727-786, 2019, arxiv.org/abs/1805.10806
A Physical Origin for Singular Support Conditions in Geometric Langlands (joint with Philsang Yoo), Communications in Mathematical Physics, Vol 368, Issue 3, Pages 985-1050, 2019, arxiv.org/abs/1707.01292
Asymptotic Freedom in the BV Formalism (joint with Brian Williams and Philsang Yoo), Journal of Geometry and Physics, Vol 123, January 2018, Pages 246-283, arxiv.org/abs/1702.05973
Geometric Langlands Twists of N=4 Gauge Theory from Derived Algebraic Geometry (joint with Philsang Yoo), Advances in Theoretical and Mathematical Physics, Vol 22, Number 3, Pages 615-708, 2018, arxiv.org/abs/1507.03048
Abelian Duality for Generalised Maxwell Theories, Mathematical Physics, Analysis and Geometry, Vol 22, Issue 22, 2019, arxiv.org/abs/1402.0890 (poster).

You can also read my thesis, which is a combination of the last two papers above, with an expository introduction and some remarks on future work.

Conferences and Workshops

I have been organizing a series of mini-conferences, so far at Amherst College in April 2024, at Boston University in November 2023 and at Northeastern University in April 2023, the New England Algebraic Topology and Mathematical Physics Seminar (NEAT MAPS), a miniconference in topology and quantum field theory aimed at researchers based in New England.
I was a co-organizer for the conference Gone Fishing 2023 at Amherst College, part of a series of conferences on Poisson geometry. The conference was centered on the theme “Poisson geometry and its derived, shifted, and higher variants”.
I have been organizing a series of summer schools on Physical Mathematics of QFT, together with Owen Gwilliam, Samantha Kirk and Matteo Lotito. We hosted events in the summer of 2022 and 2023 and will be running a third event this summer in June 2024.
In July 2021 I organised a program at ICTS in Bangalore on Quantum Fields, Geometry and Representation Theory See the link for details and registration.
In January 2019 I organised a conference on Higher Structures in Holomorphic and Topological Field Theory, together with Vasily Pestun.
In December 2014 I organised a workshop on Mathematical Aspects of Six-Dimensional Quantum Field Theory, together with David Nadler, Harrison Chen, Kevin Wray and Alex Zorn. Videos for the lectures are now available here.
In January 2012 I organised a workshop with Sam Gunningham and David Nadler on Gauge Theory and Floer Theory. 2012 Northwestern Masterclass in Gauge Theory

Seminars

From 2019 to 2022 I organised the UMass Representation Theory Seminar together with Tina Kanstrup.

In March-July 2020 I organised an online working seminar on Quantum field theory, representation theory and geometry together with Brian Williams. If you would like to join the mailing list for announcements of any future lectures, click here to sign up.

In January-March 2015 I co-organised a learning seminar on Instantons and the Nekrasov Partition Function with Philsang Yoo and Brian Williams.

In January-March 2014 I co-organised a reading seminar on papers in geometric representation theory with Aron Heleodoro. We read and discussed important papers from geometric representation theory in the last forty years.

In October-November 2013 I co-organised a learning seminar on String Topology with Brian Williams, focusing on string topology as part of a partially extended 2d topological field theory.

In April-June 2013 I co-organised a learning seminar on S-duality with Kevin Costello.

In February 2012 I organised a preseminar for the Simons Center Workshop on SUSY Gauge Theory. Hiro Tanaka compiled a partial list of notes from the participant talks here.

Lecture Series

In 2020 I recorded a lecture series on super Yang--Mills in the BV formalism, together with Brian Williams. You can find the lecture notes, videos and some exercises here.

Notes I've Written

In November 2021 I gave a talk entitled Framing Anomalies for Topological AKSZ Theories at the University of Durham.
In March 2021 I gave a talk entitled What is Supersymmetry? for "The What Is Graduate Seminar" (TWIGS) at UMass Amherst.
In March 2021 I gave a talk on Gauge Symmetry Breaking via Derived Geometry at Nottingham University.
In January 2021 I gave a talk on Geometric Langlands for Physicists as part of a joint Math/Physics department seminar at UMass (notes and video recordings for this lecture and later lectures in the series are available upon request).
In July 2019 I gave a pair of expository talks at QFT for Mathematicians at the Perimeter Institute, on Supersymmetry Algebras, and on Yang-Mills Theory and Asymptotic Freedom.
In May 2018 I gave a pair of lectures on Topological Twists of SUSY Factorization Algebras at IST, Austria.
In May 2017 I gave a talk on Vacua and Singular Supports at the Perimeter Institute.
In February 2016 I gave a talk on holomorphic Chern-Simons and (Super) Twistor Theory for a seminar on Reshetikhin-Turaev Invariants at Northwestern.
In October 2014 I gave a talk on Perturbative and Non-Perturbative Twists of Classical Field Theories at the Simons Center (video).
In October 2013 I gave a talk on the Feynman Path Integral in the Northwestern Math Graduate Student Seminar.
In April 2013 I gave a talk on Abelian Duality for Generalised Maxwell Theories at the GRASP Seminar in Berkeley.
In March 2013 I gave an expository talk on Topological Quantum Field Theory at the Northwestern Physics Brownbag Seminar.
In October 2012 I gave a talk on Dirac Quantisation at the Northwestern Math Graduate Student Seminar.
In August 2012 I gave a talk on TQFTs from Quasicoherent Sheaves on Stacks at the Categorical Representation Theory Workshop at the University of Oregon.
In March 2012 I gave a talk on Supersymmetric Lagrangians at the Simons Center Workshop on Susy Gauge Theory (video).
In May 2011 I gave a talk on The Non-abelian Hodge Correspondence for Non-compact Curves at the Talbot Workshop on Non-abelian Hodge Theory.
Notes on Symmetries in Conformal Field Theory prepared for an informal seminar at Northwestern.
Notes on Integrable Systems, the Classical Yang-Baxter Equation and the Toda Lattice prepared for a class on quantum groups at Northwestern.
Notes on The Geometry of Affine Grassmannians prepared for an informal seminar at Northwestern.
Notes on Motives and Motivic L-functions prepared for a seminar class at Northwestern.
My Cambridge part III essay on D-modules.
I have a set of notes on character sheaves which some people have found useful, but they're incomplete. E-mail me if you'd like a copy.

My notes from the 2012 Notre Dame Graduate Summer School on Topology and Field Theories.

Amherst College Mathematics and Statistics home page.