Waterloo Differential Geometry Working Seminar

Academic Year 2023-2024

1. April 16

   Aleks Milivojevic - Obstructions to almost complex structures following Massey (Abstract)

   I will report on work in progress with Michael Albanese, in which we prove statements claimed by Massey in 1961 concerning the obstructions to finding an almost complex structure on an orientable manifold (or more generally, reducing the structure group of a real vector bundle over a CW complex to the unitary group). These obstructions involve the integral Stiefel-Whitney classes – which detect the existence of integral lifts of the mod 2 Stiefel-Whitney classes, namely putative Chern classes – and relations between the Pontryagin and Chern classes. A somewhat surprising aspect of these obstructions is that they are in fact generally proper fractional parts of what one might at first expect. For example, the obstruction in degree eleven is 1/24 of the eleventh integral Stiefel-Whitney class.

2. April 9

   Lucia Martin Merchan - Hodge decomposition for Nearly Kähler manifolds (Abstract)

   Verbitsky proved that Nearly Kähler 6-dimensional manifolds satisfy Kähler-type identities. These lead to a Hodge decomposition in the compact case, and restrictions on their Hodge numbers. In this talk, we discuss a new proof for most of these results that is independent of the dimension. This is work in progress with Spiro Karigiannis, Michael Albanese and Aleks Milivojevic.

3. March 26

   Faisal Romshoo - A theoretical framework for \(H\)-strcutures (Abstract)

   For an oriented Riemannian manifold \((M^n, g)\), and Lie subgroup \(H \subset SO(n)\), a compatible \(H\)-structure on \((M^n,g)\) is a principal $H$-subbundle of the principal $SO(n)$-bundle of oriented orthonormal coframes. They can be described in terms of the sections of the homogeneous fibre bundle obtained by \(H\)-reduction of the oriented frame bundle. Examples of these structures include \(U(m)\)-structures, \(G_2\)-structures and \(\text{Spin(7)}\)-structures. In this talk, we will study a general theory for \(H\)-structures described in a paper of Daniel Fadel, Eric Loubeau, Andrés J. Moreno and Henrique N. Sá Earp titled "Flows of geometric structures" (https://arxiv.org/abs/2211.05197) .

4. March 19

   Amanda Petcu - An Introduction to (Lagrangian) Mean Curvature Flow (Abstract)

   In this talk, we will introduce the Mean Curvature Flow and explore some initial examples of the flow. We will show that in the compact case, the flow always produces singularities. We will also introduce type I and type II singularities. Finally, if time permits, we will discuss the Lagrangian Mean Curvature Flow and demonstrate that a mean curvature flow starting from a Lagrangian remains Lagrangian.

5. March 12

   Timothy Ponepal - Nijenhuis tensor of the almost complex structure on classical twistor space (Abstract)

   Let E be the bundle of anti-self dual 2-forms on an oriented Riemannian 4-manifold (M, g), with connection induced from g. Let Z be the sphere bundle of radius \(\sqrt{2}\). We show that the 6-manifold Z admits an almost complex structure J which respects the splitting \( TZ = VZ \oplus HZ\) given by the connection. In the horizontal directions, this complex structure J is defined in terms of the canonical 2-form on \(\Lambda^2 (T^* M)\), and in the vertical directions it is the standard complex structure on \(S^2\). We compute the Nijenhuis tensor of J evaluated on one horizontal and one vertical vector field.

6. March 5

   Guillermo Gallego - Multiplicative Higgs bundles, monopoles and involutions (Abstract)

   Multiplicative Higgs bundles are a natural analogue of Higgs bundles on Riemann surfaces, where the Higgs field now takes values on the adjoint group bundle, instead of the adjoint Lie algebra bundle. In the work of Charbonneau and Hurtubise, they have been related to singular monopoles over the product of a circle with the Riemann surface. In this talk we study the natural action of an involution of the group on the moduli space of multiplicative Higgs bundles, also from the point of view of monopoles. This provides a "multiplicative analogue" of the theory of Higgs bundles for real groups.

