Waterloo Differential Geometry Working Seminar

Academic Year 2024-2025

• May 1st

  Xuemiao Chen - The Sphere (Abstract)

  I will make a long story regarding the two dimensional sphere.

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• May 8th

  Spiro Karigiannis - Harvey-Lawson calibrated pluripotential theory and applications to special holonomy manifolds (Abstract)

  I will introduce some of the ideas of calibrated pluripotential theory due to Harvey-Lawson, and then discuss in some detail a recent paper of Pacini-Raffero where they use these ideas to establish analogues of Hodge decompositions and the del-delbar lemma from Kahler geometry to other special holonomy manifolds. I'll speak in both time slots, with a short break in the middle.

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• May 15th

  Justin Fus - The KKS Form and Symplectic Geometry of Coadjoint Orbits (Abstract)

  A compact Lie group acts on its Lie algebra dual via the coadjoint representation. In this talk, we will explore how the coadjoint orbits of this representation carry a natural symplectic structure called the Kirillov-Kostant-Souriau (KKS) form. The KKS form is preserved by the action. If time permits, we will show that there is a moment map for the action that coincides with the inclusion map of the orbit. A worked example for SU(2) will be performed.

  Facundo Camano - Convergence Results for Taub-NUT and Eguchi-Hanson spaces (Abstract)

  We define multi-Taub-NUT and multi-Eguchi-Hanson spaces and look at Gromov-Hausdorff convergences involving these spaces.

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• May 22nd

  Faisal Romshoo - Symmetry groups, moment maps and cohomogeneity one special Lagrangians in \( \mathbb{C}^m\) (Abstract)

  We will discuss the relationship between symmetries and moment maps as explained in arXiv:math/0008021 and how this allows us to construct cohomogeneity one special Lagrangians in \( \mathbb{C}^m\). Time permitting, we will discuss some examples of SL \(m\)-folds in \( \mathbb{C}^m\).

  Alex Pawelko - The Formal Kaehler Structure of the G2 Knot Space (Abstract)

  We will explore the usual suspects of the moduli space of knots embeddable in a G2 manifold, based upon the work of Brylinski for the analogous space corresponding to the 3-dimensional cross product. This gives an infinite-dimensional "formally Kaehler" manifold, which one can consider Kaehler reduction on. If time permits, we will gesture vaguely at considerations from gauge theory and geometric quantization that motivate many interesting questions in the case of G2 manifolds.

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• May 29th

  Kain Dineen - Linear maps preserving powers of the symplectic form (Abstract)

  Let \(\Omega\) denote the standard symplectic form on \(\mathbb{R}^{2m}\). For \(k = 1, \dots, m\), we will describe the subgroup of \(\mathrm{GL}(2m, \mathbb{R})\) which fixes \(\Omega^k\).

  Dashen Yan - Non-degenerate \(\mathbb{Z}_2\) harmonic 1-forms on \(\mathbb{R}^n\) and their geometric application (Abstract)

  The \(\mathbb{Z}_2\) harmonic 1-form arises in various compactification problems in gauge theory, including those involving \(PSL(2,\mathbb{C})\) connections and Fueter sections. In this talk, we will describe a recent construction of non-degenerate \(\mathbb{Z}_2\) harmonic 1-forms on \(\mathbb{R}^n\) for \(n \geq 3\) , and explore their relation to Lawlor’s necks—a family of special Lagrangian submanifolds in \(\mathbb{C}^n\). We will also discuss a gluing construction in which these examples are glued to a regular zero of a harmonic 1-form on a compact manifold. This yields a sequence of non-degenerate \(\mathbb{Z}_2\) harmonic 1-forms whose branching sets shrink to points. As a result, we obtain many new examples of non-degenerate \(\mathbb{Z}_2\) harmonic 1-forms on compact manifolds.

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• June 5th

  Break

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• June 12th

  Spiro Karigiannis - Unique continuation in geometry (Abstract)

  I will introduce the notion of unique continuation in geometry, closely following a survey article by Jerry Kazdan (CPAM 1988). Not all elliptic PDE exhibit the phenomenon of unique continuation, but most important elliptic PDE arising in geometry do, such as the Laplace equation, the Cauchy-Riemann equation, and the harmonic map equation.

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• June 19th

  Justin Fus - The Geometry of the Based Loop Group and Moment Maps (Abstract)

  Given a compact Lie group, we will explore a symplectic structure on the infinite-dimensional based loop group consisting of smooth maps from the circle to the Lie group with the identity as a basepoint. The maximal torus of the Lie group and the circle group together generate a Hamiltonian torus action on the loop group. Results on connectedness of level sets and convexity of the moment map, which are attempts to generalize those for finite-dimensional compact symplectic manifolds, will be previewed.

