Waterloo Differential Geometry Working Seminar

Academic Year 2022-2023

1. August 29

   Ruxandra Moraru - Hitchin pairs on higher-dimensional manifolds (Abstract)

   In this talk, we discuss generalizations of Higgs bundles on curves to higher-dimensional manifolds, consisting of pairs (E, φ) with E a holomorphic vector bundle on a complex manifold X and φ ∈ H⁰(X, End(E) ⊗ V), where V is a fixed holomorphic vector bundle on X. For example, when V = K_X, stable pairs (E, φ) correspond to solutions of the Vafa-Witten equations, and when V = T^*_X (resp. T_X), such pairs are called Higgs (resp. co-Higgs) bundles. I will give an overview of some of the known results about Hitchin pairs in higher dimensions and describe some open problems.

   Spiro Karigiannis - Formality of compact Kahler manifolds: Episode II (Abstract)

   I will continue the series of talks establishing the formality of compact Kähler manifolds, using a Riemannian geometric approach that could hopefully be adapted to other settings. In this second talk, we will define formality, and then use the del-delbar lemma on compact Kahler manifolds to establish formality. The third talk will likely be about generalizations to other compact Riemannian geometries admitting certain types of parallel differential forms.

2. August 22

   Paul Cusson - When is a closed parallelizable manifold a homogeneous space? (Abstract)

   Given a compact parallelizable real n-manifold M, the following theorem is known: M≅G/H for G a simply connected Lie group and H a discrete subgroup of G if and only if there exists n everywhere linearly independent vector fields X₁, …, X_n and constants α_ij^k such that for each 1 ≤ i < j ≤ n, [X_i, X_j] = α_ij^kX_k. This is in contrast to the holomorphic case, where a compact parallelizable complex manifold is guaranteed to have such structure constants. After proving this theorem, we will look at how this result could possibly be used as a tool for an alternative proof of the Poincaré conjecture, if one were so bold.

   Henry Li - Riemann surfaces and Fuchsian groups (Abstract)

   In this talk, we explain how Riemann surfaces of genus greater than one can be constructed as the quotient of the upper-half plane by a Fuchsian group.

3. August 8

   Jing Xuan Chen - Holomorphic bisectional curvature (continued) (Abstract)

   The Frankel conjecture states that a connected compact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to the complex projective space. Last time we proved the Frankel conjecture in dimension two. We will continue to prove the Frankel conjecture in the case of a Kähler–Einstein manifold. On Kähler–Einstein manifolds, a stronger result can be obtained, we get an isometry with the complex projective space instead of a biholomorphic equivalence.

   Spiro Karigiannis - Formality of compact Kähler manifolds: Episode I (Abstract)

   I will begin a series of talks establishing the formality of compact Kähler manifolds, using a Riemannian geometric approach that could hopefully be adapted to other settings. In this first talk, we will derive the classical Kähler identities from a Riemannian geometric viewpoint, and explore some of their consequences. This will likely be a series of 2-3 talks.

4. August 1

   Michael Albanese - The Schoen-Yau Stable Minimal Hypersurface Technique (Abstract)

   In the quest to determine whether tori admit metrics of positive scalar curvature, two powerful techniques were developed: enlargeability due to Gromov and Lawson, and the stable minimal hypersurface technique due to Schoen and Yau. Having previously discussed the former, we will introduce the latter. In particular, we will show that if a closed smooth manifold admits a positive scalar curvature metric, then so does any stable minimal hypersurface.

   Lucia Martin Merchan - Geometry of calibrated submanifolds (Abstract)

   Given a Riemmanian manifold M endowed with calibration α, the condition that a submanifold L of M is calibrated is a first order condition. By contrast, its geometric data, given by the second fundamental form A and the induced tangent and normal connections ∇ on TL and D on NL, respectively, is second order information. In this talk, we characterize the conditions imposed on the geometric data (A,∇,D) when a submanifold L is calibrated with respect to a calibration α on M which is parallel. After that, we apply our results to some interesting calibrated geometries, such as Kähler, Calabi-Yau, G₂ and Spin(7) manifolds.

