Oct 24, 2022
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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09:30 AM - 10:30 AM
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Overview of Some Appearances of Multivalued Solutions in Differential Geometry
Simon Donaldson (Imperial College, London)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- 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 co-associative fibred G2 manifolds and research directions which these suggest.
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10:30 AM - 11:00 AM
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Break
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11:00 AM - 12:00 PM
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K3 Surfaces as Gauge-Theoretic Moduli Spaces
Max Zimet (Stanford University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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I will explain a novel construction of K3 surfaces as moduli spaces of singular equivariant instantons on a 4-torus. This gives the first hyper-Kahler quotient construction of a compact non-toroidal 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 Donaldson-Uhlenbeck-Yau theorem that operates in this setting, some of its consequences (such as non-emptiness 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 Yang-Mills functional, and I will describe the Fredholm theory for the heat operator that undergirds the proof of the short- and long-time existence and regularity of the flow. While these results are all proved for singular connections on a 4-torus, they are expected to generalize to enable the study of vast new classes of gauge-theoretic 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
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Lunch
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02:00 PM - 03:00 PM
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Vafa-Witten Invariants of Projective Surfaces - Overview
Martijn Kool (Universiteit Utrecht)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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On a four-manifold underlying a complex smooth projective surface S, Tanaka-Thomas gave an algebro-geometric definition of the SU(r) Vafa-Witten partition function. When S has a non-zero holomorphic 2-form, the partition function is essentially topological with a rich modular structure. The algebro-geometric viewpoint provides new insights into Vafa-Witten theory such as (1) new mathematical verifications of S-duality, (2) K-theoretic refinement, (3) higher rank expressions with relations to the Rogers-Ramanujan continued fraction.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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03:30 PM - 04:30 PM
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Holomorphic Floer Theory and Donaldson-Thomas Invariants
Pierrick Bousseau (University of Georgia)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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Holomorphic Floer theory is the analog of Floer theory for holomorphic symplectic manifolds. This topic has been recently explored, from several different perspectives, by Kontsevich-Soibelman and Doan-Rezchikov. In a different direction, I will in this talk present a conjectural picture relating holomorphic Floer theory of complex integrable systems to Donaldson-Thomas invariants. In physical terms, we will discuss a relation between the BPS spectrum of an N=2 4-dimensional field theory and the enumerative geometry of the corresponding Seiberg-Witten integrable system.
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04:30 PM - 06:20 PM
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Reception
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Oct 25, 2022
Tuesday
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09:30 AM - 10:30 AM
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All Gravitational Instantons from Monopole Moduli Spaces
Sergey Cherkis
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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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 Yang-Mills instantons).
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10:30 AM - 11:00 AM
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Break
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11:00 AM - 12:00 PM
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Homological Link Invariants from Mirror Symmetry
Mina Aganagic (University of California, Berkeley)
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- Location
- SLMath: Eisenbud 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(1|1) by an arbitrary Lie algebra. The theories can be solved exactly.
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12:00 PM - 02:00 PM
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Lunch
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02:00 PM - 03:00 PM
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Existence and Non-Existence Results of Z2 Harmonic 1-Forms
Siqi He (Academy of Mathematics and Systems Science)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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Z2 harmonic 1-forms 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 1-forms 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 1-forms. In this talk, we will discuss a method to construct examples of Z2 harmonic 1-forms. We will also discuss the relationship between Z2 harmonic 1-forms and special Lagrangian geometry and present a non-existence result based on the work of Abouzaid-Imagi.
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03:00 PM - 03:30 PM
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Afternoon Tea
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03:30 PM - 04:30 PM
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Holomorphic Floer Theory and the Fueter Equation
Aleksander Doan (University of Cambridge; University College London)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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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 infinite-dimensional version of the Fukaya-Seidel 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 Fukaya-Seidel 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
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09:30 AM - 10:30 AM
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Computations of Instanton Knot Floer Homology with Local Coefficients
Tomasz Mrowka (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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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 n-strand 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
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Break
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11:00 AM - 12:00 PM
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Concentrating Local Solutions of the Two-Spinor Seiberg-Witten Equations
Gregory Parker (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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Given a compact 3-manifold $Y$ 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 two-spinor Seiberg-Witten 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 two-spinor Seiberg-Witten solutions on Y.