  Spiro Karigiannis - Unique continuation in geometry (conclusion) (Abstract)

  I will finish discussing the paper by Jerry Kazdan on unique continuation in geometry. I will try to make this second talk self-contained, by stating the various estimates which we derived in my first talk, and continuing the proof from there.

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  June 26th

  Break

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• July 3rd

  Facundo Camano - A Gromov—Hausdorff Convergence Result for the Moduli Space of Singular Monopoles (Abstract)

  I will introduce singular monopoles on \(\mathbb{R}^3 \) and their moduli space. We will then focus on \(\text{U}(2) \) singular monopoles, which have known explicit expressions, and look at the Gromov—Hausdorff convergence of the moduli space as one singularity is sent off towards infinity.

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• July 10

  Faisal Romshoo - Constructing calibrated submanifolds through evolution equations (Abstract)

  I will talk about how we can construct examples of calibrated submanifolds using the techniques of evolution equations. We will begin by defining the ideas involved in coming up with these evolution equations and then look at some of the examples of calibrated submanifolds that are constructed this way, following arXiv:math/0008021 , arXiv:math/0008155 , and arXiv:math/0401123 .

  Xuemiao Chen - On the space of lines (Abstract)

  I will make a story about the space of oriented lines in the three dimensional Euclidean space.

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• July 17

  Paul Cusson - Vector bundles over a complex torus (Abstract)

  We will cover basic results about the complex geometry of a complex torus \(X\), followed by a discussion of holomorphic vector bundles over \(X\). An immediate result due to Hodge theory is the existence of complex bundles that don't admit holomorphic structures when the complex dimension of \(X\) is at least 2. We will thus focus on bundles whose Chern classes lie in the diagonal of the Hodge diamond and ask which ones can be holomorphic.

  Kaleb Ruscitti - Embedding a family of moduli spaces of \(\mathrm{SL}(2,\mathbb{C})\) bundles into projective spaces (Abstract)

  The moduli space of polystable degree-0 \(\mathrm{SL}(2,\mathbb{C})\) bundles on a compact connected Riemann surface of genus \(g>=2\) is a Kähler manifold, and an open subset of the moduli space of semi-stable bundles, which is a projective variety of dimension \(3g-3\). Biswas and Hurtubise constructed a toric degeneration of this moduli space, meaning a family of moduli spaces over \(\mathbb{C}\) whose fiber over 0 is a toric variety. The toric variety has a moduli interpretation as a space of framed parabolic bundles. In this talk, I will describe the family and then describe how one can embed the entire family into \(\mathbb{P}^N x \mathbb{C}\). This is the key step in a current project I am working on, about relating different geometric quantizations of the moduli space of \(\mathrm{SL}(2,\mathbb{C})\) bundles.

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  July 24th

  Break

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• July 31st

  Amanda Petcu - Cohomogeneity one solitons of the hypersymplectic flow (Abstract)

  Given a hypersymplectic manifold \(X^4\), one can give a flow of hypersymplectic structures that evolve according to the equation \(\partial_t \underline{\omega} = d(Q d^*(Q^{-1} \underline{\omega}))\) where \(\underline{\omega}\) is the triple that gives the hypersymplectic structure and \(Q\) is a \(3 \times 3\) symmetric matrix. In this talk we let \(X^4\) be \(\mathbb{R}^4\) with an \(\text{SO}(4)\) action, and the hypersymplectic triple be \(\omega_k = \frac{i}{2} \partial_k \bar{\partial}_k (h_k)\) where \(h_k\) are functions that depends solely on the radial coordinate. We will examine how the triple evolves under this flow and given this initial structure, determine all possible solitons of the hypersymplectic flow.

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  August 7th

  Break

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• Aug 14th

  Alex Pawelko - Gerbes of Coassociative Submanifolds and the first Chern class (Abstract)

  We will define (bundle) gerbes, a generalization of principal \(S^1\)-bundles, and define connections on gerbes, whose corresponding forms over a trivialization are 2-forms and whose curvatures are 3-forms. Then, we will build a gerbe from coassociative submanifolds of a \(G_2\) manifold, and study its analogue of the first Chern class, an integral cohomology class in degree 3.

  Xuemiao Chen - Space of lines-II (Abstract)

  I will continue to talk about some related constructions on the space of oriented lines in the three dimensional Euclidean space.

