Mathematical physics, including mathematics, is a research area where novel mathematical techniques are invented to tackle problems in physics, and where novel mathematical ideas find an elegant physical realization. Historically, it would have been impossible to distinguish between theoretical physics and pure mathematics. Often spectacular advances were seen with the concurrent development of new ideas and fields in both mathematics and physics. Here one might note Newton's invention of modern calculus to advance the understanding of mechanics and gravitation. In the twentieth century, quantum theory was developed almost simultaneously with a variety of mathematical fields, including linear algebra, the spectral theory of operators and functional analysis. This fruitful partnership continues today with, for example, the discovery of remarkable connections between gauge theories and string theories from physics and geometry and topology in mathematics.
Format results

11 talksCollection NumberC20029
Talk

Projective elliptic genera and applications
National University of Singapore 
Topological Modular Forms and Quantum Field Theory
Perimeter Institute for Theoretical Physics 
Equivariant elliptic cohomology with integral coefficients
Utrecht University 
The de Rham model for elliptic cohomology from physics
Harvard University 
Quasisymmetric characteristic numbers for Hamiltonian toric manifolds
Johns Hopkins University 
Codes, vertex algebras and topological modular forms
Ruhr University Bochum 

Elliptic characteristic classes, bow varieties, 3d mirror duality
University of North Carolina  Chapel Hll


PSI 2019/2020  Statistical Physics (Kubiznak)
3 talksCollection NumberC19036Talk

PSI 2019/2020  Statistical Physics  Lecture 1
Charles University 
PSI 2019/2020  Statistical Physics  Lecture 2
Charles University 
PSI 2019/2020  Statistical Physics  Lecture 3
Charles University


QFT for Mathematicians
25 talksCollection NumberC19023Talk

Lecture 1: Factorization Algebras and the General Structure of QFT
Seoul National University 
Lecture 1: Supersymmetric Quantum Mechanics and All That
Durham University 
TA Session: 0d QFT and Feynman diagrams
Dalhousie University 
Lecture 1: Boundary Conditions and Extended Defects
Perimeter Institute for Theoretical Physics 
Lecture 2: Factorization Algebras and the General Structure of QFT
Perimeter Institute for Theoretical Physics 
TA Session: Supersummetry Algebras
University of Massachusetts Amherst 
Lecture 3: Factorization Algebras and the General Structure of QFT
Seoul National University 
Lecture 2: Supersymmetric Quantum Mechanics and All That
Durham University


Cohomological Hall Algebras in Mathematics and Physics
19 talksCollection NumberC19018Talk

An introduction to Cohomological Hall algebras and their representations
Kansas State University 
Gauge theory, vertex algebras and COHA
Perimeter Institute for Theoretical Physics 
Networks of intertwiners, 3d theories and superalgebras
University of Edinburgh 
COHA of surfaces and factorization algebras
University of Tokyo 
Yangians from cohomological Hall algebras
University of Edinburgh 
Algebraic structures of T[M3] and T[M4]
California Institute of Technology (Caltech)  Division of Physics Mathematics & Astronomy 
Categorification of 2d cohomological Hall algebras
University of Tokyo 
Short starproducts for filtered quantizations
Massachusetts Institute of Technology (MIT)


Topological Holography Course (Costello)
8 talksCollection NumberC19017Talk

Topological Holography Course  Lecture 1
Perimeter Institute for Theoretical Physics 
Topological Holography Course  Lecture 2
Perimeter Institute for Theoretical Physics 
Topological Holography Course  Lecture 3
Perimeter Institute for Theoretical Physics 
Topological Holography Course  Lecture 5
Perimeter Institute for Theoretical Physics 
Topological Holography Course  Lecture 6
Perimeter Institute for Theoretical Physics 
Topological Holography Course  Lecture 7
Perimeter Institute for Theoretical Physics 
Topological Holography Course  Lecture 8
Perimeter Institute for Theoretical Physics 
Topological Holography Course  Lecture 9
Perimeter Institute for Theoretical Physics


Higher Algebra and Mathematical Physics
21 talksCollection NumberC18024Talk


N=1 supersymmetric vertex algebras of small index
Perimeter Institute for Theoretical Physics 
Geometric Langlands: Comparing the views from CFT and TQFT
Deutsches ElektronenSynchrotron DESY 
Cutting and gluing branes
University of California, Berkeley 
The lowenergy TQFT of the generalized double semion model
University of Texas  Austin 

Moduli of connexions on open varieties
Paul Sabatier University 
The Duistermaat–Heckman distribution for the based loop group
University of Toronto