7. February 13

   Timothy Ponepal - The flow of the horizontal lift of a vector field (Abstract)

   Let E be a vector bundle over a manifold M, and let $\nabla$ be a connection on E. Given a vector field $X$ on $M$, the connection determines its horizontal lift X ^h, which is a vector field on the total space of E. We will show that the flow of X ^h is related to parallel transport with respect to $\nabla$. If time permits, we will show that in the special case when E is a rank 3 oriented real vector bundle with fibre metric, the flow of X ^h preserves the cross product on the fibres.

8. February 6

   Spiro Karigiannis - An exercise in Riemannian geometry (or how to make a Riemannian geometric omelet without breaking any eggs) (Abstract)

   I will describe a particular class of Riemannian metrics on the total space of a vector bundle, depending only on one natural coordinate $r$, and which are thus of cohomogeneity one. Such metrics arise frequently in the study of special holonomy, By carefully thinking before diving in, one can extract many useful formulas for such metrics without needing to explicitly compute all of the Christoffel symbols and the curvature. For example, these include the rough Laplacian of a function or of a vector field which are invariant under the symmetry group. If time permits, I will explain why I care about such formulas, as they are ingredients in the study of cohomogeneity one solitons for the isometric flow of $\mathrm{G}_2$-structures

9. January 31

   Faisal Romshoo - Some computations with gauge tranformations on a G2 manifold (Abstract)

   Given a (torsion-free) G ₂ -manifold $(M, \varphi, g)$ and a gauge transformation $P: TM \rightarrow TM$, we want to look at the $G_2$ structures $\Tilde{\varphi} = P^*g$ and explore the conditions for it to be torsion-free. In this talk, we will start in a more general setting with a Riemannian manifold $(M, g)$ and obtain an expression for the tensor $B(X, Y) = \tilde{\nabla}_X Y -\nabla_X Y$ before moving on to computing the full torsion tensor $\tilde{T}_{pq}$ in the case when $M$ is a $G_2$ manifold.

10. January 16

    Benoit Charbonneau - Deformed Hermitian-Yang-Mills equation (Abstract)

    The Deformed Hermitian-Yang-Mills equation has been an intense topic of study in the recent past. I will describe the equation, the concept of central charge pertinent in this story, and various conjectures and progress that has been made.

11. November 21

    Spiro Karigiannis - Flows of G2 structures (Abstract)

    A G2-structure is a special type of 3-form on an oriented 7-manifold, which determines a Riemannian metric in a nonlinear way. The best class of such 3-forms are those which are parallel with respect to their induced Levi-Civita connections, which is a fully non-linear PDE. More generally, the torsion of a G2-structure is a 2-tensor which quantifies the failure of a G2-structure to be parallel. It is natural to consider geometric flows of G2-structures as a means of starting with a G2-structure with torsion and (hopefully) improving it in some way along the flow. I will begin with an introduction to all of these ideas, and try to survey some of the results in the field. Then I will talk about recent joint work with Dwivedi and Gianniotis to study a large class of flows of G2-structures. In particular, we explicitly describe all possible second order differential invariants of a G2-structure which can be used to construct a quasi-linear second order flow. Then we find conditions on a subclass of these general flows which are amenable to the deTurck trick for establishing short-time existence and uniqueness.

12. November 7

    Frederik Benirschke (University of Chicago) - An introduction to bi-algebraic geometry, with a view towards strata of differentials (Abstract)

    I will introduce some of the main ideas in bi-algebraic geometry, focusing on the case of algebraic tori and Abelian varieties. Afterward, we explore these ideas for a natural bi-algebraic structure on the moduli space of marked points on the sphere that arises from integrating differential forms.

13. October 31

    Lucia Martin Merchan - Hodge decomposition for Nearly Kähler 6-manifolds (Abstract)

    In this talk we discuss the paper of M. Verbitsky (arXiv:math/0510618) where he finds Kähler identities for Nearly Kähler 6-manifolds. From that, he deduces a Hodge decomposition in the compact case, as well as some restrictions on their refined Betti numbers.