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• January 15th

  Spiro Karigiannis - Infinitesimal deformations of \(G\)-structures (Abstract)

  I will introduce the setting of \(G\)-structures on an oriented Riemannian \(n\)-manifold, where \(G\) is a closed Lie subgroup of \(\mathrm{SO}(n)\). These can be understood in terms of global sections of the \(\mathrm{SO}(n)/G\) bundle which is the quotient of the \(\mathrm{SO}(n)\)-prinicipal bundle of oriented orthonormal frames by the free action of \(G\). We will define the intrinsic torsion of a \(G\)-structure, and explain how to describe infinitesimal deformations of \(G\)-structures. If time permits, we will discuss a Dirichlet energy type of functional on the space of \(G\)-structures, whose critical points are called harmonic \(G\)-structures. This condition includes the torsion-free \(G\)-structures but is more general. These ideas were developed recently by Fowdar, Loubeau, Moreno, Sa Earp building on earlier work in the \(\mathrm{G}_2\) and \(\mathrm{Spin}(7)\) cases by myself from 2006-2007.

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• January 22nd

  Faisal Romshoo - Topological calibrations and their moduli spaces (Abstract)

  We discuss an approach to deformation problems of geometric structures laid out in https://arxiv.org/abs/math/0112197 by Ryushi Goto. In particular, we will explore the cohomological conditions under which the moduli space of the geometric structures become smooth manifolds of finite dimension. As an application, we will prove the unobstructedness of \(\mathrm{G}_2\) structures and if time permits, of \(\mathrm{Spin}(7)\)-structures as well.

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• January 29th

  Amanda Petcu - Cohomogeneity one solitons of the hypersymplectic flow (Abstract)

  Given a manifold \(X^4 \times \mathbb{T}^3\) where \(X^4\) is hypersymplectic, one can give a flow of hypersymplectic structures that evolve according to the equation \(\partial_t \underline{\omega} = d(Q d^*(Q^{-1} \underline{\omega}))\) where \(\underline{\omega}\) is the triple that gives the hypersymplectic structure and \(Q\) is a \(3 \times 3\) symmetric matrix that relates the symplectic forms \(\omega_i\) to one another. We will let \(X^4\) be \(\mathbb{R}^4\) with a cohomogeneity one action and explain what it means to be a soliton for the hypersymplectic flow and examine a (potentially hyperkähler) metric that comes from this set-up.

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• February 5th

  Kain Dineen - Gromov's non-squeezing theorem (Abstract)

  I will discuss Gromov's non-squeezing theorem. We will prove the affine version of the theorem and discuss a potential generalization of it for maps preserving some power of the symplectic form. We will then discuss the general non-squeezing theorem and, as an application, prove the classical rigidity result that the symplectomorphism group of any symplectic manifold is \(C^0\)-closed in the diffeomorphism group.

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• February 12th

  Roberto Albesanio - From division to extension (Abstract)

  The \(L^2\) extension theorem of Ohsawa and Takegoshi, and the \(L^2\) division theorem of Skoda are two fundamental results in complex analytic geometry. They are also intimately related: in fact, Ohsawa showed that a version of the latter can be proved as a corollary of the former. I will explain the main idea of Ohsawa and how, conversely, a version of the \(L^2\) extension theorem can be obtained as an easy corollary of a Skoda-type \(L^2\) division theorem with bounded generators.

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• February 26th

  Facundo Camano - Bows to Singular Monopoles (Abstract)

  We will discuss an approach to constructing singular monopoles on \(\mathbb{R}^3\) by Sergey Cherkis. We begin with the background and the traditional approach to constructing singular monopoles via the Nahm equations. We then talk about bows and their representations, along with the resulting moduli spaces and their self-dual instantons. We apply the bow approach to construct self-dual instantons on the multi-Taub-NUT space and then exploit Kronheimer's correspondence to obtain singular monopoles.

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• March 5th

  Paul Cusson - Holomorphic vector bundles over an elliptic curve (Abstract)

  We'll go over the classification of holomorphic vector bundles over an elliptic curve, with a focus on the rank 1 and 2 cases. For the case of line bundles, we'll show that the space of degree 0 line bundles is isomorphic to the elliptic curve itself. The classification of rank 2 bundles rests on the existence of two special indecomposable 2-bundles of degree 0 and 1, which we will describe in detail. The general case for higher ranks would then follow essentially inductively.

