This biweekly online colloquium features geometers and physicists presenting current research on a wide range of topics in the interface of the two fields. The talks are aimed at a broad audience. They will take place via Zoom on alternate Mondays at 3pm Eastern, noon Pacific, 4pm BRT. Each session features a 60 minute talk, followed by 15 minutes for questions and discussion. You may join the meeting 15 minutes in advance. Questions and comments may be submitted to the moderator via the chat interface during the talk, or presented in person during the Q&A session. These colloquia will be recorded and will be available (linked from this page) asap after the event.
As an alternative to Zoom, you may watch a live stream of the lecture at our: YouTube streaming site.
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Organizing committee: Tudor Dimofte, Ron Donagi, Dan Freed, Sheldon Katz, Dave Morrison, Andy Neitzke.
(Indexed at researchseminars.org.)
Go to Past Talks (2020). Past Talks (2021)May 17, 2021 
Richard Thomas (Imperial College)Nonabelian DT theory from abelian DT theoryAbstract: Fix a CalabiYau 3fold X. Its DT invariants count stable bundles and sheaves on X. Joyce's generalised DT invariants count semistable sheaves on X. I will describe ongoing work with Soheyla Feyzbakhsh with the eventual aim of writing the generalised DT invariants in any rank r in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X. Along the way we express rank r DT invariants in terms of rank 0 invariants counting D4D2D0 branes. These invariants are predicted by Sduality to be governed by (vectorvalued mock) modular forms. Based partly on arXiv:2007.03037 and arXiv:2103.02915 .

June 7, 2021 
Zohar Komargodski (SCGP, Stony Brook)

June 21, 2021 
no meeting, due to Strings 2021

July 12, 2021 
Sarah Harrison (McGill)

July 26, 2021 

January 11, 2021 
Mike Freedman (Microsoft Research/UCSB)Quantum Codes and Systolic freedomAbstract: In work with Hastings we find a twoway street between quantum error correcting codes and Riemannian manifolds. A recent advance in coding theory allows us to produce the first example of a manifolds with Z_2power lawsystolic freedom. Specifically we find, for any e>0, a sequence of appropriately scaled 11D Riemannian manifolds M_i, so that for any dual 4 and 7 dimensional Z_2cycles, X_i and Y_i, resp. $ Vol_4(X_i)*Vol_7(Y_i) > (Vol_11(M_i))^(5/4e) $. Video of lecture OR Video of lecture (alternate version, same content as the other)

January 25, 2021 
Albrecht Klemm (Bonn)CalabiYau modularity and Feynman GraphsAbstract: Using the GKZ system for the primitive cohomology of an infinite series of complete intersection CalabiYau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana integrals with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel \hat bclass evaluation in the ambient spaces of the mirror, while the imaginary part of the amplitude in this regime is determined by the Γbclass of the mirror CalabiYau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κconstants, which determine the behaviour of the amplitudes, when the momentum square equals the sum of the masses squared, in terms of zeta values. We find the exact differential equation for the graph integrals with arbitrary value for the dimensional regularisation (d\epsilon) parameter and extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogenous) PicardFuchs differential ideal for arbitrary masses. Using a recent padic analysis of the periods we determine the value of the maximal cut equal mass fourloop amplitude at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins. Video of lecture OR Video of lecture (alternate version, same content as the other) 
February 8, 2021 
Nikita Nekrasov (SCGP, Stony Brook)Beauty of the defectsAbstract: Surface and pointlike defects in supersymmetric gauge theories in four dimensions are studied with applications to quantum/classical correspondence. In particular, the GIL formula for the taufunction of Painleve VI is explained using the blowup method in the context of the BPS/CFT correspondence, while the spin chain generalisation of KharchevLebedev wavefunction of periodic Toda chain is obtained via wallcrossing. Based on several works, in particular on papers in collaboration with Saebyeok Jeong; Norton Lee; Oleksandr Tsymbaliuk; as well as S. Jeong and N. Lee Video of lecture OR Video of lecture (alternate version, same content as the other) 
February 22, 2021 
Tamas Hausel (IST Austria)Mirror symmetry for Langlands dual Higgs bundles at the tip of the nilpotent coneAbstract: I will explain what we can prove and what we conjecture about the mirror of Hecke transformed Hitchin section motivated by symmetry ideas of KapustinWitten. The talk is based on arXiv:2101.08583 joint with Hitchin. Video of lecture OR Video of lecture (alternate version, same content as the other) 
March 8, 2021 
