Waterloo Differential Geometry Working Seminar

Academic Year 2023-2024

Winter ❄ Schedule - Tuesdays 2:30 pm - 4:00 pm. In person in MC 5403.
Summer ☼ Schedule - Wednesdays 1:00 pm - 4:00 pm in MC 5417.

August 21st

Speaker #1 from 1:00 PM - 2:15 PM: Jacques Van Wyk - An Introduction to Generalised Geometry (Abstract)

Generalised geometry is a field in differential and complex geometry in which one views the direct sum $TM \oplus T^*M$ instead of $TM$ as the bundle associated to a manifold $M$. Generalised geometry has seen great success in acting as a unifying framework in which structures defined on $TM$ and $T^*M$ can be viewed as specific instances of structures defined on $TM \oplus T^*M$. For example, almost complex structures and pre-symplectic structures can both be viewed as generalised almost complex structures, a certain kind of automorphism of $TM \oplus T^*M$.

In this talk, I will give an introduction to generalised geometry. I will show $TM \oplus T^*M$ comes with an intrinsic non-degenerate bilinear form. I will introduce the Dorfman bracket on $\Gamma(TM \oplus T^*M)$, an analogue of the Lie bracket, which together with the aforementioned bilinear form gives $TM \oplus T^*M$ the structure of a Courant algebroid. I will define generalised almost complex structures in this setting, and show how almost complex structures and pre-symplectic structures can be viewed as generalised almost complex structures. I will introduce generalised metrics and generalised connections, and if time permits, I will discuss integrability of generalised almost complex structures in terms of generalised connections, and/or discuss the analogue of the Levi-Civita connection and what complications it comes with.

Speaker #2 from 2:30 PM - 3:45 PM: Kyrylo Petruvshyn - TBD (Abstract)
TBD
August 14th

Speaker #1 from 1:00 PM - 2:15 PM: Spencer Whitehead - An introduction to the Nahm transform and construction of instantons on tori (Abstract)
A Nahm transform recognizes the moduli space of instantons in some setting as an isometric 'dual space'. In this sense the Nahm transform is a 'nonlinear Fourier transform'. In this talk, I will give an introduction to Nahm transforms, sketching from two different points of view the classical Nahm transform of hermitian bundles over 4-tori. Along the way, we will develop a zoo of instanton examples in all ranks using constructions from differential and complex geometry.

Speaker #2 from 2:30 PM - 3:45 PM: Lucia Martin Merchan - Closed G2 manifolds with finite fundamental group (Abstract)
In this talk, we construct a compact closed $G_2$ manifold with $b_1=0$ using orbifold resolution techniques. Then, we study some of its topological properties: fundamental group, cohomology algebra, and formality.
August 7th

Speaker #1 from 1:00 PM - 2:15 PM: Alex Pawelko - Strongly Nondegenerate Forms and their Associated Structures on Higher Knot Spaces (Abstract)
Given a Riemannian manifold M with special holonomy, one obtains a distinguished parallel differential form on it, called a strongly nondegenerate form. In this talk, we show how such a form gives rise to an almost Kaehler structure on the space of higher-dimensional knots in M, and how its (infinite-dimensional) symplectic geometry corresponds to the calibrated geometry of M. We will then discuss the special case when M is Calabi-Yau, in which we obtain additional holomorphic analogues of these results.

Speaker #2 from 2:30 PM - 3:45 PM: Faisal Romshoo - The Moduli Space of Torsion-free G2 Structures (Abstract)
The moduli space of torsion-free $\mathrm{G}_2$ structures on a compact $7$-manifold forms a non-singular smooth manifold of dimension $b^3(M)$. In this talk, we will see Joyce's proof of this fact.
July 31st

Speaker #1 from 1:00 PM - 2:15 PM: Utkarsh Bajaj - An introduction to the Mckay correspondence (Abstract)
The McKay correspondence is a bijection between the finite subgroups of $SL(2,C) $ and the Dynkin diagrams of the type $ A_r, D_r, E_6, E_7, E_8.$ One bijection takes a subgroup $G$, constructs the orbit space $ C^2/G $, resolves the singularities by inserting Riemann spheres multiple times, sees how the spheres intersect, and then constructs a graph to represent this information. Another bijection constructs irreducible representations of $ G$. We will see how these bijections are related.

Speaker #2 from 2:30 PM - 3:45 PM: Filip Milidrag -The relation between the Wythoff construction and abstract polytopes (Abstract)
In this talk we will use the Wythoff construction of a geometric polytope to describe its face lattice and then use this to make the connection between geometric polytopes and the notion of an abstract polytope. We will then go on to speak a bit about abstract polytopes and some related definitions.
July 24th

Speaker from 1:00 PM - 2:15 PM: Max Schult - Nahm's equations and rational maps (Abstract)
We study solutions to Nahm's equations on a bounded open interval up to gauge equivalence and relate them to rational maps on the projective line.
July 17th

Speaker #1 from 1:00 PM - 2:15 PM: Paul Cusson - Holomorphic Rank Two Vector Bundles Over Complex Projective Spaces (Abstract)
In order to approach the still open problem of whether all complex vector bundles over $\mathbb{CP}^n$ for $n \geq 4$ admit a holomorphic structure, we look at the simplest cases, that of rank 2 bundles over $\mathbb{CP}^4$. The Horrocks-Mumford bundle, an indecomposable holomorphic example, will be studied in more depth.