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• March 12th

  Faisal Romshoo - A canonical form theorem for elements of \(\mathfrak{spin}(7)\) (Abstract)

  We will first demonstrate the maximal torus theorem at the Lie algebra level for the exceptional Lie algebra \(\mathfrak{g}_2\) by proving a canonical form theorem for the elements of \(\mathfrak{g}_2\) following arXiv:2209.10613. Then, we will proceed to prove a canonical form theorem for the elements of the Lie algebra \(\mathfrak{spin}(7)\).

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• March 19th

  Amanda Petcu - A hypersymplectic structure on \(\mathbb{R}^4\) with an \(\text{SO(4)}\) action (Abstract)

  Given a hypersymplectic manifold \(X^4\), one can give a flow of hypersymplectic structures that evolve according to the equation \(\partial_t \underline{\omega} = d(Q d^*(Q^{-1} \underline{\omega}))\) where \(\underline{\omega}\) is the triple that gives the hypersymplectic structure and \(Q\) is a \(3 \times 3\) symmetric matrix. In this talk we let \(X^4\) be \(\mathbb{R}^4\) with an \(\text{SO}(4)\) action, and the hypersymplectic triple be \(\omega_k = \frac{i}{2} \partial_k \bar{\partial}_k (h)\) where \(h\) is a function that depends solely on the radial coordinate. The flow of the hypersymplectic triple then descends to a single flow of the derivatives of the function \(h\). We will examine this flow, as well as solitons of the hypersymplectic flow in this set up. Furthermore, the triple \(\underline{\omega}\) gives rise to a Riemannian metric \(g_{HS}\). We will conclude with a discussion about the Riemann and Ricci curvature tensors that are derived from this metric.

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• March 26th

  Kain Dineen - Symplectic Capacities and Rigidity (Abstract)

  As an application of Gromov's non-squeezing theorem, we'll prove that the symplectomorphisms (and anti-symplectomorphisms) of \((\mathbb{R}^{2m}, \omega_0)\) are exactly the diffeomorphisms that additionally preserve the capacity of every compact ellipsoid. If time permits, then we will use this to prove that if a sequence of symplectomorphisms of any symplectic manifold \((M, \omega)\) converges in the \(C^0\)-sense to a diffeomorphism \(\psi\), then \(\psi^*\omega = \pm \omega\).

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• April 2nd

  Jacques Gideon Van Wyk - Curvature Tensors of Generalised Levi-Civita Connections (Abstract)

  In generalised geometry, there is no unique Levi-Civita connection: Given a generalised metric on a Courant algebroid, there are infinitely many torsion-free generalised connections compatible with the generalised metric. In this seminar, we will discuss the various curvature tensors associated to such a metric, and to what extent these tensors depend on the metric rather than the connection.

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• April 9th

  Utkarsh Bajaj - TBD (Abstract)

  TBD

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• September 18th

  Aleksandar Milivojevic - Formality in rational homotopy theory (Abstract)

  I will introduce the notion of formality of a manifold and will discuss some topological implications of this property, together with a computable obstruction to formality called the triple Massey product. I will then survey a conjecture relating formality and the existence of special holonomy metrics.

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• September 25th (Special Geometry and Topology Seminar)

  Lucia Martin Merchan - About formality of compact manifolds with holonomy \(\mathrm{G}_2\) (Abstract)

  The connection between holonomy and rational homotopy theory was discovered by Deligne, Griffiths, Morgan, and Sullivan, who proved that compact Kähler manifolds are formal. This led to the conjecture that compact manifolds with special and exceptional holonomy should also be formal. In this talk, I will discuss my recent preprint arXiv:2409.04362, where I disprove the conjecture for holonomy \(\mathrm{G}_2\) manifolds.

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• October 2nd

  Paul Cusson - The Kodaira embedding theorem and background material (Abstract)

  The Kodaira embedding theorem is a crucial result in complex geometry that forms a nice bridge between differential and algebraic geometry, giving a necessary and sufficient condition for a compact complex manifold to be a smooth projective variety, that is, a complex submanifold of a complex projective space. The material and proof will follow the exposition in Griffiths & Harris's classic textbook.

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• October 9th

  Jacques Van Wyk - “The” Generalised Levi-Civita Connection (Abstract)

  I will discuss the notions of generalised metrics and generalised connections in generalised geometry. A generalised connection has an associated torsion tensor, so one may ask, if given a generalised metric \(G\), whether there is a torsion-free connection \(D\) compatible with \(G\); this is the analogue of the Levi-Civita connection. We will see that there are infinitely many such connections \(D\), that is, there is no unique “generalised Levi-Civita connection,” a striking difference from the situation for Riemannian geometry.