14. October 24

    Amanda Petcu - Partial progress on a conjecture of Donaldson by Fine and Yao (Part 2) (Abstract)

    Given a compact hypersymplectic manifold X⁴, Donaldson conjectured that the hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperkähler structure. Fine and Yao consider a manifold with closed G₂-structure that is set up as 𝕋³ × X⁴. They examine the G₂-Laplacian flow under in this setting and give a flow of hypersymplectic structures which evolve according to the equation ∂_t ω = d(Q d^*(Q^-1 ω), where ω is the triple that gives the hypersymplectic structure and Q is a 3 × 3 symmetric matrix that relates the symplectic forms ω_i to one another. Lotay—Wei have established long time existence of the G₂-Laplacian flow provided the velocity of the flow remains bounded. Fine—Yao use this extension theorem in their setup and manage to improve it by proving long time existence of the hypersymplectic flow provided the torsion tensor T remains bounded. Furthermore, one can relate the scalar curvature and torsion tensor of manifold with closed G₂-structure and thus they conclude long time existence for the hypersymplectic flow provided the scalar curvature remains bounded. In this talk we will go over some details from this paper by Fine—Yao.

15. October 17

    Aleksandar Milivojevic - Formality and non-zero degree maps (Abstract)

    I will talk about a recent result with J. Stelzig and L. Zoller showing that formality is preserved under non-zero degree maps. Namely, if the domain of a non-zero degree map is formal, then so is the target. Some geometric applications are the formality of closed orientable manifolds with large first Betti number admitting a metric with non-negative Ricci curvature, and alternative proofs of the formality of positive quaternion-Kähler manifolds, and of singular complex varieties satisfying rational Poincaré duality.

16. October 3

    Spiro Karigiannis - A curious system of second order nonlinear PDEs for U(m)-structures on manifolds (Abstract)

    Compact Kähler manifolds possess a number of remarkable properties, such as the Kähler identities, the ∂∂-lemma, and the relation between Betti numbers and Hodge numbers. I will discuss an attempt in progress to generalize some of these ideas to more general compact U(m)-manifolds, where we do not assume integrability of the almost complex structure nor closedness of the associated real (1,1)-form. I will present a system of second order nonlinear PDEs for such a structure, of which the Kähler structures form a trivial class of solutions. Any compact non-Kähler solutions to this second order system would have properties that are formally similar to the above-mentioned properties of compact Kähler manifolds, including relations between cohomological (albeit non-topological) data. This is work in progress with Xenia de la Ossa (Oxford) and Eirik Eik Svanes (Stavanger).

17. September 26

    Michael Albanese - Generalised Kähler-Ricci Solitons (Abstract)

    The notion of a Kähler-Ricci soliton arises from the study of the Kähler-Ricci flow. They can only exist on certain manifolds, namely Fano manifolds. In this restricted case, there is an equivalent formulation which is no longer equivalent in the non-Fano case - these are called Generalised Kähler-Ricci solitons. I intend to discuss both Kähler-Ricci solitons and Generalised Kähler-Ricci solitons, as well as some differences between them.

18. September 19

    Anton Iliashenko - A special class of harmonic map submersions with calibrated fibres (Abstract)

    Say we have a conformally horizontal submersion between two Riemannian manifolds of dimensions n and k respectively. If the domain admits a closed (n-k)-calibration form, then we can define a special type of maps which we call conformally calibrated (or Smith) maps. We will show that these maps are k-harmonic and we will see a couple of examples.

19. September 12

    Amanda Petcu - Partial progress on a conjecture of Donaldson by Fine and Yao (Abstract)

    Given a compact hypersymplectic manifold X⁴, Donaldson conjectured that the hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperkähler structure. Fine and Yao consider a manifold with closed G₂-structure that is set up as 𝕋 × X⁴. They examine the G₂-Laplacian flow under in this setting and give a flow of hypersymplectic structures which evolve according to the equation ∂_t ω = d(Q d^*(Q^-1 ω), where ω is the triple that the hypersymplectic structure and Q is a 3 × 3 symmetric matrix that relates the symplectic forms ω_i to one another. Lotay—Wei have established long time existence of the G₂-Laplacian flow provided the velocity of the flow remains bounded. Fine—Yao use this extension theorem in their setup and manage to improve it by proving long time existence of the hypersymplectic flow provided the torsion tensor T remains bounded. Furthermore, one can relate the scalar curvature and torsion tensor of manifold with closed G₂-structure and thus they conclude long time existence for the hypersymplectic flow provided the scalar curvature remains bounded. In this talk we will go over some details from this paper by Fine—Yao.