Speaker #2 from 2:30 PM - 3:45 PM: Robert Cornea - Some Calculations of Stable Pairs on $P^2$ (Abstract)
We consider what is known as Wild Vafa-Witten bundles on $P^2$. These are holomorphic vector bundles on $P^2$ along with a section $ \Phi\in H^0(End(E)\otimes\mathcal{O}(d)) $ called a Higgs field. We consider pairs $(E,\Phi)$ that are stable for a suitable stability condition. With this we consider special rank two holomorphic bundles called “Schwarzenberger bundles” and compute how many stable pairs exist. Using deformation theory, we show that this is the dimension of the tangent space at a point $(E,\Phi) $ in the moduli space of stable Wild Vafa-Witten bundles on $ P^2 $.
July 3rd * The seminar on this day will feature Canadian Undergraduate Mathematics Conference practice talks *

Practice talk #1 from 1:00 PM - 1:30 PM: Alex Pawelko - Quantifying the Fundamental Theorem of Calculus (Abstract)
The Fundamental Theorem of Calculus is amazingly cool - in one dimension it says that we can compute integrals with antiderivatives, and better yet: any continuous function always has an antiderivative! Unfortunately, that second part isn't true in higher dimensions - and it's not the fault of our functions, but rather the fault of the geometry of our spaces. This leads to an amazing idea: if we can quantify "how much" the FTC fails, we can use this to study the geometry of higher dimensions!

Practice talk #2 from 1:45 PM - 2:15 PM: Filip Milidrag - Regular Polytopes and their Wythoff Constructions (Abstract)
In this talk we will introduce the notion of a Wythoff construction and use it to generate n-dimensional polytopes. Then we will show that every regular polytope admits such a construction and we’ll see some consequences of this.

Practice talk #3 from 2:30 PM - 3:00 PM: Utkarsh Bajaj - Surfaces Formed by Discrete Quotients (Abstract)
Let a finite subgroup G of square complex matrices act on $C^n$. Then, the space $C^n/G$ denotes the set of orbits under the group action, where 2 points in C^n are the "same" if one can go to other via an element of G. What does this space look like? Believe it or not, it's usually homeomorphic to a surface defined by $f = 0$, where f is a polynomial. We'll prove this.

Practice talk #4 from 3:15 PM - 3:45 PM: Kyrylo Petruvshyn - Hyperbolic geometry and its tessellations (Abstract)
This talk I will begin with the history of hyperbolic geometry, statement of its axioms and is place among other geometries. The models of hyperbolic plane will be described. I will show how the notions of parallelism and perpendicularity behave on hyperbolic plane. Then I will show tessellations on the hyperbolic plane, including regular $\{n,3\} $for $n>=6$ and apeirogonal tiling.
June 26th

Speaker from 1:00 PM - 2:15 PM: Max Schult - Twistor spaces of oriented Riemannian 4-manifolds (Abstract)
We give the construction of an almost complex structure on the total space of the sphere bundle in the bundle of anti-self-dual 2-forms on an oriented Riemannian 4-manifold and derive an integrability condition.
June 19th

Speaker #1 from 1:00 PM - 2:15 PM: Faisal Romshoo - The Ebin Slice Theorem (Abstract)
The Ebin Slice Theorem shows the existence of a "slice" for the action of the group of diffeomorphisms $ \textrm{Diff}(M) $ on the space of Riemannian metrics $\mathcal{R}(M) $ for a closed smooth manifold $M $. We will see a proof of the existence of a slice in the finite-dimensional case and if time permits, we will go through the generalization of the proof to the infinite-dimensional setting.

Speaker #2 from 2:30 PM - 3:45 PM: Jacques Van Wyk - Bi-Lagrangian Structures on Symplectic Manifolds (Abstract)
We study symplectic manifolds equipped with bi-Lagrangian structures, that is, a pair of complementary Lagrangian distributions of the manifold. We discuss a natural integrability condition for these structures, and show how they relate to para-almost Hermitian and para-Kahler structures.
June 12th

Speaker #1 from 1:00 PM - 2:15 PM: Benoit Charbonneau - Maple for differential geometry (Abstract)
While we are certainly competent to do with pen and paper the myriad of computations required by our research, refereeing and our supervision work, I find that using tools can improve speed and accuracy and reduce frustration. I will share some principles and illustrate using Maple, including packages useful for differential geometry: difforms, DifferentialGeometry, and Clifford. Code displayed for this presentation can be found at $\href{https://git.uwaterloo.ca/bcharbon/maple-demos}{https://git.uwaterloo.ca/bcharbon/maple-demos}$