Thomas Dumitrescu (UCLA)2Group Global Symmetry in Quantum Field TheoryAbstract: Higherform generalizations of global symmetries play an important role in Quantum Field Theory (QFT). In general, symmetries of different form degrees need not be independent; instead, they can form a higher group. In this talk I will illustrate this phenomenon by explaining why many simple Lagrangian QFTs in four and six dimensions enjoy 2group global symmetries. I will then apply this understanding to deduce new general results about (typically nonLagrangian) SCFTs in six dimensions. Video of lecture OR Video of lecture (alternate version, same content as the other) 
March 22, 2021 
David BenZvi (U. Texas, Austin)ElectricMagnetic Duality between Periods and LfunctionsAbstract: I will describe joint work with Yiannis Sakellaridis and Akshay Venkatesh, in which ideas originating in quantum field theory are applied to a problem in number theory. A fundamental tool in number theory, the relative Langlands program, is centered on the representation of Lfunctions of Galois representations as integrals of automorphic forms. However, the data that naturally index these period integrals (spherical varieties for a reductive group G) and the Lfunctions (representations of the Langlands dual group G^) don't seem to line up, making the search for integral representations somewhat of an art. We present an approach to this problem via the KapustinWitten interpretation of the [geometric] Langlands correspondence as electricmagnetic duality for 4dimensional supersymmetric gauge theory. Namely, we rewrite the relative Langlands program as duality in the presence of boundary conditions. As a result the partial correspondence between periods and Lfunctions is embedded in a natural duality between Hamiltonian actions of the dual groups. Video of lecture OR Video of lecture (alternate version, same content as the other) 
April 5, 2021 
Arnav Tripathy (Harvard)K3 metricsIt has long been an open problem to explicitly produce a Ricciflat metric on a (nontoroidal) compact manifold. I'll discuss two approaches to this problem, related by a version of 3d mirror symmetry, for K3 manifolds. This is joint work with M. Zimet. Video of lecture OR Video of lecture (alternate version, same content as the other) 
April 19, 2021 
Cody Long and Cumrun Vafa (Harvard)NonHolomorphic Cycles and NonBPS Black BranesAbstract: We discuss extremal nonBPS black holes and strings arising in Mtheory compactifications on CalabiYau threefolds, obtained by wrapping M2 branes on nonholomorphic 2cycles and M5 branes on nonholomorphic 4cycles. Using the attractor mechanism we compute the black hole mass and black string tension, leading to a conjectural formula for the asymptotic volumes of connected, locally volumeminimizing representatives of nonholomorphic, evendimensional homology classes in the threefold, without knowledge of an explicit metric. In the case of divisors we find examples where the volume of the representative corresponding to the black string is less than the volume of the minimal piecewiseholomorphic representative, predicting recombination for those homology classes and leading to stable, nonBPS strings. We also show how to compute the central charges of nonBPS strings in Ftheory via a nearhorizon AdS3 limit in 6d which, upon compactification on a circle, account for the asymptotic entropy of extremal nonsupersymmetric 5d black holes (i.e., the asymptotic count of nonholomorphic minimal 2cycles). Video of lecture (starts late, unfortunately) OR Video of lecture (alternate version, same content as the other, but starts on time) 
May 3, 2021 
Fei Yan (Rutgers)Two tales of networks and quantizationAbstract: I will describe two quantization scenarios. The first scenario involves the construction of a quantum trace map computing a new link "invariant" (with possible wallcrossing behavior) for links L in a 3manifold M, where M is a Riemann surface C times a real line. This construction computes familiar link invariants in a new way, moreover it unifies that computation with the computation of protected spin characters counting ground states with spin for line defects in 4d N=2 theories of classS. Certain networks on C play an important role in the construction. The second scenario concerns the study of Schroedinger equations and their higher order analogues, which could arise in the quantization of SeibergWitten curves in 4d N=2 theories. Here similarly certain networks play an important part in the exact WKB analysis for these Schroedingerlike equations. At the end of my talk I will also try to sketch a possibility to bridge these two quantization scenarios. The first part of the talk is based on joint work with A. Neitzke; the final sketch is based on discussions with D. Gaiotto, G. Moore and A. Neitzke. Video of lecture OR Video of lecture (alternate veureion, same content as the other) 
April 13, 2020 
Edward Witten (IAS)Volumes and Random MatricesAbstract: I will describe recent results relating twodimensional gravity and supergravity; volumes of moduli spaces of Riemann surfaces and super Riemann surfaces; and random matrix ensembles. See https://arxiv.org/abs/1903.11115 by Saad, Shenker, and Stanford; https://arxiv.org/abs/1907.03363 by Stanford and me. 
