Oct 24, 2022
Monday

09:15 AM  09:30 AM


Welcome

 Location
 MSRI: Simons Auditorium, Online/Virtual
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09:30 AM  10:30 AM


Overview of Some Appearances of Multivalued Solutions in Differential Geometry
Simon Donaldson (State University of New York, Stony Brook)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
An interesting development, over the past decade or so, has been the appearance of “multivalued” objects (such as functions, forms, spinors) in various branches of differential geometry. These include gauge theory, calibrated geometry and adiabatic limits of structures on manifolds. The developments extend more classical constructions in the case of Riemann surfaces. In the first part of the talk we review, in outline, some of the geometric and analytic aspects of these developments (related to other talks in this workshop). In the last part we focus in more detail on the adiabatic description of coassociative fibred G2 manifolds and research directions which these suggest.
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10:30 AM  11:00 AM


Break

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11:00 AM  12:00 PM


K3 Surfaces as GaugeTheoretic Moduli Spaces
Max Zimet (Stanford University)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
I will explain a novel construction of K3 surfaces as moduli spaces of singular equivariant instantons on a 4torus. This gives the first hyperKahler quotient construction of a compact nontoroidal manifold and yields explicit formulae for K3 metrics near torus orbifold limits. I will also describe a novel Fredholm theory — and its development using interesting ideas from microlocal analysis — for Laplacians and Dirac operators constructed out of these very singular connections which enables the construction. I will next introduce a variant of the DonaldsonUhlenbeckYau theorem that operates in this setting, some of its consequences (such as nonemptiness of our moduli spaces), and the novel notion of stability for these singular connections that enters into the theorem. This theorem is proved by studying the gradient flow equation for the YangMills functional, and I will describe the Fredholm theory for the heat operator that undergirds the proof of the short and longtime existence and regularity of the flow. While these results are all proved for singular connections on a 4torus, they are expected to generalize to enable the study of vast new classes of gaugetheoretic moduli spaces consisting of objects with severe singularities in codimension at least three. Based on joint work with Arnav Tripathy and Andras Vasy.
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12:00 PM  02:00 PM


Lunch

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02:00 PM  03:00 PM


VafaWitten Invariants of Projective Surfaces  Overview
Martijn Kool (Universiteit Utrecht)

 Location
 MSRI: Simons Auditorium, Online/Virtual
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 Abstract
On a fourmanifold underlying a complex smooth projective surface S, TanakaThomas gave an algebrogeometric definition of the SU(r) VafaWitten partition function. When S has a nonzero holomorphic 2form, the partition function is essentially topological with a rich modular structure. The algebrogeometric viewpoint provides new insights into VafaWitten theory such as (1) new mathematical verifications of Sduality, (2) Ktheoretic refinement, (3) higher rank expressions with relations to the RogersRamanujan continued fraction.
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03:00 PM  03:30 PM


Afternoon Tea

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03:30 PM  04:30 PM


Holomorphic Floer Theory and DonaldsonThomas Invariants
Pierrick Bousseau (University of Georgia)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
Holomorphic Floer theory is the analog of Floer theory for holomorphic symplectic manifolds. This topic has been recently explored, from several different perspectives, by KontsevichSoibelman and DoanRezchikov. In a different direction, I will in this talk present a conjectural picture relating holomorphic Floer theory of complex integrable systems to DonaldsonThomas invariants. In physical terms, we will discuss a relation between the BPS spectrum of an N=2 4dimensional field theory and the enumerative geometry of the corresponding SeibergWitten integrable system.
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04:30 PM  06:20 PM


Reception

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Oct 25, 2022
Tuesday

09:30 AM  10:30 AM


All Gravitational Instantons from Monopole Moduli Spaces
Sergey Cherkis

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
Initial examples of gravitational instantons of every type emerged from the study of monopoles. In this talk we aim to show that in fact all gravitational instantons can be realized as moduli spaces of monopoles (and YangMills instantons).
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10:30 AM  11:00 AM


Break

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11:00 AM  12:00 PM


Homological Link Invariants from Mirror Symmetry
Mina Aganagic (University of California, Berkeley)