5. July 18

   Caleb Suan (UBC) - Conifold Transitions, Balanced Metrics, and Stability (Abstract)

   Conifold transitions are a general procedure for constructing new complex 3-folds from a compact Kähler Calabi—Yau 3-fold by contracting disjoint (-1,-1)-curves and smoothing the resulting ODP singularities. Though we begin with a Kähler manifold, the manifolds that we get may not stay Kähler, which may suggest that we should seek a more "general notion of Kähler". In this talk, we outline the geometry of conifold transitions and discuss results in this direction such as the balanced metrics of Fu—Li—Yau and the more recent stability results of Collins—Picard—Yau.

   Aïssa Wade - On the stability of symplectic leaves of Poisson manifolds (Abstract)

   A regular symplectic foliation on a smooth manifold M is a (regular) foliation F on M together with a foliated 2-form ω whose restriction to each leaf S is a symplectic form ω_S. Any regular symplectic foliation determines a regular Poisson structure on M. However a more interesting class of Poisson structures is that of Poisson structures corresponding to irregular symplectic foliations (i.e. the dimensions of the leaves vary). In this talk, I will present some results on the stability problem for compact leaves of Poisson manifolds. First I will review Poisson manifolds and their symplectic leaves. Then, I will explain the geometric data that encode the Poisson structure in a tubular neighborhood of a symplectic leaf S. Finally, I will explain some cohomology criterion for stability of symplectic leaves.

6. July 11

   Jacques Van Wyk - Fundamentals of Elliptic Partial Differential Operators (Abstract)

   In differential geometry, there is a very nice class of operators taking sections of one bundle to another known as elliptic partial differential operators. In this talk, I will prove some fundamental results about these operators, intending to give some idea for why these operators are so nice. This will come in two parts: first I will give a brief overview of the theory of (elliptic) pseudodifferential operators on ordinary multivariate functions from the perspective of Fourier analysis; then, I will show how this theory can be extended to elliptic operators between bundles. If time permits, I will show how these results can be used to prove a strong generalisation of the Hodge Decomposition Theorem.

   Christopher Lang - Revisiting Symmetric Hyperbolic Monopoles (Abstract)

   In this talk, I will present a new derivation of the symmetry equations for spherically symmetric hyperbolic monopoles. This new proof sidesteps previously needed hypotheses and also provides a new constraint on the structure group. It is also easy to modify for other gauge-theoretic objects like Euclidean monopoles and instantons.

7. July 4

   Lucia Martin Merchan - A compact closed G₂ manifold with b₁=0 (Abstract)

   The presence of a G₂-holonomy metric on a 7-dimensional manifold yields a closed G₂ structure, and some topological properties, such as b₁=0. However, there aren't examples of manifolds with a closed G₂ structure satisfying these topological properties that do not admit a metric with holonomy G₂. In this talk, we construct a compact closed G₂ manifold with b₁=0 using orbifold resolution techniques. Later, we compare it with the already-known manifolds with holonomy G₂.

   Michael Albanese - Enlargeable Manifolds and Positive Scalar Curvature (Abstract)

   In the late 70's, the question of whether tori could admit metrics of positive scalar curvature was being tackled by Gromov and Lawson, and independently by Schoen and Yau. The former duo developed the notion of enlargeability to settle the question in the negative. In this talk I will define enlargeability and indicate its relationship to the existence of positive scalar curvature metrics.

8. June 27

   Amanda Petcu - The G₂ Laplacian flow and Laplacian solitons (Abstract)

   This talk will introduce the G₂-Laplacian flow for closed G₂ structures on a compact manifold. We will show its relationship with a natural volume functional and discuss the main properties of the flow such as short-time existence and the evolution of the induced metric and volume form. Moreover, we will introduce the soliton solutions of the Laplacian flow, proving a non-existence result except in the case of a torsion-free G₂ structure.

   Spiro Karigiannis - The deTurck trick demystified (Part II) (Abstract)

   This is a continuation of last week's talk. The Ricci flow is not strictly parabolic due to diffeomorphism invariance. This makes it harder to prove short-time existence of the flow. I will explain the “deTurck trick” to break the diffeomorphism invariance (also called gauge-fixing). This produced a modified flow that is strictly parabolic. One then shows that a solution to the modified flow can be transformed into a solution of the original flow.