Gauge Theory, Geometric Langlands and Vertex Operator Algebras
11 talksCollection NumberC18004Talk

Gauge Theory, Geometric Langlands, and All That
Institute for Advanced Study (IAS)  School of Natural Sciences (SNS) 
Overview of the global Langlands correspondence
University of WisconsinMilwaukee 
Gauge theory, vertex algebras and quantum Geometric Langland dualities
Perimeter Institute for Theoretical Physics 

Introduction to local geometric Langlands
The University of Texas at Austin 




Talk

Semisimple Hopf algebras and fusion categories
Universidad de los Andes 
The Hopf C*algebraic quantum double models  symmetries beyond group theory
Freie Universität Berlin 
Modular categories and the Witt group
Radboud Universiteit Nijmegen 
Topological Quantum Computation
Texas A&M University 
Gapped phases of matter vs. Topological field theories
Perimeter Institute for Theoretical Physics 
An Introduction to Hopf Algebra Gauge Theory
University of ErlangenNuremberg 
Kitaev lattice models as a Hopf algebra gauge theory
University of ErlangenNuremberg 
Topological defects and highercategorical structures
Karlstad University


Quantum Field Theory on Manifolds with Boundary and the BV Formalism
12 talksCollection NumberC17013Talk

Perturbative BVBFV theories on manifolds with boundary
University of Zurich 
Gactions in quantum mechanics (and spectral sequences and the cosmological constant)
University of Edinburgh 
Perturbative BVBFV theories on manifolds with boundary Part 2
University of Zurich 
Degenerate Field Theories and Boundary Theories
Seoul National University 
Bulkboundary BV quantization for 21 theories
Boston University 
A link between AdS/CFT and Koszul duality
Perimeter Institute for Theoretical Physics 
Poisson Sigma Model with symplectic target
National Institute for Nuclear Physics 
Vertex algebras and BV master equation
Tsinghua University


String Theory for Mathematicians  Kevin Costello
4 talksCollection NumberC17014Talk

String Theory for Mathematicians  Lecture 1
Perimeter Institute for Theoretical Physics 
String Theory for Mathematicians  Lecture 2
Perimeter Institute for Theoretical Physics 
String Theory for Mathematicians  Lecture 3
Perimeter Institute for Theoretical Physics 
String Theory for Mathematicians  Lecture 7
Perimeter Institute for Theoretical Physics


Hitchin Systems in Mathematics and Physics
18 talksCollection NumberC17001Talk

Critical points and spectral curves
University of Oxford 
Generalizing Quivers: Bows, Slings, Monowalls
University of Arizona 
Holomorphic symplectic Morita equivalence and the generalized Kahler potential
University of Toronto 
Nahm transformation for parabolic harmonic bundles on the projective line with regular residues
Budapest University of Technology and Economics 
A mathematical definition of 3d indices
University of Edinburgh 
Perverse Hirzebruch ygenus of Higgs moduli spaces
Institute of Science and Technology Austria 
Motivic Classes for Moduli of Connections
University of Southern California 
BPS algebras and twisted character varieties
University of Edinburgh


Deformation Quantization of Shifted Poisson Structures
17 talksCollection NumberC16005Talk

Formal derived stack and Formal localization
Laboratoire de Physique Théorique, IRSAMC, Université Paul Sabatier 
An overview of derived analytic geometry
Institut de Mathématiques de Jussieu 
Categorification of shifted symplectic geometry using perverse sheaves
University of Oxford 
Shifted structures and quantization
University of Pennsylvania 
What is the Todd class of an orbifold?
University of Wisconsin–Madison 
Singular support of categories
University of WisconsinMilwaukee 
Symplectic and Lagrangian structures on mapping stacks
Universität Wien 
The Maslov cycle and the Jhomomorphism
Boston College


Elliptic Cohomology and Physics
11 talksCollection NumberC20029 
PSI 2019/2020  Statistical Physics (Kubiznak)
3 talksCollection NumberC19036PSI 2019/2020  Statistical Physics (Kubiznak) 

Cohomological Hall Algebras in Mathematics and Physics
19 talksCollection NumberC19018This workshop will bring together leading mathematicians and physicists interested in the Cohomological Hall algebra as it appears in the study of moduli spaces and in gauge and string theory.