 Location
 MSRI: Simons Auditorium, Online/Virtual
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The knot categorification problem is to find the theory that categorifies quantum link invariants which works uniformly with respect to the choice of a Lie algebra, and originates from geometry and physics. The solution comes from a new relation between homological mirror symmetry and representation theory. The symplectic geometry side of mirror symmetry is a theory generalizing Heegard–Floer theory. The generalization corresponds to replacing gl(11) by an arbitrary Lie algebra. The theories can be solved exactly.
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12:00 PM  02:00 PM


Lunch

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02:00 PM  03:00 PM


Existence and NonExistence Results of Z2 Harmonic 1Forms
Siqi He (Academy of Mathematics and Systems Science)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
Z2 harmonic 1forms was introduced by Taubes as the boundary appearing in the compactification of the moduli space of flat SL(2,C) connections. Although from gauge theory aspect, Z2 harmonic 1forms should exist widely, it is challenging to explicitly construct examples of them. Besides the curvature condition, there seems to have more obstruction of the existence of Z2 harmonic 1forms. In this talk, we will discuss a method to construct examples of Z2 harmonic 1forms. We will also discuss the relationship between Z2 harmonic 1forms and special Lagrangian geometry and present a nonexistence result based on the work of AbouzaidImagi.
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03:00 PM  03:30 PM


Afternoon Tea

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03:30 PM  04:30 PM


Holomorphic Floer Theory and the Fueter Equation
Aleksander Doan (University of Cambridge; University College London)

 Location
 MSRI: Simons Auditorium, Online/Virtual
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 Abstract
I will discuss an idea of constructing a category associated with a pair of holomorphic Lagrangians in a hyperkahler manifold, or, more generally, a manifold equipped with a triple of almost complex structures I,J,K satisfying the quaternionic relation IJ =JI= K. This category can be seen as an infinitedimensional version of the FukayaSeidel category associated with a Lefschetz fibration. While many analytic aspects of this proposal remain unexplored, I will argue that in the case of the cotangent bundle of a Lefschetz fibration, our construction recovers the FukayaSeidel category. This talk is based on joint work with Semon Rezchikov and has many connections to recent work of Bousseau.
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Oct 26, 2022
Wednesday

09:30 AM  10:30 AM


Computations of Instanton Knot Floer Homology with Local Coefficients
Tomasz Mrowka (Massachusetts Institute of Technology)

 Location
 MSRI: Simons Auditorium, Online/Virtual
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 Abstract
Instanton knot Floer homology admits an interesting deformation (unavailable without the knot). The deformation is a Floer homology with local coefficients. Though more complicated than the vanilla version they turn out to be easier to compute. We’ll describe some of computations of these modules. A key computation is the trivial nstrand braid in S1xS2, generalizing work of Ethan Street. This is joint work with Peter Kronheimer.
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10:30 AM  11:00 AM


Break

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11:00 AM  12:00 PM


Concentrating Local Solutions of the TwoSpinor SeibergWitten Equations
Gregory Parker (Stanford University)

 Location
 MSRI: Simons Auditorium, Online/Virtual
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 Abstract
Given a compact 3manifold $Y$ and a $\mathbb Z_2$harmonic spinor $(\mathcal Z_0, A_0,\Phi_0)$ with singular set $\mathcal Z_0$, I will explain a construction of a family of local solutions to the twospinor SeibergWitten equations parameterized by $\epsilon \in (0,\epsilon_0)$ on tubular neighborhoods of $\mathcal Z_0$. These solutions concentrate in the sense that the $L^2$norm of the curvature near $\mathcal Z_0$ diverges as $\epsilon\to 0$, and after renormalization they converge locally to the original $\mathbb Z_2$harmonic spinor. I will explain how these model solutions are used in a gluing construction showing a $\mathbb Z_2$harmonic spinor satisfying mild assumptions necessarily arises as the limit of a sequence of twospinor SeibergWitten solutions on Y.
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12:00 PM  02:00 PM


