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September 4th
Spiro Karigiannis - Organizational Meeting (Abstract)
We will plan the speakers for the rest of the semester.
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September 11
Aleksandar Milivojevic - Realizing topological data by closed almost complex manifolds (Abstract)
I will talk about the topological obstructions to placing an almost complex structure on a smooth manifold. I will then discuss how the vanishing of these obstructions is in many cases sufficient to realize a given rational homotopy type (with a choice of cohomology classes) by an almost complex manifold (with those cohomology classes as its rational Chern classes).
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September 18th
Alex Pawelko - Calibrated Geometry of a Strongly Nondegenerate Knot Space (Abstract)
We will discuss a modification of Lee-Leung's work of the Kaehler structure on the knot space that allows one to define an infinite-dimensional analogue of G2 manifolds, then explore their calibrated geometry.
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September 25th
Jacques Van Wyk - The Clutching Construction (Abstract)
The clutching construction is a technique in differential topology to construct fibre bundles over spheres. I will explain how the clutching construction works, and how it can be used to define symplectic fibre bundles over spheres.
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October 2nd
Faisal Romshoo - Constructing examples of Smith maps (Abstract)
I will start off by giving some background on Smith maps, which are special k-harmonic maps between two Riemannian manifolds. Smith maps have deep connections with both calibrated geometry and conformal geometry. I will then discuss my current work, where I am trying to construct explicit examples of Smith immersions.
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October 9th
Paul Cusson - Various moduli spaces of monopoles (Abstract)
We will go over, in increasing generality, results classifying various classes of Euclidean \(SU(n)\)-monopoles, starting with the \(n=2\) case. We will see that these moduli spaces are described using spaces of rational maps from the projective line to flag varieties.
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October 16th
Pause
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October 23
Alexander Cole Teeter - Slice Knots and Knot Concordance (Abstract)
In this talk, we explore an overview of the interplay between Knot Theory and Four Dimensional Topology. Specifically, we look at both Topologically and Smoothly Slice Knots, which are Knots in \(S^{3}\) that bound (smoothly) embedded disks in \(B^{4}\). We explore some of the techniques in the proof that the conway knot is not smoothly slice,and look at some of the ideas involved the construction of exotic \(\mathbb{R}^{4}\)s using such knots.
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October 30
Amanda Petcu - Stability and Lyapunov Functions (Abstract)
When working with a nonlinear system of differential equations, finding explicit, closed-form solutions can be difficult. A tool in such situations is to determine the stability of the equilibrium points of the system. This analysis allows us to predict the long-term behavior of the system by examining its trajectories and how they behave near an equilibrium point: specifically, do they remain bounded in some compact set, converge to the point, or escape to infinity? In this talk, we will discuss Lyapunov's Direct Method, a technique that allows us to determine the stability of an equilibrium point without explicitly solving the differential equations.
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November 8th
Pause
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November 13th
Paul Cusson - Moduli spaces of monopoles part 2 (Abstract)
We continue discussing Euclidean \(SU(n)\)-monopoles, now in the case \(n \geq 3\), and we aim to describe their moduli spaces using spaces of rational maps from the projective line to flag varieties.
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November 20th
Facundo Camano - Dimensional Reduction of \(S^1\)-Invariant Instantons on the Multi-Taub-NUT (Abstract)
In this talk I will discuss the dimensional reduction of \(S^1\)-invariant instantons on the multi-Taub-NUT space to singular monopolos on \(\mathbb{R}^3\). I will first introduce the multi-Taub-NUT space, followed up by a discussion on \(S^1\)-equivariant principal bundles. Next, I will go over the natural decomposition of \(S^1\)-invariant connections into horizontal and vertical pieces, and then show how the self-duality equation reduces to the Bogomolny equation under said decomposition. I will then show how the smoothness of the instanton over the NUT points determines the asymptotic conditions for the singular monopole. Finally, I will go over the reverse construction: starting with a singular monopole on \(\mathbb{R}^3\) and building up to an \(S^1\)-invariant instanton on the multi-Taub-NUT space.
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November 27th
Kaleb Ruscitti - Correspondence between logarithmic connections and framed parabolic bundles on the blow up of a nodal Riemann surface (Abstract)
In this seminar I will explain how a Mehta-Seshadri type correspondence between logarithmic connections and parabolic vector bundles works for a specific setting of interest. That is the blow up of the complex curve \(xy=t\) at the nodal point
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December 4th
Spiro Karigiannis - Schwarz Lemma for Smith maps (Abstract)
I will discuss a generalized Schwarz Lemma for Smith maps, proved recently by Broder-Iliashenko-Madnick, and explain how it generalizes the classical Schwarz Lemma from complex analysis.
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December 11th
Alex Pawelko - Riemannian Geometry of Knot Spaces (Abstract)
We will review the construction of knot spaces of manifolds, specifically over \(G_2\) and \(\text{Spin}(7)\) manifolds. We will then see an explicit construction of the Levi-Civita connection of the knot space, and see what this can tell us about the torsion of the induced special geometric structures on knot spaces of \(G_2\) and \(\text{Spin}(7)\) manifolds.
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