9. June 20

   Jing Xuan Chen - Holomorphic bisectional curvature (Abstract)

   The bisectional curvature on a Kähler manifold (M,J) is defined as H(X,Y)=R(X,JX,JY,Y) for unit vectors X,Y. We will see what we can prove about a compact connected Kähler manifold if we assume that it has positive bisectional curvature.

   Spiro Karigiannis - The deTurck trick demystified (Abstract)

   The Ricci flow is not strictly parabolic due to diffeomorphism invariance. This makes it harder to prove short-time existence of the flow. I will explain the “deTurck trick” to break the diffeomorphism invariance (also called gauge-fixing). This produced a modified flow that is strictly parabolic. One then shows that a solution to the modified flow can be transformed into a solution of the original flow.

10. June 13

    Amanda Petcu - The G₂ Laplacian flow (Abstract)

    In this talk, we will introduce the G₂ Laplacian flow and attempt to show short-time existence and uniqueness following the work of Bryant and Xu.

    Recording and Discussion: Sir Michael Atiyah - A Panoramic View of Mathematics (Abstract)

    This is a pre-recorded lecture from a conference at the University of Cambridge in 2010. The talk can be found here: https://sms.cam.ac.uk/media/753316.
    
    The original abstract is the following:
    
    Climbing a mountain is strenuous and hazardous, but the view from the top can be spectacular and makes it all worth while. There is a clear analogy with mathematics. In both cases one has to train properly, collect the right tools and gear, practice on the lower slopes, examine maps and do some background reading. After that it is a matter of hard work, patience and skill. I will look back on my 60 years of mathematics, describing what views I have seen from the heights and what challenges lie ahead for the next generation. There are many more mountain ranges to explore.

11. June 6

    Xuemiao Chen - Boundary value problems for G₂ holonomy equation and mapping problems for 3 forms on 5-d manifolds (Abstract) (Reference)

    A result of Donaldson says: with the set-up motivated by Hitchin’s variational point of view, the G₂ holonomy equation with boundary is elliptic mod related diffeomorphisms. Applications include the existence of a G₂-cobordism between small deformations of Calabi-Yau threefolds. Similar discussions regarding boundary value problems can be also done for complex Calabi-Yau threefolds. However, it is not elliptic. Donaldson and Lehmann considered a variant of this problem called the mapping problem which asks: when could a three-form on a 5-d manifold be the pull-back of the real part of the holomorphic 3-form on a Calabi-Yau threefold under some embedding? For this, they define a class of closed strongly pseudo-convex 3-form corresponding to the CR geometry and prove that the perturbative version of this problem can be solved if a finite dimensional obstruction vector space vanishes. We will give an introduction to these results.
    https://arxiv.org/abs/1802.09694, https://arxiv.org/abs/2210.16208

    Shubham Dwivedi - Parabolic frequency on Ricci flows (Abstract) (Reference)

    We will define a parabolic frequency function on Reimannian manifolds with metrics evolving by Ricci flows. We’ll prove the frequency monotonicity along the flow and use it to prove unique continuation theorems for certain 2nd order operators.
    Parabolic Frequency on Ricci Flows

12. May 23

    Ravi Mudaliar - A Categorical Equivalence between Compact Connected Riemann Surfaces and Function Fields in 1 variable (Abstract)

    I will discuss the interplay between analytic and algebraic data in the case of Riemann Surfaces. That is, to each Riemann Surface we can associate an algebraic object (function field) that completely encodes its analytic structure and given a function field we can associate a unique Riemann Surface (up to isomorphism) that completely encodes its algebraic structure. Throughout this construction I will also be proving some well-known facts about Riemann surfaces, such as the fact every compact connected Riemann Surface is an algebraic curve.

13. May 16

    Jacques Van Wyk - The Hodge Decomposition Theorem (Abstract)

    There is a generalisation of the ordinary Laplacian Δ = -Σ_i∂²/∂x_i² to an operator Δ, called the Laplace-Beltrami operator, on the space of differential forms of a manifold M. The Hodge Decomposition Theorem states that the equation Δω = α has a solution ω in the smooth k-forms on M if and only if the k-form α is orthogonal (in a suitable sense) to the space of harmonic k-forms (those for which Δη = 0). In this talk, we present the Hodge Decomposition Theorem, derive some of its consequences, and give as much of a proof of it as time permits.