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• October 16th (Special Geometry and Topology Seminar)

  Viktor Majewski (Humboldt-Universität zu Berlin)- Resolutions of \(\textrm{Spin}(7)\)-Orbifolds (Abstract)

  In Joyce's seminal work, he constructed the first examples of compact manifolds with exceptional holonomy by resolving flat orbifolds. Recently, Joyce and Karigiannis generalised these ideas in the \(\mathrm{G}_2\) setting to orbifolds with \(\mathbb{Z}_2\)-singular strata. In this talk I will present a generalisation of these ideas to \(\textrm{Spin}(7)\) orbifolds and more general isotropy types. I will highlight the main aspects of the construction and the analytical difficulties.

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• October 23rd

  Faisal Romshoo - Special Lagrangian Geometry (Abstract)

  I will talk about special Lagrangian submanifolds in \(\mathbb{C}^m\), which have garnered considerable interest in several areas in differential geometry and theoretical physics. In particular, I will describe some examples of special Lagrangian submanifolds explicitly.

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• October 30th

  Zev Friedman - \(N\)-cohomologies on non-integrable almost complex manifolds (Abstract)

  I will define an \(N\)-cohomology and compute some interesting examples, showing the different isomorphism classes on certain almost complex manifolds.

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• November 6th

  Facundo Camano - Gromov-Hausdorff Convergence (Abstract)

  I will introduce Hausdorff and Gromov-Hausdorff distances on metric spaces. We will look at examples of calculating distances and convergent sequences of metric spaces. We will end off with proving Gromov's precompactness theorem and a few pathological examples of convergence stemming from the result.

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• November 13th

  Francisco Villacis - Convexity of Toric Moment Maps (Abstract)

  Toric moment maps are arguably the nicest family of moment maps in symplectic geometry. A classical theorem from the 80s state that the images of these moment maps are convex polytopes, which was proven independently by Atiyah, and Guillemin and Sternberg. In this talk I will go through Atiyah's slick proof of the convexity theorem using Morse theory, and if time permits I will talk about other results in this area.

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• November 20th

  Alex Pawelko - Prequantum Line Bundles and Geometric Quantization (Abstract)

  Prequantum line bundles are objects in symplectic geometry that play a somewhat analogous role to holomorphic line bundles in complex geometry. In this talk, we will discuss the existence of prequantum line bundles, examples of them, and their uses in symplectic geometry, most notably in geometric quantization.

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• November 27th

  Spiro Karigiannis - A tale of two Lie groups (Abstract)

  The classical Lie group \(\mathrm{SO}(4)\) is well-known to possess a very rich structure, relating in several ways to complex Euclidean spaces. This structure can be used to construct the classical twistor space \(Z\) over an oriented Riemannian \(4\)-manifold \(M\), which is a \(6\)-dimensional almost Hermitian manifold. Special geometric properties of \(Z\) are then related to the curvature of \(M\), an example of which is the celebrated Atiyah-Hitchin-Singer Theorem. The Lie group \(\mathrm{Spin}(7)\) is a particular subgroup of \(\mathrm{SO}(8)\) determined by a special \(4\)-form. Intriguingly, \(\mathrm{Spin}(7)\) has several properties relating to complex Euclidean spaces which are direct analogues of \(\mathrm{SO}(4)\) properties, but sadly (or interestingly, depending on your point of view) not all of them. I will give a leisurely introduction to both groups in parallel, emphasizing the similarities and differences, and show how we can nevertheless at least partially succeed in constructing a "twistor space" over an \(8\)-dimensional manifold equipped with a torsion-free \(\mathrm{Spin}(7)\)-structure. (I will define what those are.) This is joint work with Michael Albanese, Lucia Martin-Merchan, and Aleksandar Milivojevic. The talk will be accessible to a broad audience.

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• December 4th

  Pause

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• December 11th

  Zev Friedman - \(4\)-dimensional \(\mathrm{U(m)}\)-structures (Abstract)

  We will define the modified deRham operator \(D\) on \(U(m)\)-structures, and prove that \(D^2=0\) is equivalent to \(d \omega=0\) in the \(4\)-dimensional case.

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• December 18th

  Faisal Romshoo - Constructing associatives in \(7\)-manifolds (Abstract)

  We will revisit the classical examples of Special Lagrangians invariant under some group \(G \subset \mathrm{SU(n)}\) using a new method and check if we can use the same method to construct associative submanifolds, which are a type of calibrated \(3\)-submanifolds in \(7\)-manifolds, in \(\mathbb{R}^7\).