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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.
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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.
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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.
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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.
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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 \( P^2 \).
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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.
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June 26th
Speaker from 1:00 PM - 2:15 PM: Max Schult - Twistor spaces of oriented Riemannian 4-manifolds
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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.
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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.
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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.
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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!
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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.
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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.
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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.
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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.