14. May 9

    Michael Albanese - Hyperbolic Manifolds and Complex Structures (Abstract)

    Closed surfaces of genus at least two admit a hyperbolic metric and a compatible complex structure. Following a paper of Gauduchon, we will see this is never the case in higher dimensions: a closed hyperbolic manifold of dimension 2m > 2 does not admit a compatible complex structure. The proof uses a quantity involving scalar curvature and Weyl curvature which is similar to the Yamabe constant of a conformal class.

    Shengda Hu - Volume for a generalized metric (Abstract)

    We will discuss an attempt in defining volume for a generalized metric.

15. May 2

    Anton Iliashenko - Betti numbers of nearly G₂ and nearly Kähler manifolds with Weyl curvature bounds (Abstract)

    We use the Weitzenböck formulas to get information about the Betti numbers of nearly G₂ and nearly Kähler manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature.

    Amanda Petcu - Some Calculations Regarding G₂ and the Isometric Flow (Part 2) (Abstract)

    In the paper the authors consider the following setup for a 7-dimensional manifold M. Given a hypersymplectic structure ω on X⁴ they consider the manifold M=X⁴×T³ where T³=S¹×S¹×S¹. With this setup, they consider a closed 3-form φ on M that gives a G₂ structure. This is due to the hypersymplectic structure on X⁴. In the case where X⁴ is compact the authors deform the hypersymplectic structure on X⁴ to a hyperkahler triple. Then the G₂-structure φ on M=X⁴×T³ has vanishing torsion forms when ω is a hyperkahler triple implying that φ determines a G₂-metric. In this talk, I will loosen the conditions on X⁴ to pre-hypersymplectic and compute the forms φ and *φ=ψ. We will also compute the four torsion forms for M and determine what conditions are needed in order for the torsion forms to vanish. Finally given the full torsion tensor T of M, we will compute Div(T) in terms of the four torsion forms and determine what conditions we might need in order for Div(T) to vanish.

16. April 25

    Spiro Karigiannis - Relation between self-duality and conformal structures in dimension 4 (Abstract)

    We present the full details of the following classical fact, which is well-known ‘folklore’ but not easy to find in the literature. Let V be an oriented 4-dimensional real vector space. There is a symmetric bilinear form Q on the 6-dimensional vector space W = Λ² V^*, which is well-defined up to scale. This bilinear form has signature (3,3). We show that there is a one-to-one correspondence between conformal classes of metrics on V (that is, inner products up to positive scale) and maximal spacelike (positive-definite) subspaces of W. If time permits, we relate these ideas to the hypersymplectic structures that Amanda talked about last week.

17. April 17 Note this is on Monday at 2:30 pm - 3:30 pm in MC 5417

    Amanda Petcu - Some Calculations Regarding G₂ and the Isometric Flow I (Abstract) (Reference)

    In the paper the authors consider the following setup for a 7-dimensional manifold M. Given a hypersymplectic structure ω on X⁴ they consider the manifold M=X⁴×T³ where T³=S¹×S¹×S¹. With this setup, they consider a closed 3-form φ on M that gives a G₂ structure. This is due to the hypersymplectic structure on X⁴. In the case where X⁴ is compact the authors deform the hypersymplectic structure on X⁴ to a hyperkahler triple. Then the G₂-structure φ on M=X⁴×T³ has vanishing torsion forms when ω is a hyperkahler triple implying that φ determines a G₂-metric. In this talk, I will loosen the conditions on X⁴ to pre-hypersymplectic and compute the forms φ and *φ=ψ. We will also compute the four torsion forms for M and determine what conditions are needed in order for the torsion forms to vanish. Finally given the full torsion tensor T of M, we will compute Div(T) in terms of the four torsion forms and determine what conditions we might need in order for Div(T) to vanish.
    J.Fine and C. Yao, ``Hypersymplectic 4-manifolds, the G₂-Laplacian flow, and extension assuming bounded scalar curvature'', Duke University Press, vol. 167, no. 18, 2018, doi: 10.1215/00127094-2018-0040