April 27, 2020 
Kevin Costello (Perimeter Institute)Topological strings, twistors, and SkyrmionsAbstract: It has long been known that holomorphic field theories on twistor space lead to "physical" field theories on Minkowski space. In this talk I will discuss a type I (unoriented) version of the topological B model on twistor space. The corresponding theory on Minkowski space is a sigmamodel with target the group SO(8). This is a variant of the Skyrme model that appears as the lowenergy effective theory of mesons in QCD. (The group SO(8) appears because of the GreenSchwarz mechanism in the topological string). The origin of this model in the topological string implies many remarkable properties. For one thing, the model is, in a certain sense, integrable. Further, although the Lagrangian is powercounting nonrenormalizable, counterterms at all loops can be uniquely fixed. 
May 11, 2020 
Mark Gross (Cambridge)Intrinsic Mirror SymmetryAbstract: I will talk about joint work with Bernd Siebert, proposing a general mirror construction for log CalabiYau pairs, i.e., a pair (X,D) with D a "maximally degenerate" boundary divisor and K_X+D=0, and for maximally unipotent degenerations of CalabiYau manifolds. We accomplish this by constructing the coordinate ring or homogeneous coordinate ring respectively in the two cases, using certain kinds of GromovWitten invariants we call "punctured invariants", developed jointly with Abramovich and Chen. 
May 18, 2020 
Miranda Cheng (Univ. of Amsterdam/National Taiwan University)Quantum Modularity from 3ManifoldsAbstract: Quantum modular forms are functions on rational numbers that have rather mysterious weak modular properties. Mock modular forms and false theta functions are examples of holomorphic functions on the upperhalf plane which lead to quantum modular forms. Inspired by the 3d3d correspondence in string theory, a new topological invariants named homological blocks for (in particular plumbed) threemanifolds have been proposed a few years ago. My talk aims to explain the recent observations on the quantum modular properties of the homological blocks, as well as the relation to logarithmic vertex algebras. The talk will be based on a series of work in collaboration with Sungbong Chun, Boris Feigin, Francesca Ferrari, Sergei Gukov, Sarah Harrison, and Gabriele Sgroi. 
June 1, 2020 
Davide Gaiotto (Perimeter Institute)Integrable Kondo problems and affine Geometric LanglandsAbstract: I will present some work on integrable line defects in WZW models and their relation to 4d CS theory, the IM/ODe correspondence and affine generalizations of Geometric Langlands constructions. 
June 15, 2020 
Maxim Kontsevich (IHES)Spacetime analyticity in QFTAbstact: I will talk on a joint work with Graeme Segal. We propose a new axiomatics for unitary quantum field theory which includes both Lorentzian and Euclidean signatures for curved spacetime manifolds. The key to the definition is certain open domain in the space of complexvalued symmetric bilinear forms on a real vector space. The justification comes from holomorphic convexity (lower bound) and from higher gauge theories (upper bound). 
June 22, 2020 
Anton Kapustin (Cal Tech)From gapped phases of matter to Topological Quantum Field Theory and back againAbstract: I will review the connection between gapped phases of matter and Topological Quantum Field Theory (TQFT). Conjecturally, this connection becomes 11 correspondence if one restricts to a special class of phases and TQFTs (namely, invertible ones). A related conjecture is that the space of all lattice Hamiltonians describing ShortRange Entangled phases of matter is an infinite loop space. These conjectures predict that the space of lattice Hamiltonians has nontrivial cohomology in particular dimensions. We test this by constructing closed differential forms on the space of gapped lattice Hamiltonians following a suggestion by Kitaev. These differential forms can be regarded as a highercategorical generalization of the curvature of the Berry connection and correspond to WessZuminoWitten forms in field theory. 
