Waterloo Differential Geometry Working Seminar

Academic Year 2021-2022

1. August 17

   Paul Mcauley - Vertical and Horizontal Spaces (Abstract)

   For a given Riemannian manifold (M,g), a vector bundle E over M, we can define the vertical space VE which is a submanifold of TE. Given a connection on E we can then define the horizontal space HE which is a submanifold of TE. These spaces give us a fibre metric on TE and then we can look at the Levi-Civita connection in terms of these vertical and horizontal spaces.

   Shengda Hu - Spinors for split signature pairing (Abstract)

   We describe some explicit constructions of spinor modules when the pairing has split signature.

2. August 10

   Amanda Petcu - Isometric immersions part 4 (Abstract)

   This talk will bring an end to our series on isometric immersions. We will define a totally geodesic immersion and a minimal submanifold. To finish we will introduce and prove the three fundamental equations: Gauss' equation, Ricci's equation, and Codazzi's equation.

   Spiro Karigiannis - Some observations about associative 3-planes in R7 and the Lie algebra g2 (Abstract)

   I will show that given an associative 3-plane in R7, it determines a Lie subalgebra of so(7) isomorphic to so(3) \oplus so(3). Some possible applications will be discussed.

3. July 27

   Amanda Petcu - Isometric immersions part 3 (Abstract)

   We will continue with our discussion of the second fundamental form and introduce Gauss' equation and compute some examples. If we have time, we might discuss the normal connection and the normal curvature and perhaps finish off with the three fundamental equations: Gauss' equation, Ricci's equation, and Coddazzi's equation.

   Anton Iliashenko - Pinching estimates for curvature operators (Abstract)

   This talk is based on the paper "Curvature Operators: Pinching Estimates and Geometric Examples" by Jean-Pierre Bourguignon and Hermann Karcher. The curvature tensor R can be thought of as a symmetric map acting on 2-forms and also on symmetric 2-tensors. We will see how bounds on the sectional curvature of R give us bounds on eigenvalues of both of those operators.

4. July 20

   Recording and Discussion: Cliff Taubes - An AMS recorded lecture (1995) (Abstract)

   We will watch an AMS recorded lecture from 1995 by Cliff Taubes about gauge theory, Donaldson invariants, and Seiberg-Witten invariants. This talk is just under two hours long. We will start at 9:30am and possibly take a short break halfway through. The video can be found here: http://www.math.stonybrook.edu/Videos/Einstein/455-19950101-Taubes.html

5. July 6

   Recording and Discussion: Sven Hirsch (Duke University) - Spacetime harmonic functions and applications to relativity (2021)

   Recording and Discussion: Vladimir Arnold (Steklov Mathematical Institute of Russian Academy of Sciences) - Statistics of the Morse theory of smooth functions (2008) (Abstract)

   On a two-dimensional sphere there are exactly 17746 topologically different Morse functions with 4 saddles. This result, based on combinatorics of random graphs, was obtained only a couple of years ago (during the research of Hilbert's 16th problem in real algebraic geometry on topological classification of polynomials). [In 2007], the American mathematician L. Nikolaescu proved the Arnold conjecture that the number of such functions with T saddles on a two-dimensional sphere grows T to degree 2T (using methods going back to quantum field theory and to the mirror symmetry theory of physics). The paper describes these studies and their counterparts for the theory of smooth functions on other manifolds, e.g. for functions on a torus and for trigonometric polynomials with a fixed Newton diagram of a given Coxter affine group.

6. June 29

   Catalina Quincosis Martínez - One way to tell a simple polytope from its graph (Abstract)

   The Perles conjecture is a question about reconstruction and representation of convex polytopes. In this talk, we will focus on the result for simple polytopes, presenting a short and algorithmic proof due to Gil Kalai. The reconstruction of a simple polytope from its graph was first done by Blind and Mani, and recent work has been aimed at finding a polynomial-time algorithm to accomplish it.