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12:00 PM - 02:00 PM
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Lunch
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Oct 27, 2022
Thursday
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09:30 AM - 10:30 AM
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Disconnected Gauge Groups and Categorical Symmetry
Mathew Bullimore (University of Durham)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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There has been recent renewed interest in four-dimensional 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
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Break
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11:00 AM - 12:00 PM
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Hyperkahler Metrics Near Semi-Flat Limits
Laura Fredrickson (University of Oregon)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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We make rigorous (a generalization of) the formalism of Gaiotto, Moore, and Neitzke for constructing hyperkahler manifolds near semi-flat 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 Gromov-Hausdorff collapse of hyperkahler manifolds to semi-flat limits and completes the Strominger-Yau-Zaslow 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 non-compact 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 semi-flat limit. This talk is based on joint work with Arnav Tripathy and Max Zimet.
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12:00 PM - 02:00 PM
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Lunch
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02:00 PM - 03:00 PM
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Counting Solutions of Kapustin-Witten Equations on a Three-Manifold Times a Line from Physics Dualities
Pavel Putrov (Abdus Salam International Centre for Theoretical Physics)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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In my talk I will summarize results about some explicit physics predictions of counts of solutions of Kapustin-Witten equations on a closed three-manifold times a line or a half-line. In particular I will describe some conjectural relations between these counts for the case of a half-line and WRT/CGP invariants of 3-manifolds. In the case of a full line, I will describe how one can obtain explicit predictions using Stokes phenomenon in the analytically continued Chern-Simons theory for a class of 3-manifolds.
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03:00 PM - 03:30 PM
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Afternoon Tea
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03:30 PM - 04:30 PM
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On the Geometry of $G_2$-Monopoles and the Donaldson--Segal Program
Ákos Nagy (University of California, Santa Barbara)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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$G_2$-monopoles are special solutions to the Yang--Mills--Higgs equation on (noncompact) $G_2$-manifolds, similar to the 3-dimensional 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 state-of-art 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
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09:30 AM - 10:30 AM
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Hyperkahler Mirror Symmetry
Arnav Tripathy (Harvard University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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The broad contours of mirror symmetry for Calabi-Yau manifolds are well-understood by now: classical complex-geometric information is exchanged for a symplectic-geometric 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 Ricci-flat metric will have two expansions: (i) in terms of gauge theory on a four-torus, 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
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Break
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11:00 AM - 12:00 PM
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Path Integral Derivations of Vafa-Witten and K-Theoretic Donaldson Invariants
Jan Manschot (Trinity College)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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I will discuss 5-dimensional N=1 super Yang-Mills compactified on X times S^1, with X a smooth, compact, oriented 4-manifold. 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 R-expansion of the path integral correspond to the index of a Dirac operator on moduli spaces of instantons and monopoles, or more generally K-theoretic 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 wall-crossing formula for 4-manifolds with b_2^+=1. I will also discuss the evaluation of path integrals for 4-manifolds 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
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Lunch
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02:00 PM - 03:00 PM
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Some Remarks on Yang–Mills Type Equations in Higher Dimensions
Richard Wentworth (University of Maryland)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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The talk consists of two parts. First, I will discuss a variant of anti-self-dual 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
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Afternoon Tea
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03:30 PM - 04:30 PM
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Towards a Monopole Fueter Floer Homology
Saman Habibi Esfahani (Duke University; MSRI / Simons Laufer Mathematical Sciences Institute (SLMath))
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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In this talk, we propose a monopole Fueter invariant of 3-manifolds, motivated by the Donaldson-Segal program.
The theory of Yang-Mills connections, and in particular instantons, revolutionized the study of 4-manifolds. Monopoles appear as the dimensional reduction of instantons to 3-manifolds. 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 higher-dimensional spaces. The most interesting examples appear on Calabi-Yau 3-folds and G2-manifolds. Donaldson and Segal hinted at the idea of defining invariants of Calabi-Yau 3-folds by counting monopoles on these manifolds. These monopole invariants, conjecturally, are related to the special Lagrangians, similar to the Taubes’ theorem, which relates the Seiberg-Witten and Gromov invariants of symplectic 4-manifolds.
Motivated by this conjecture, we propose numerical invariants of 3-manifolds 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 3-manifolds. A major difficulty in defining these invariants is related to the non-compactness problems. We prove partial results in this direction, examining the different sources of non-compactness, and proving some of them, in fact, do not occur.
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