18. April 4

    Lucia Martin-Merchan - Formality of Joyce's manifolds (Part 2) (Abstract)

    The topological notion of formality was first related to Riemannian geometry by a Theorem of Deligne, Griffiths, Morgan, and Sullivan, which states that compact Kähler manifolds are formal. Therefore, manifolds with holonomy SU(n) and Sp(n) also have this property. However, it is still an open question to determine whether manifolds with exceptional holonomy are formal. I will discuss the paper of M. Amann and I. Taimanov (https://www.arxiv.org/abs/2012.10915) showing that a Joyce's example is formal by a direct computation of its minimal model.

19. March 28

    Lucia Martin-Merchan - Formality of Joyce's manifolds (Part 1) (Abstract)

    The topological notion of formality was first related to Riemannian geometry by a Theorem of Deligne, Griffiths, Morgan, and Sullivan, which states that compact Kähler manifolds are formal. Therefore, manifolds with holonomy SU(n) and Sp(n) also have this property. However, it is still an open question to determine whether manifolds with exceptional holonomy are formal. I will discuss the paper of M. Amann and I. Taimanov (https://www.arxiv.org/abs/2012.10915) showing that a Joyce's example is formal by a direct computation of its minimal model.

20. March 21

    Christopher Lang - Hyperbolic monopoles with continuous symmetries (part 2) (Abstract)

    Few examples of hyperbolic monopoles exist. By modifying previous work of mine with collaborators, we will discuss a structure theorem for generating highly symmetric hyperbolic monopoles. We will briefly cover general geometric details discussed in my previous talk and focus more on the use of representation theory to generate monopoles and examine some examples generated by the method.

21. March 14

    Michael Albanese - The Hitchin-Thorpe Inequality (Abstract)

    Every two-dimensional manifold admits a metric of constant curvature by the Uniformisation theorem. In higher dimensions, one could try to generalise this fact by asking for the existence of a metric of constant scalar curvature, constant Ricci curvature, or constant sectional curvature (these all coincide in dimension two). Each condition is more restrictive than the last. There is a plethora of constant scalar curvature metrics on every manifold, while constant sectional curvature metrics rarely exist, so focus often turns to metrics of constant Ricci curvature, also known as Einstein metrics. In dimensions five and above, there are no known examples of manifolds which fail to admit such a metric. This is in stark contrast to dimension four where the Hitchin-Thorpe inequality provides a topological obstruction. We will give the proof of this inequality, following Hitchin's original paper "Compact four-dimensional Einstein manifolds".

22. March 7

    Eric Boulter - Co-isotropic foliations in holomorphic symplectic manifolds (Abstract) (Reference)

    In this talk, we examine a natural foliation on certain sub-manifolds of a holomorphic symplectic manifold. We discuss conditions on this foliation which guarantee that the leaf space is Hausdorff and smooth, and we show that the leaf space naturally inherits a holomorphic symplectic structure. We will also work through several examples where the leaf space of the foliation can be computed directly. The main reference for this talk is "Foliations on hypersurfaces in holomorphic symplectic manifolds" by Justin Sawon (available at arXiv:0812.3939).
    https://arxiv.org/abs/0812.3939

23. February 28

    Paul Cusson - Holomorphic maps between Riemann surfaces (Abstract)

    Let M,N be Riemann surfaces. When is every continuous map f:M→N homotopic to a holomorphic map? This question, which deals with a special case of the so-called Oka-principle, is the subject of Winkelmann's paper "The Oka-Principle for Mappings Between Riemann Surfaces". In this talk, we will discuss the results of this paper, which completely classifies the pairs of Riemann surfaces such that the above question has a positive answer. If time permits, we will look at some cases of higher dimensional complex manifolds for which the Oka-principle holds.