   Paul Mcauley - Lifting the Metric into the Tangent Space (Abstract)

   For a Riemannian manifold (M,g), we will look at splitting the tangent space of TM into two parts, the vertical space and the horizontal space. We will then look at a metric on TM induced by g which makes the vertical and horizontal spaces orthogonal. We will also investigate the Levi-Civita connection of this metric.

7. June 22

   Amanda Petcu - Isometric Immersions, Part 1 (Abstract)

   This series of talks will cover Chapter 6 of Manfredo Do Carmo's book 'Riemannian Geometry'. In this specific talk we will introduce the notion of an isometric immersion and the second fundamental form. We will see some examples as well as some properties of isometric immersions and the creation of the second fundamental form for a Riemannian manifold.

   Shengda Hu - Generalized symmetry from Hermitian line bundles (Abstract)

   We describe how the generalized curvatures of a Hermitian line bundle correspond to generalized symmetries via Lax flow of a generalized metric.

8. June 15

   Anton Iliashenko - Some results about Riemannian submersions (Abstract)

   We will go through some results about Riemannian submersions from the Arthur L. Besse's book on Einstein manifolds. For example, we will show that a surjective Riemannian submersion is harmonic iff the fibers are minimal, and some others as the time permits

   Recording and Discussion: Jean-Pierre Bourguignon (President, European Research Council) - What is a spinor? (Abstract)

   This was the title of the lecture Sir Michael gave in September 2013 at IHES on the occasion of the farewell conference for my retirement as Director. This was most appropriate as I learned a lot from him about this subject. It is true that mathematicians struggled for a long time to get acquainted with spinors. It is in sharp contrast with the fact that physicists adopted them without hesitation as soon as Paul-Adrien Maurice Dirac showed they were essential to formulate a quantum equation invariant under the Poincaré group. Indeed spinors have a number of features that make them both subtle and powerful to deal with mathematical problems. Of great importance are of course the natural differential operators universally defined on spinor fields, namely the Dirac and the Penrose operators. The purpose of the lecture is to revisit historical steps taken to master these objects, explore their remarkable geometric content and present some mathematical problems on which they shed light.
   
   Note: This is a *pre-recorded lecture* which was given as part of the Maxwell Institute's "Atiyah Lecture Series" in Edinburgh on January 11, 2021. We watched the first half of the lecture two weeks ago, and we will watch the rest of it this time. As we did last time, we will pause the video at any moment to ask each other questions and hopefully get meaningful answers from each other. We watched the first half in Fall 2021, but never finished it, and there's been a lot of turnover in our audience, so we'll just watch the whole talk from the beginning.
   
   The video is available here: https://www.icms.org.uk/events/2021/inaugural-atiyah-lecture-jean-pierre-bourguignon.

9. June 8

   Amanda Petcu - Introduction to Kähler Geometry, Part 2 (Abstract)

   We will continue with the basics of complex geometry to build the necessary knowledge for studying Kähler geometry. More precisely we will talk about holomorphic forms and vector fields and, certain theorems that relate to integrability of the manifold and almost complex structures.

   Spiro Karigiannis - McLean's second variation formula revisited, Part 2 (Abstract)

   Last time we discussed calibrations satisfying a Harvey-Lawson identity, and focused on the associative and coassociative cases. We also derived the general formula for determining the infinitesimal deformations of calibrated submanifolds, and worked out the coassociative case. This time we'll work out the associative case, including understanding how the normal bundle is a Clifford bundle with an associated Dirac operator. Then we'll give the Van Le - Vanzura proof of McLean's second variation formula in general.

10. June 1

    Daren Cheng - A strong stability condition on minimal submanifolds and its implications, Part 2 (Abstract)

    I will continue where I left off last time with the paper by Tsai and Wang (J. Reine Angew. Math., 2020) and finish presenting the proof of their uniqueness theorem for strongly stable minimal submanifolds. Then I will go over a few examples of strongly stable minimal submanifolds that they discuss in the paper. In most of these examples, the submanifold is calibrated to begin with.

