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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 - TBD (Abstract)
TBD
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February 26th
Facundo Camano - TBD (Abstract)
TBD
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March 5th
Paul Cusson - TBD (Abstract)
TBD
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March 12th
Faisal Romshoo - TBD (Abstract)
TBD
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March 19th
Amanda Petcu - TBD (Abstract)
TBD
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March 26th
Kain Dineen - TBD (Abstract)
TBD
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April 2nd
Jacques Gideon van Wyk - 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 \(\mathbb{R}^7\).