June 29, 2020 
No meeting due to Strings 2020

July 6, 2020 
Nima ArkaniHamed (IAS)Spacetime, Quantum Mechanics and Clusterhedra at InfinityAbstract: Elementary particle scattering is perhaps the most basic physical process in Nature. The data specifying the scattering process defines a "kinematic space", associated with the onshell propagation of particles out to infinity. By contrast the usual approach to computing scattering amplitudes, involving path integrals and Feynman diagrams, invokes auxilliary structures beyond this kinematic spacelocal interactions in the interior of spacetime, and unitary evolution in Hilbert space. This description makes spacetime locality and quantummechanical unitarity manifest, but hides the extraordinary simplicity and infinite hidden symmetries of the amplitude that have been uncovered over the past thirty years. The past decade has seen the emergence of a new picture, where scattering amplitudes are seen as the answer to an entirely different sort of mathematical question involving "positive geometries" directly in the kinematic space, making surprising connections to total positivity, combinatorics and geometry of the grassmannian, and cluster algebras. The hidden symmetries of amplitudes are made manifest in this way, while locality and unitarity are seen as derivative notions, arising from the "factorizing" boundary structure of the positive geometries. This was first see in the story of "amplituhedra" and scattering amplitudes in planar N=4 SYM theory. In the past few years, a similar structure has been seen for nonsuperysmmetric "biadjoint" scalar theories with cubic interactions, in any number of dimensions. The positive geometries through to oneloop order are given by "cluster polytopes"generalized associahedra for finitetype cluster algebraswith a simple description involving "dynamical evolution" in the kinematic space. Extending these ideas involves understanding cluster algebras associated with triangulations of general Riemann surfaces. These cluster algebras are infinite, reflecting the infinite action of mapping class group. One of the manifestations of this infinity is that the "gvector fan" of the cluster algebra is not spacefilling, making it impossible to define cluster polytopes, and obstructing the connection with positive geometries and scattering amplitudes. Remarkably, incorporating noncluster variables, associated with closed loops in the Riemann surfaces, suggests a natural way of modding out by the mapping class group, canonically compactifying the cluster complex, and associating it with "clusterhedron" polytopes. Clusterhedra are conjectured to exist for all surfaces, providing the positive geometry in kinematic space for scattering amplitudes in the biadjoint scalar theory to all loop orders and all orders in the 1/N expansion. In this talk I will give a simple, selfcontained overview of this set of ideas, assuming no prior knowledge of scattering amplitudes or cluster algebras.

July 13, 2020 
Mina Aganagic (UC Berkeley)Knot categorification from mirror symmetry, via string theoryAbstract: I will describe two approaches to categorifying quantum link invariants which work uniformly for all simple Lie algebras, and originate from geometry and string theory. A key aspect of both approaches is that it is manifest that decategorification gives the quantum link invariants one set out to categorify. Many ingredients that go into the story have been found by mathematicians earlier, but string theory spells out how they should be put together for a uniform framework for knot categorification. The first approach is based on derived categories of coherent sheaves on resolutions of slices in affine Grassmannians. Some elements of it have been discovered by mathematicians earlier and others are new. The second approach is perhaps more surprising. It uses symplectic geometry and is related to the first by two dimensional (equivariant) mirror symmetry. Unlike previous symplectic geometry based approaches, it produces a bigraded homology theory. In both cases, mirror symmetry, and techniques developed to understand it play a crucial role. I will explain the string theory origin of the two approaches, and the relation to another string theory based approach, due to Witten. 
July 20, 2020 
Greg Moore (Rutgers)Breaking News About, Topologically Twisted Rank One N=2* Supersymmetric YangMills Theory On FourManifolds, Without SpinAbstract: I will report on work in progress with Jan Manschot. We generalize previous results concerning a topological theory in four dimension that generalizes both the Donaldson invariants and the VafaWitten invariants. In contrast with previous studies we include an arbitrary background spinc structure with connection. The Coulomb branch measure involves nonholomorphic topological couplings to the background spinc connection. (This violates some folklore). Using some novel identities for the $N=2*$ prepotential, the Coulomb branch integral can be evaluated explicitly using the theory of mock modular and Jacobi forms. For $b_2^+>1$ the path integral can be written explicitly in terms of SeibergWitten invariants and modular functions of the ultraviolet coupling. We discuss the orbit of partition functions of the three rank one $N=2*$ theories under the action of Sduality. Video of Lecture 
July 27, 2020 
No meeting due to StringMath 2020

August 3, 2020 
Sakura SchaferNameki (Oxford)5d SCFTs: Symmetries and Moduli SpacesAbstract: I will report on recent developments in 5d SCFTs, studying their global symmetries, 0 and higherform, Mtheory on a canonical singularity. We provide a geometric characterization of the Coulomb and Higgs branch moduli spaces and connect this to recent work on magnetic quivers in 3d. 