    Spiro Karigiannis - McLean's second variation formula revisited (Abstract)

    I will discuss a paper of the same title by Van Le and Vanzura. The authors revisit the second variation formula for calibrated submanifolds, originally due to McLean, especially in the associative and Cayley settings. They give simpler proofs, modulo some results of Gayet and Ohst relating the normal bundles in these settings to twisted spinor bundles. I will try to review those results as well.

11. May 25

    Guest Speaker: Tommaso Pacini (University of Torino) - One theorem, three points of view (live via Zoom) (Abstract)

    Consider the standard 2-sphere. Rotations around the z-axis generate a family of orbits, containing a unique maximal orbit: the equator (a geodesic). The 2-sphere is Kähler, and rotations are isometries. It turns out that the above picture generalizes to this broad context (which includes for example toric manifolds), leading to an interesting interplay between (i) curvature and volume, (ii) potential theory and Riemannian submersions, (iii) Riemannian, complex and symplectic geometry. We will also discuss extensions to the recent potential theory on calibrated manifolds due to Harvey-Lawson.

    Amanda Petcu - An Introduction to Kähler Geometry: Part 1 (Abstract)

    This talk will focus on the background knowledge needed to begin studying Kähler Geometry. In particular we will focus on building knowledge of complex geometry. We will introduce complex manifolds and almost complex manifolds, as well as holomorphic forms and vector fields and, holomorphic vector bundles. We will examine a few examples and theorems of integrable structures and holomorphic structures. Lastly, we might even talk about Kähler metrics if I get to it in time.

12. May 18

    Daren Cheng - A strong stability condition on minimal submanifolds and its implications, Part 1 (Abstract)

    I will talk about the paper by C.-J. Tsai and M.-T. Wang (J. Reine Angew. Math., 2020) in which they introduce the class of "strongly stable" minimal submanifolds and generalize to these their earlier work on the uniqueness and dynamical stability (with respect to the mean curvature flow) of the zero section as a minimal submanifold in the total space of certain vector bundles equipped with special holonomy metrics. I will begin by describing the questions they want to address in the paper. Then, I will focus on the first of their two main theorems, which states that in an arbitrary Riemannian manifold, every strongly stable, closed minimal submanifold has a tubular neighborhood which contains no other closed minimal submanifolds of at least the same dimension. The proof boils down to showing that the squared distance to the given submanifold has a certain convexity property.

    Rutgers Geometric Analysis Conference - Live Stream (Details)

    We will live stream the Rutgers Geometric Analysis Conference. The speaker is Scott Wilson, and the talk is about formality of compact complex manifolds. The full schedule for this conference is here.

13. May 11

    Spiro Karigiannis - Calibrated subbundles of ℝ⁷, part II (Abstract)

    I will continue, and hopefully conclude, the discussion of calibrated subbundles in ℝ⁷. We will quickly review the previous talk and then focus on the coassociative case, which corresponds to negative superminimal surfaces.

    Spencer Whitehead - Generalized Pythagorean Lutes (Abstract)

    The lute of Pythagoras is a well-known self-similar pattern filling the plane with regular pentagons and pentagrams. This talk describes a generalization to higher dimensions and classification of such objects, yielding lutes corresponding to the dodecahedron in three dimensions and the 120-cell in four dimensions. A semi-formal discussion of these results is presented, together with Zome models and computer images of the new lutes.

    Anton Iliashenko - Instantons on Flat Spaces (Abstract)

    This talk is based on the paper by Jason D. Lotay and Thomas Bruun Madsen "Instantons on Flat Space: Explicit Constructions". Currently, the idea of generalisation of the (anti-)self-duality conditions to higher dimensions is getting a lot of attention. In this talk, we take the basic (or BPST) instanton on ℝ⁴ with its flat hyperkähler structure and show that it has natural generalisations to ℝ⁷ and ℝ⁸ viewed as flat G₂- and Spin(7)-manifolds, respectively.

14. April 28

    Shengda Hu - Riemannian Metrics and Generalized Geometry (Abstract)

    We will discuss some computations involving Riemannian metrics in the context of generalized geometry. We will try to illustrate various constructions involving metric connections and their curvatures.