Lunch

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Oct 27, 2022
Thursday

09:30 AM  10:30 AM


Disconnected Gauge Groups and Categorical Symmetry
Mathew Bullimore (University of Durham)

 Location
 MSRI: Simons Auditorium, Online/Virtual
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 Abstract
There has been recent renewed interest in fourdimensional gauge theories with disconnected gauge groups. Such theories admit a rich spectrum extended topological operators generating global symmetries that go beyond the paradigm of groups and have a natural description in terms of higher fusion categories. I will present some of these ideas and comment on potential applications to geometric Langlands for disconnected groups and enumerative problems.
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10:30 AM  11:00 AM


Break

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11:00 AM  12:00 PM


Hyperkahler Metrics Near SemiFlat Limits
Laura Fredrickson (University of Oregon)

 Location
 MSRI: Simons Auditorium, Online/Virtual
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 Abstract
We make rigorous (a generalization of) the formalism of Gaiotto, Moore, and Neitzke for constructing hyperkahler manifolds near semiflat limits. In particular, we account for the effects of complicated (e.g., densely wall crossing) BPS spectra and singular fibers. This provides a general framework for proving results about GromovHausdorff collapse of hyperkahler manifolds to semiflat limits and completes the StromingerYauZaslow conception of mirror symmetry for hyperkahler manifolds at the level of hyperkahler geometry (as opposed to only constructing one complex structure). We characterize the dependence of the periods of the three canonical Kahler forms on the natural parameters of the construction, and in particular prove that for noncompact manifolds this dependence is affine linear. Specializing to the case of moduli spaces of weakly parabolic SU(2) Higgs bundles on a sphere with four punctures, we prove that this construction yields all such manifolds which are sufficiently close to the semiflat limit. This talk is based on joint work with Arnav Tripathy and Max Zimet.
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12:00 PM  02:00 PM


Lunch

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02:00 PM  03:00 PM


Counting Solutions of KapustinWitten Equations on a ThreeManifold Times a Line from Physics Dualities
Pavel Putrov (Abdus Salam International Centre for Theoretical Physics)

 Location
 MSRI: Simons Auditorium, Online/Virtual
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 Abstract
In my talk I will summarize results about some explicit physics predictions of counts of solutions of KapustinWitten equations on a closed threemanifold times a line or a halfline. In particular I will describe some conjectural relations between these counts for the case of a halfline and WRT/CGP invariants of 3manifolds. In the case of a full line, I will describe how one can obtain explicit predictions using Stokes phenomenon in the analytically continued ChernSimons theory for a class of 3manifolds.
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03:00 PM  03:30 PM


Afternoon Tea

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03:30 PM  04:30 PM


On the Geometry of $G_2$Monopoles and the DonaldsonSegal Program
Ákos Nagy (University of California, Santa Barbara)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
$G_2$monopoles are special solutions to the YangMillsHiggs equation on (noncompact) $G_2$manifolds, similar to the 3dimensional BPS monopoles.
Donaldson and Segal conjectured that these gauge theoretic objects have a close relationship to the geometry of the underlying $G_2$structure. Intuitively, $G_2$monopoles with “large mass” are predicted to concentrate around coassociative submanifolds.
In this talk, I will introduce the relevant concepts to this conjecture, in particular, $G_2$monopoles and coassociative submanifolds, in more detail. Then I will explain what the stateofart is, with an emphasis on the work I have done (or planning on doing) with my collaborators.
This is a joint project with Gonçalo Oliveira, Daniel Fadel, Saman Habibi Esfahani, and Lorenzo Foscolo.
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Oct 28, 2022
Friday

09:30 AM  10:30 AM


Hyperkahler Mirror Symmetry
Arnav Tripathy (Harvard University)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video