August 10, 2020 
Ben Webster (Waterloo)3d mirror symmetry and its discontentsAbstract: One of the central topics of the interaction between QFT and math is mirror symmetry for 2d theories. This theory has a more mysterious and exotic friend one dimension higher, sometimes called 3d mirror symmetry, which relates two 3dimensional theories with N=4 supersymmetry. For roughly a decade, I struggled to understand this phenomenon without understanding what most of the words in the previous sentence meant. Eventually, I wised up and based on work of Braverman, Finkelberg, Nakajima, Dimofte, Gaiotto, Hilburn and others, I actually did learn a little bit, and will now try to explain to you what I learned. This knowledge has some interesting payoffs in the mathematics related to 3d theories, such as an understanding of Bezrukavnikov and Kaledin's noncommutative resolutions of the Coulomb branch, and explaining a lot of interesting Koszul dualities between category O's. 
August 17, 2020 
Marco Gualtieri (Univ. of Toronto)Branes in symplectic groupoidsAbstract: After reviewing coisotropic Abranes in symplectic manifolds and their role in mirror symmetry and geometric quantization, I will explain how the problem of holomorphic quantization of Poisson brackets may be recast, and in some cases solved, as a problem of computing morphisms between coisotropic branes in symplectic groupoids. This is joint work with Francis Bischoff and Joshua Lackman. 
August 24, 2020 
Laura Fredrickson (Stanford/U. Oregon)The asymptotic geometry of the Hitchin moduli spaceAbstract: Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmuller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkahler metric. An intricate conjectural description of its asymptotic structure appears in the work of physicists GaiottoMooreNeitzke and there has been a lot of progress on this recently. I will discuss some recent results using tools coming out of geometric analysis which are wellsuited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures. 
September 14, 2020 
David Jordan (Edinburgh)Quantum geometric Langlands as a fully extended TFTI will survey several recent works realizing Betti geometric Langlands and its quantization as fully extended TFT's. In physics terms this is most closely related to the KapustinWitten twist of N=4 d=4 SYM at generic values of \Psi. I will outline numerous applications to quantum topology, most notably to a conjecture of Witten on finitedimensionality of skein modules, a conjecture of BonahonWong concerning skein modules at rootofunity parameters, a proposal of BenZvi concerning cluster varieties and factorization homology, and an appearance of the typeA spherical DAHA from a oncemarked torus. Finally, I will present a novel conjectural appearance of Sduality in the study of skein modules of 3manifolds. 
September 21, 2020 
Lauren Williams (Harvard)How is the hypersimplex related to the amplituhedron?Abstract: In 1987, GelfandGoreskyMacPhersonSerganova made a beautiful connection between the geometry of the Grassmannian and convex polytopes, via the moment map; the moment map image of the Grassmannian Gr(k,n) is a polytope known as the hypersimplex Delta(k,n). In 2013, motivated by the desire to give a geometric basis for the computation of scattering amplitudes in N=4 SYM, ArkaniHamed and Trnka introduced the amplituhedron A(n,k,m) as the image of the positive Grassmannian Gr+(k,n) under a linear map Z from R^n to R^{k+m} which is totally positive. While the case m=4 is most relevant to physics, the amplituhedron makes sense for any m. In my talk I will explain some strange parallels between the positroidal subdivisions of the hypersimplex Delta(k+1,n) and the m=2 amplituhedron A(n,k,2). One link is provided by the positive tropical Grassmannian. Attributions: based on joint works with Tomek Lukowski, Matteo Parisi, and David Speyer. Disclaimer: I'm neither a geometer nor a physicist. 
