Waterloo Differential Geometry Working Seminar

Speaker: Spiro Karigiannis

Title: Organizational Meeting

Abstract: We will plan the speakers for the rest of the semester.

Speaker: Viktor Majewski

Title: Non-Existence of Smooth Full-Holonomy Cayley Fibrations

Abstract: In this talk, I complete the proof that every Cayley fibration of a compact torsion-free \(\textrm{Spin(7)}\)-manifold with full holonomy must have singular fibers. This confirms a long-standing expectation in the study of calibrated fibrations with exceptional holonomy. Last semester, I showed that using the rigidity of the \(\textrm{Spin(7)}\)-structure to reduce any hypothetical nonsingular Cayley fibration to two possible topological configurations. This time, I exclude both by proving a new spinnability theorem for smooth fiber bundles over simply-connected \(4\)-manifolds with fiber homeomorphic to the elliptic surfaces \(E(2)\) or \(E(4)\). The \(E(4)\) case, which constitutes the main new difficulty, is established by combining families Seiberg–Witten theory with parametrized equivariant homotopy theory and equivariant \(K\)-theory.

Speaker: Paul Cusson

Title: Spectral data of \(\mathrm{SU}(2)\)-monopoles with symmetry group \(S^1\)

Abstract: We will prove the existence and uniqueness of a Euclidean \(\mathrm{SU(2)}\)-monopole with a given charge, center, \(S^1\) symmetry group, and axis of symmetry. This will be proven using a theorem of Hitchin regarding the equivalence of \(\mathrm{SU(2)}\)-monopoles and the cohomological data obtained from spectral curves.

Speaker: Facundo Camano

Title: \(\mathrm{SU}(2)\) is lying to us.

Abstract: We will compute the index of the deformation operator of a BPS monopole on \(\mathbb{R}^3\). During this computation, we will encounter how classical index results fall short in the presence of symmetry breaking. A hybridization of the \(b\)-calculus and scattering calculus will be the remedy to bypass the symmetry breaking and perform the computation.

Reading week

Speaker: Benoit Charbonneau

Title: TBA

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Speaker: Spencer Kelly

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Speaker: Faisal Romshoo

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Speaker: Francisco Villacis

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Speaker: Jacques Van Wyk

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Speaker: Faisal Romshoo

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Speaker: Danial Ghamari

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Speaker: Spencer Kelly

Title: TBA

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Speaker: Danial Ghamari

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