 Abstract
The broad contours of mirror symmetry for CalabiYau manifolds are wellunderstood by now: classical complexgeometric information is exchanged for a symplecticgeometric expansion defined in terms of a mirror manifold. In the hyperkahler context, one may expect to make still stronger statements. In this talk, I will explain how to do so in the case of the K3 manifold (and some degenerations thereof). Namely, the exact Ricciflat metric will have two expansions: (i) in terms of gauge theory on a fourtorus, and (ii) as corrected by a list of enumerative invariants. In particular, these two constructions are the topics of the talks by M. Zimet and L. Fredrickson earlier in the workshop; I will review these constructions before explaining how they are related by this strong form of mirror symmetry. This talk surveys a series of joint works with L. Fredrickson, S. Kachru, A. Vasy, and M. Zimet.
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10:30 AM  11:00 AM


Break

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11:00 AM  12:00 PM


Path Integral Derivations of VafaWitten and KTheoretic Donaldson Invariants
Jan Manschot (Trinity College)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
I will discuss 5dimensional N=1 super YangMills compactified on X times S^1, with X a smooth, compact, oriented 4manifold. After a partial topological twist along X, the theory is locally independent of the metric on X, while it does depend on the radius R of S^1. The coefficients of the Rexpansion of the path integral correspond to the index of a Dirac operator on moduli spaces of instantons and monopoles, or more generally Ktheoretic Donaldson invariants. I will evaluate path integrals using two methods: 1) the quantum mechanics of the theory reduced to S^1 and 2) the low energy effective theory reduced to X. Both methods reproduce the same wallcrossing formula for 4manifolds with b_2^+=1. I will also discuss the evaluation of path integrals for 4manifolds with b_2^+>1 using method 2. Our results agree with those for algebraic surfaces by Gottsche, Nakajima and Yoshioka (2006) and Gottsche, Kool and Williams (2021). This talk is based on work in progress with H. Kim, G. Moore, R. Tao and X. Zhang.
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12:00 PM  02:00 PM


Lunch

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02:00 PM  03:00 PM


Some Remarks on Yang–Mills Type Equations in Higher Dimensions
Richard Wentworth (University of Maryland)

 Location
 MSRI: Simons Auditorium, Online/Virtual
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 Abstract
The talk consists of two parts. First, I will discuss a variant of antiselfdual connections in higher dimensions that is motivated by calibrated geometry and multipolarizations in Kaehler geometry. Such connections are shown to satisfy a version of Uhlenbeck weak compactness. In the case of Hermitian manifolds, we prove analogs of the Donaldson–Uhlenbeck–Yau and nonabelian Hodge theorems. Second, I describe an extension of the Donaldson–Uhlenbeck–Yau theorem to normal projective varieties. Taken together, the two parts give an analytic proof of Miyaoka’s version of the Bogomolov–Gieseker inequality, with a sharp result in the case of equality. This is joint work with Xuemiao Chen.
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03:00 PM  03:30 PM


Afternoon Tea

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03:30 PM  04:30 PM


Towards a Monopole Fueter Floer Homology
Saman Habibi Esfahani (Duke University; MSRI / Simons Laufer Mathematical Sciences Institute (SLMath))

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
In this talk, we propose a monopole Fueter invariant of 3manifolds, motivated by the DonaldsonSegal program.
The theory of YangMills connections, and in particular instantons, revolutionized the study of 4manifolds. Monopoles appear as the dimensional reduction of instantons to 3manifolds. The moduli spaces of monopoles on R3 form ALF hyperkähler manifolds. An interesting feature of the monopole equation is that it can be generalized to certain higherdimensional spaces. The most interesting examples appear on CalabiYau 3folds and G2manifolds. Donaldson and Segal hinted at the idea of defining invariants of CalabiYau 3folds by counting monopoles on these manifolds. These monopole invariants, conjecturally, are related to the special Lagrangians, similar to the Taubes’ theorem, which relates the SeibergWitten and Gromov invariants of symplectic 4manifolds.
Motivated by this conjecture, we propose numerical invariants of 3manifolds by counting Fueter sections on hyperkähler bundles with fibers modeled on the moduli spaces of monopoles on R3. More ambitiously, one would hope this would result in a Floer theoretic invariant of 3manifolds. A major difficulty in defining these invariants is related to the noncompactness problems. We prove partial results in this direction, examining the different sources of noncompactness, and proving some of them, in fact, do not occur.
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