Topological Holography Course (Costello)
8 talksCollection NumberC19017Topological Holography Course (Costello) 
Higher Algebra and Mathematical Physics
21 talksCollection NumberC18024Higher algebra has become important throughout mathematics physics and mathematical physics and this conference will bring together leading experts in higher algebra and its mathematical physics applications. In physics the term algebra is used quite broadly any time you can take two operators or fields multiply them and write the answer in some standard form a physicist will be happy to call this an algebra. Higher algebra is characterized by the appearance of a hierarchy of multilinear operations (e.g. A_infty and L_infty algebras). These structures can be higher categorical in nature (e.g. derived categories cosmology theories) and can involve mixtures of operations and cooperations (Hopf algebras Frobenius algebras etc.). Some of these notions are purely algebraic (e.g. algebra objects in a category) while others are quite geometric (e.g. shifted symplectic structures). An early manifestation of higher algebra in highenergy physics was supersymmetry. Supersymmetry makes quantum field theory richer and thus more complicated but at the same time many aspects become more tractable and many problems become exactly solvable. Since then higher algebra has made numerous appearances in mathematical physics both high and lowenergy. A telltale sign of the occurrence of higher structures is when classification results involve cohomology. Group cohomology appeared in the classification of condensed matter systems by the results of Wen and collaborators. Altland and Zirnbauer s "tenfold way" was explained by Kitaev using Ktheory. And Kitaev's 16 types of vortexfermion statistics were classified by spin modular categories. All these results were recently enhanced by the work of Freed and Hopkins based on cobordism theory. In high energy physics cohomology appears most visibly in the form of "anomalies". The ChernSimons anomaly comes from the fourth cohomology class of a compact Lie group and the 5brane anomaly is related to a certain cohomology class of the Spin group. The classification of conformal field theories involves the computation of all algebras objects in certain monoidal categories which is a type of nonabelian cohomology. Yet another important role for higher algebra in mathematical physics has been in the famous Langlands duality. Langlands duality began in number theory and then became geometry. It turned into physics when Kapustin and Witten realized geometric Langlands as an electromagnetic duality in cN=4 super YangMills theory. Derived algebra higher categories shifted symplectic geometry cohomology and supersymmetry all appear in Langlands duality. The conference speakers and participants drawn from both sides of the Atlantic and connected by live video streams will explore these myriad aspects of higher algebra in mathematical physics.

Gauge Theory, Geometric Langlands and Vertex Operator Algebras
11 talksCollection NumberC18004The workshop will explore the relation between boundary conditions in fourdimensional gauge theory the Geometric Langlands program and Vertex Operator Algebras.

Hopf Algebras in Kitaev's Quantum Double Models: Mathematical Connections from Gauge Theory to Topological Quantum Computing and Categorical Quantum Mechanics
18 talksCollection NumberC17029The Kitaev quantum double models are a family of topologically ordered spin models originally proposed to exploit the novel condensed matter phenomenology of topological phases for faulttolerant quantum computation. Their physics is inherited from topological quantum field theories, while their underlying mathematical structure is based on a class of Hopf algebras. This structure is also seen across diverse fields of physics, and so allows connections to be made between the Kitaev models and topics as varied as quantum gauge theory and modified strong complementarity. This workshop will explore this shared mathematical structure and in so doing develop the connections between the fields of mathematical physics, quantum gravity, quantum information, condensed matter and quantum foundations.

Quantum Field Theory on Manifolds with Boundary and the BV Formalism
12 talksCollection NumberC17013In the past five years their have a been number of significant advances in the mathematics of QFT on manifolds with boundary. The work of Cattaneo, Mnev, and Reshitihkinbeyond setting rigorous foundationshas led to many computable and salient examples. Similarly, the work of Costello (specifically projects joint with Gwilliam and Si Li) provides a framework (and deformation/obstruction) for the observable theory of such theories with boundary/defects. There are related mathematical advances: constructible factorization algebras and higher category theory as pioneered by Lurie and the collaboration of Ayala, Francis, and Tanaka. The goal of the workshop is to bring together the leading experts in this multifaceted subject.The structure of the workshop will be such as to maximize the exchange of knowledge and collaboration. More specifically, the morning sessions will consist of several lecture series, while the afternoons will be reserved for research working groups. The mornings will communicate the essential ideas and techniques surrounding bulkboundary correspondences and perturbative AKSZ theories on manifolds with boundary/corners. The afternoons will be research driven and focus on specific problems within the following realms: the interaction of renormalization with cutting/pasting, aspects of the AdS/CFT correspondence, cohomological approaches to gravity, and the observable/defect theory of AKSZ type theories.

String Theory for Mathematicians  Kevin Costello
4 talksCollection NumberC17014String Theory for Mathematicians  Kevin Costello 
Hitchin Systems in Mathematics and Physics
18 talksCollection NumberC17001Hitchin Systems in Mathematics and Physics

Deformation Quantization of Shifted Poisson Structures
17 talksCollection NumberC16005Deformation Quantization of Shifted Poisson Structures