Waterloo Differential Geometry Working Seminar

Academic Year 2025-2026

• September 8th

  Spiro Karigiannis - Organizational Meeting (Abstract)

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

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• January 22nd

  Spencer Kelly - Proper Group Actions and the Slice Theorem in Finite Dimensions (Abstract)

  In this talk we will begin by reviewing important properties of group actions on manifolds, and characteristics of proper actions. We then define isotropy and orbit types, discuss the slice theorem (on finite dimensional manifolds), and go over non-trivial examples of slice bundles. This will set us up to conclude with the principal orbit theorem and the stratification of the orbit space.

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• January 29th

  Viktor Majewski - Dirac Operators on Orbifold Resolutions (Abstract)

  In this talk we discuss Dirac operators along degenerating families of Riemannian manifolds that converge, in the Gromov-Hausdorff sense, to a Riemannian orbifold. Such degenerations arise naturally when analysing the boundary of Teichmüller spaces of special Riemannian metrics as well as moduli spaces appearing in gauge theory and calibrated geometry. Here sequences of smooth geometric structures on Riemannian manifolds may converge to an orbifold limit. To understand and control these degenerations, we introduce smooth Gromov-Hausdorff resolutions of orbifolds, that are, smooth families \((X_t,g_t)\), which collapse to the orbifold \((X_0,g_0)\) as \(t\to 0\).
  
  The central analytic problem addressed in this paper is to understand the behaviour of Dirac operators along such resolutions, in particular in collapsing regimes where classical elliptic estimates fail. We develop a uniform Fredholm theory for the family of Dirac operators on the Gromov-Hausdorff resolution. Using weighted function spaces, adiabatic analysis, and a decomposition of \(X_t\) into asymptotically conical fibred (ACF), conically fibred (CF) and conically fibred singular (CFS), we obtain uniform realisations of the model operators and prove a linear gluing exact sequence relating global and local (co)kernels. As a consequence, we construct uniformly bounded right inverses for \(D_t\), and derive an index additivity formula.
  
  The theory developed here provides the analytic foundation for nonlinear gluing problems in gauge theory and special holonomy geometry, including torsion-free \(G\)-structures, instantons, and calibrated submanifolds of Riemannian manifolds close to an orbifold limit.

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• February 5th

  TBD - TBD (Abstract)

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• February 12th

  Paul Cusson - TBD (Abstract)

  TBD

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• February 19th

  Pause

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• February 26th

  Facundo Camano - TBD (Abstract)

  TBD

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• March 5th

  Faisal Romshoo - TBD (Abstract)

  TBD

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• March 12th

  Amanda Petcu - TBD (Abstract)

  TBD

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• March 19th

  Spencer Kelly - TBD (Abstract)

  TBD

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• March 26th

  Viktor Majewski - Filling Holes in the \(\mathrm{Spin}(7)\)-Teichmüller Space and String Cohomology (Abstract)

  In this talk, I apply the analytic results from the first talk to study the boundary of the \(\mathrm{Spin}(7)\) Teichmüller space. Using compactness results for Ricci-flat metrics together with known examples of \(\mathrm{Spin}(7)\) manifolds, it is known that \(\mathrm{Spin}(7)\) orbifolds with \(\mathrm{SU}(n)\) isotropy arise as boundary points of the moduli space.
  
  Building on the resolution scheme for \(\mathrm{Spin}(7)\) orbifolds that I discussed in 2024, and which I will briefly review, we show how this boundary can be removed by requiring \(\mathrm{Spin}(7)\) orbifolds to encode information about their resolutions. In this way, the Teichmüller space is enlarged to include orbifold limits together with their compatible resolutions, thereby filling in the boundary.
  
  Finally, we explain how this perspective is related to a \(\mathrm{Spin}(7)\) analogue of the crepant resolution conjecture from string cohomology, providing a geometric interpretation of the obstruction complex discussed in the linear gluing analysis in the first talk.

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• April 2nd

  Faisal Romshoo - TBD (Abstract)

  TBD

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• April 9th

  Alex Pawelko - TBD (Abstract)

  TBD

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• April 16th

  Paul Cusson - TBD (Abstract)

  TBD

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• April 23rd

  Facundo Camano - TBD (Abstract)

  TBD

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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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