Speaker #2 from 2:30 PM - 3:45 PM: Michael Albanese - Local Conformal Flatness and Weyl Curvature (Abstract)
A Riemannian manifold is locally conformally flat if each point admits a neighborhood in which the metric is conformal to a flat metric. In dimension at least 4, a Riemannian manifold is locally conformally flat if and only if it has vanishing Weyl curvature. We will give the proof of this theorem and explain what changes in lower dimensions.
June 5th * The seminar on this day will be in MC 5501 *

Speaker #1 from 1:00 PM - 2:15 PM: Filip Milidrag - The Classification of Irreducible Discrete Reflection Groups (Abstract)
In this talk we will make a correspondence between irreducible discrete reflection groups and associated connected Coxeter diagrams. Then we will use this to classify all connected Coxeter diagrams and by extension every irreducible discrete reflection group.

Speaker #2 from 2:30 PM - 3:45 PM: Utkarsh Bajaj - Klein's icosahedral function (Abstract)
Can we define a rational function on the sphere? Sure we can. Can we define a rational function on the sphere so that it is invariant under the rotational symmetries under the icosahedron? Yes - by embedding the icosahedron in the Riemann sphere (and then doing some algebra). We then show how this beautiful function reveals connections between the symmetries of the icosahedron and the E8 lattice - the lattice that gives the most efficient packing of spheres in 8 dimensions!
May 29th

Speaker #1 from 1:00 PM - 2:15 PM: Alex Pawelko - Symmetry Reduction and the Quest for G2 Moment Maps (Abstract)
We present an overview of the classical theory of moment maps from symplectic geometry and their use within the Marsden-Weinstein-Mayer theory of symplectic reduction, with an emphasis on the Lie theoretic considerations that arise. If time permits, we will then discuss some attempts to generalize moment maps to the setting of G2 manifolds.

Speaker #2 from 2:30 PM - 3:45 PM: Paul Marriott - Statistics and Geometry: We don't talk any more. (Abstract)
George Bernard Shaw once said Britain and America are two counties separated by a common language. Perhaps the same can be said for Statistics and Geometry. This talk gives a high-level overview of a recent graduate course which explored the relationship between Statistics and Geometry. It looks at what the disciplines have in common but also where there are points of substantive difference. The talk will review the long history of geometric tools finding a place in statistical practice and will highlight modern developments using ideas from convex, differential and algebraic geometry and showing applications in Neuroscience.
May 22nd

Speaker #1 from 1:00 PM - 2:15 PM: Lucia Martin Merchan - A Grassmannian bundle over a Spin(7) manifold (Abstract)
In this talk we study the geometry of the fiber bundle $G(2,M)$ of oriented 2-planes on a Riemannian manifold $ (M,g)$ with a Spin(7) structure. More precisely, we construct an almost complex structure and we discuss how to compute its torsion when the holonomy of g is contained in Spin(7).

Speaker #2 from 2:30 PM - 3:45 PM: Anton Iliashenko - Bubble Tree (Abstract)
We motivate and construct the bubble tree for solutions to conformally invariant equations. Next, in the context of harmonic maps we prove the No Neck Energy lemma which gives us Stability and the Bubble Tree Convergence Theorem. Finally, we mention an application which is Gromov-Witten Invariants.
May 15th * The seminar on this day will be in MC 5479 *

Speaker #1 from 1:00 PM - 2:15 PM: Spiro Karigiannis - The linear algebra of 2-forms in 4-dimensions part II (Abstract)
I will present some important facts about the linear algebra of 2-forms in 4 dimensions, which everyone should know. We start with classical results about self-dual and anti-self dual 2-forms, and then proceed to discuss "hypersymplectic" structures in 4d à la Donaldson. Then we put all this on an oriented Riemannian 4-manifold.

Speaker #2 from 2:30 PM - 3:45 PM: Xuemiao Chen - Compact Riemann surfaces of low genus(Abstract)
I will make a 75-minute story regarding compact Riemann surface of low genus.
May 6th

Speaker #1 from 1:00 PM - 2:15 PM: Spiro Karigiannis - The linear algebra of 2-forms in 4-dimensions part I (Abstract)
I will present some important facts about the linear algebra of 2-forms in 4 dimensions, which everyone should know. We start with classical results about self-dual and anti-self dual 2-forms, and then proceed to discuss "hypersymplectic" structures in 4d à la Donaldson. Then we put all this on an oriented Riemannian 4-manifold.

Speaker #2 from 2:30 PM - 3:45 PM: Benoit Charbonneau - Coxeter groups and Clifford Algebras (Abstract)
If one wants to understand representation theory of the rotation group of the icosahedron, or of its lift to Sp(1), it is extremely useful to be able to compute things intelligently. It turns out that instead of using matrices, it is much better to play with Clifford Algebras. I’ll explain those concepts and illustrate them.
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.
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.
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) .
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.