October 5, 2020 
Natalie Paquette (IAS)BorcherdsKacMoody algebras, 2d strings, & other curiositiesAbstract: In this talk we will extol the virtues of compactifying critical string theory down to few noncompact spacetime dimensions (particularly two). These string vacua possess rich groups of dualities. BPSsaturated quantities, which mathematically are described by automorphic forms, are invariant under such duality transformations. Further, such BPS states can furnish representations of interesting algebras, such as infinitedimensional Lie algebras. In this talk, we explore some particularly nice, concrete examples, which employ holomorphic super vertex operator algebras in our `compactification' theories. The BPS states in these models organize into representations of algebras, which we prove are (new) examples of BorcherdsKacMoody superalgebras. Video of Lecture 
October 19, 2020 
Djordje Radicevic (Brandeis)The LatticeContinuum Correspondence in Quantum MechanicsAbstract: It is very well known that longdistance correlation functions of many lattice systems can be calculated from continuum QFTs. Making this correspondence more precise  identifying continuum operators that correspond to individual lattice operators, or exhibiting the lattice origins of subtler continuum phenomena like operator product expansions  has proven quite daunting. In this talk, I will report on recent progress in this direction, using quantum mechanics (QFT in 0+1 dimensions) as an example. I will show how a finite but large quantum system can be systematically reduced to an Ersatz continuum theory, using both Hamiltonian and path integral formalisms. Along the way I will point out the lattice origins of several familiar continuum concepts, including contact terms, scale invariance, and the distinction between compact and noncompact theories. I will also stress the limitations imposed on the emergent continuum theory by its lattice progenitor  for instance, any supersymmetric continuum theory emerging from a finite theory must have a vanishing Witten index. Video of Lecture 
November 2, 2020 
Christopher Beem (Oxford)HallLittlewood Chiral Rings and Derived Higgs BranchesAbstract: I will discuss a relatively novel algebraic structure arising in fourdimensional N=2 superconformal field theories: the HallLittlewood Chiral Ring (HLCR). The HLCR is an enhancement of the more familiar Higgs branch chiral ring (which encodes the Higgs branch of the moduli space of vacua as an algebraic variety). The HLCR in gauge theories is constructed as the cohomology of a kind of BRST complex, which allows it to be identified with the ring of functions on the derived Higgs branch (in the sense of derived algebraic geometry). I will describe the solution of the HLCR cohomology problem for a large class of Lagrangian theories (the class S theories of type A1), which illustrate some interesting phenomena. This talk is based on work in progress with Diego Berdeja Suárez. Video of Lecture 
November 16, 2020 
Sergei Gukov (Cal Tech)Rokhlin, quantum groups, and BPS statesAbstract: What do ADO polynomials, cobordism invariants, and affine Grassmannians have in common? We will discuss how these seemingly different objects can be put under one roof of a BPS qseries that, on the one hand, can be thought of as a 3d analogue of the VafaWitten partition function and, on the other hand, is associated to quantum groups at generic q where Verma modules with arbitrary complex weights play an important role. Video of Lecture 
November 30, 2020 
Sourav Chatterjee (Stanford)Strong mass gap implies quark confinementAbstract: I will show that if a lattice gauge theory has exponential decay of correlations under arbitrary boundary conditions (which I call strong mass gap), and the gauge group has a nontrivial center, then Wilson's area law holds. Video of Lecture OR Video of Lecture (alternate version, same content as the other) 
December 14, 2020 
Harold Williams (USC)Canonical Bases for Coulomb BranchesAbstract: Following work of KapustinSaulina and GaiottoMooreNeitzke, one expects halfBPS line defects in a 4d N=2 field theory to form a monoidal category with a rich structure. In general, a mathematical definition of this category is not known. In this talk we discuss an algebrogeometric proposal in the case of gauge theories with polarizable matter. The proposed category is the heart of a nonstandard tstructure on the dg category of coherent sheaves on the derived BravermanFinkelbergNakajima space of triples. We refer to its objects as Koszulperverse coherent sheaves, as this tstructure interpolates between the perverse coherent tstructure and certain tstructures appearing in the theory of Koszul duality (specializing to these in the case of a pure gauge theory and an abelian gauge theory, respectively). As a byproduct, this defines a canonical basis in the associated quantized Coulomb branch by passing to classes of irreducible objects. This is joint work with Sabin Cautis. Video of Lecture OR Video of Lecture (alternate version, same content as the other) 