24. February 21

    Reading Week (no talk)

25. February 14

    No talk (postponed to February 28)

26. February 8 (Extra Talk: 3-4 pm in MC 5479)

    Spiro Karigiannis - The geometry of G₂ manifolds: a marriage of non-associative algebra and non-linear analysis (Abstract)

    This is a practice run for a colloquium talk that I am giving at Florida International University during reading week. I want to make sure it is pitched at the right level for a general colloquium, and also that it does not go over the allotted time of 50 minutes. So I would welcome any useful feedback that you can give me. However, it should also be a useful introduction to G₂ manifolds for graduate students who are not experts on the subject. Here is the official abstract: I will spend the first part of the talk giving a gentle introduction to G₂ manifolds, with a special emphasis on their similarities and differences with Kahler manifolds. In the second part, I will focus on some recent work with Daren Cheng (Miami) and Jesse Madnick (Oregon) on the analytic properties of a special class of maps into G₂ manifolds, which are analogues of J-holomorphic maps in Kahler geometry. The talk should be accessible to a general mathematical audience.

27. February 7

    Anton Iliashenko - The third Betti number of nearly Kahler 6-manifolds (Abstract)

    On a Riemannian manifold, Weizenbock formulas relate the usual Laplacian and the rough Laplacian along with Riemannian curvature. We consider 6-dimensional nearly Kahler manifolds and look at how the Weizenbock formula for 3-forms simplifies. We will see that a new curvature-type operator shows up which acts on symmetric 2-tensors. Assuming compactness, it will be possible to conclude the vanishing of the 3rd Betti number from a specific bound on this new operator.

28. January 31

    Spiro Karigiannis - Nearly Kahler 6-manifolds have SU(3)-structures (Abstract) (Reference)

    If (M, g, J) is an almost Hermitian manifold, then it is called nearly Kahler if the covariant derivative of J is skew in its two T^*M arguments. Something special happens in dimension 6, where it turns out that a (strictly) nearly Kahler structure is equivalent to an SU(3)-structure satisfying certain conditions. In particular, we must have c₁(M, J) = 0. I will discuss a relatively recent paper by Giovanni Russo, which explains this equivalence in detail.
    https://arxiv.org/abs/2002.04673v3

29. January 24

    Hanming Liu - Heegaard Floer Homology (Abstract)

    This will be an introduction to Heegaard Floer homology. We will aim to present its definition and state some topological applications.

30. January 17

    Xuemiao Chen - Bogomolov inequality (Abstract)

    After introducing some necessary basics, we will present Miyaoka's elegant proof of the Bogomolov inequality for slope semistable bundles over projective smooth surfaces.

31. December 6

    Paul Cusson - A proof of the Newlander-Nirenberg Theorem (Abstract)

    The Newlander-Nirenberg Theorem is a result that determines exactly when an almost complex manifold is actually a complex manifold. That is, given an almost complex structure J on a manifold, it is integrable if and only if a certain PDE is satisfied, namely the vanishing of the Nijenhuis tensor. We will look at a proof of this theorem given by Malgrange, which uses results from the theory of nonlinear systems of elliptic PDEs.

32. November 29

    Michael Albanese - Almost Complex Four-Manifolds with no Complex Structure (Abstract)

    Which manifolds admit a complex structure? This is a very difficult problem with no concrete answer in general. To make some progress, it is useful to consider an intermediate structure known as an almost complex structure, whose existence is much easier to detect. Every complex manifold has an almost complex structure, so one is naturally led to consider the converse: does a manifold which admits an almost complex structure also admit a complex structure? This question has a positive answer in dimension two, a negative answer in dimension four, and is completely open in higher dimensions. We will discuss the results related to these conclusions and focus on why four dimensions is special.

33. November 22

    Brady Ali Medina - Lifting a co-Higgs field to a Poisson structure (Abstract)

    A quadratic Poisson structure on a holomorphic vector bundle V with coisotropic fibers induces a co-Higgs field on V. We could ask ourselves what does it take to recover the quadratic Poisson structure from its co-Higgs field. It has been proven by Mykola M. that under certain conditions a co-Higgs field on a rank-2 vector bundle V over P¹ lifts to a Poisson structure on V. In this talk, we are going to show that it is possible to extend this result for a co-Higgs field on a rank-2 vector bundle V over a complex manifold.

34. November 15

    Anton Iliashenko - Using Weitzenböck formulas to get Betti numbers using bounds on curvature operators(Abstract)

    On a Riemannian manifold there is a Weitzenböck identity which is a relationship between the Laplacian, the Bochner Laplacian and the curvature. We will consider this formula in the nearly G₂ setting and see how bounds on the Weyl Tensor or Sectional Curvature can give us certain information about the Betti numbers.

35. November 8

    Christopher Lang - Hyperbolic Monopoles with Continuous Symmetries (Abstract) (Reference)

    We examine hyperbolic monopoles with continuous symmetries and develop a structure theorem which generates spherically symmetric hyperbolic monopoles. To do this, we modify the steps laid out in a collaborative paper of mine wherein we proved a similar structure theorem for Euclidean monopoles. We discuss how these steps may be applied to other gauge theoretic objects.
    https://arxiv.org/abs/2102.01657

36. November 1

    Spiro Karigiannis - Cohomologies on almost complex manifolds and their applications (Abstract)

    We define three cohomologies on an almost complex manifold (M, J), defined using the Nijenhuis-Lie derivations induced from the almost complex structure J and its Nijenhuis tensor N, regarded as vector valued forms on M. One of these can be applied to distinguish non-isomorphic non-integrable almost complex structures on M. Another one, the J-cohomology, is familiar in the integrable case but we extend its definition and applicability to the case of non-integrable almost complex structures. The J-cohomology encodes whether a complex manifold satisfies the “del-delbar-lemma”, and more generally in the non-integrable case the J-cohomology encodes whether (M, J) satisfies a generalization of this lemma. We also mention some other potential cohomologies on almost complex manifolds, related to an interesting question involving the Nijenhuis tensor. This is joint work with Ki Fung Chan and Chi Cheuk Tsang.

37. October 25

    Lucia Martin-Merchan - A compact non-formal closed G₂ manifold with b₁=1 (Abstract)

    A G₂ structure on a 7-dimensional Riemannian manifold (M,g) is determined by a stable of 3-form ɸ. It is said to be closed if dɸ=0 and torsion-free if ɸ is parallel. The purpose of the talk is understanding certain topological properties of compact manifolds with a closed G₂ structure that cannot be endowed with any torsion-free G₂ structure. Namely, we outline the construction of such a manifold that is non-formal and has first Betti number b₁=1.

38. October 18

    Amanda Petcu - An Introduction to Calibrated Geometry (Abstract)

    We will start by using Dominic Joyce's book on Calibrated Geometry 'Riemannian holonomy groups and calibrated geometry' to give a nice introduction to calibration forms and calibrated submanifolds. We will then go through some sections of the paper by Harvey and Lawson titled "Calibrated geometries" to tie calibrated geometry into different areas of geometry. Ideally, we would get to talking about (special) Lagrangian submanifolds if we have the time.

39. October 11

    Reading Week (no talk)

40. October 4

    Xuemiao Chen - Restriction of slope semi-stable bundles (Abstract)

    Given a slope semi-stable bundle over a projective manifold, a classical theorem by Mehta and Ramanathan states that its restriction to a generic high degree hyper-surface is still slope semi-stable. This plays a key role in Donaldson’s proof for the existence of Hermitian-Einstein metrics for stable vector bundles over projective manifolds. We will discuss Flenner’s proof of the restriction theorem, which could give an effective estimate about how large the degree is needed.

41. September 27

    Benoit Charbonneau - Hyperkähler structure of bow varieties (Abstract)

    My aim is to discuss some of the background material necessary to understand the recent paper of Roger Bielawski, Yannic Borchard, and Sergey Cherkis titled “Deformations of instanton metrics” (https://arxiv.org/abs/2208.14936).

42. September 20

    Paul Cusson - Infinite Flag Manifolds and Vector Bundle Filtrations (Abstract)

    In this talk we will look at infinite flag manifolds as classifying spaces of vector bundle filtrations, which generalizes the classifying property of infinite Grassmannians for vector bundles. The main tool for this study is the fact that real and complex flag manifolds are in general iterated fiber bundles with Grassmannian components. As such, this allows one to efficiently write down the homotopy groups and cohomology rings of flag manifolds in terms of those of the Grassmannians. Bundle extension problems and obstruction theoretic results will be looked at if time allows.

43. September 13

    Organizational Meeting