15. April 21

    Spiro Karigiannis - Calibrated subbundles of ℝ⁷ (Abstract)

    One can view ℝ⁷ as the total space of the bundle E = Λ²-(ℝ⁴) of anti-self-dual 2-forms on ℝ⁴. In this way we can describe the standard flat G₂-structure in terms of 4-dimensional geometry. Given an oriented surface M² in ℝ⁴, the restriction of E to M decomposes as a direct sum of a line bundle and a rank 2 bundle. We determine conditions on the immersion of M² in ℝ⁴ that are equivalent to the total spaces of the subbundles being (respectively) associative and coassociative submanifolds. This is (very old) work of myself, Ionel, and Min-Oo from 2005. I will also discuss several ways in which it has already been generalized and ways in which it can potentially still be generalized further.

16. April 14

    Christopher Lang - The Spectral Curve of a SU(2) Monopole (Part 2): Identifying Subbundles (Abstract)

    We will be following Hitchin's 1982 paper, Monopoles and Geodesics, continuing from the last talk. This time, we find two holomorphic subbundles of the holomorphic vector bundle from the previous talk and identify them. Time permitting, we will define the spectral curve and discuss its properties.

17. April 7

    Dennis The - Simply-transitive CR real hypersurfaces in C³ (Abstract)

    Holomorphically (locally) homogeneous CR real hypersurfaces M³ in C² were classified by Elie Cartan in 1932. A folklore legend tells that an unpublished manuscript of Cartan also treated the next dimension M⁵ in C³ (in conjunction with his study of bounded homogeneous domains), but no paper or electronic document currently circulates. Starting from around the year 2000, significant progress was made on this 5-dimensional classification problem, and the classification was completed in 2020. I will discuss the final aspects of this effort, in particular giving a survey of my joint work with Doubrov & Merker in which we used a novel approach to settle the simply-transitive, Levi non-degenerate classification.

18. March 24

    Daren Cheng - Incompressible minimal surfaces and topological consequences of positive scalar curvature (Part 4) (Abstract)

    Suppose M is a closed 3-manifold whose fundamental group contains a copy of the fundamental group of T² and fix a metric h on M. Given a metric g on T², in the previous talk I described how Schoen and Yau (Annals, 1979) produced an energy minimizer among maps from (T², g) into (M, h) that induce the same π₁-action as a given smooth map from T² to M, which we can choose to be π₁-injective thanks to the assumption on π₁(M). In this talk I will explain the second stage of the minimization process, in which one varies the domain metric g. It is to prove convergence in this step that the π₁-injectivity of the maps found above is needed. The result is that (M, h) admits a branched immersed stable minimal torus. As mentioned last time, this prevents the ambient metric h from having positive scalar curvature, modulo ruling out the branch points, which we will not address in the talk. Instead, time permitting, I will elaborate on some of the regularity results used in the process for energy-minimizing maps from two-dimensional domains.

19. March 17

    Amanda Petcu - Hamiltonian Structures for Evolution Equations Describing Pseudo-Spherical Surfaces (Abstract)

    The calculation of conservation laws for a differential equation has been a problem of interest for many researchers. The conservation laws that arise naturally from physics such as conservation of mass and momentum are but a drop in a bucket. This is why we are very interested in algorithms that could provide an infinite sequence of conservation laws for certain classes of evolution equations. This seminar will explore two classes of evolution equations for which there exist algorithms that create an infinite hierarchy of conservation laws for the equation. The first class are evolution equations that describe pseudo-spherical surfaces. The second class are evolution equations which admit a multi-Hamiltonian structure. Since these two classes of evolution equations share this property, the question of whether or not there exist evolution equations that describe pseudo-spherical surfaces and also admit a multi-Hamiltonian structure will be explored in the case of the KdV equation and another quintic evolution equation.

20. March 3

    Christopher Lang - The Spectral Curve of a SU(2) Monopole (Part 1): A Holomorphic Vector Bundle (Abstract)

    In this talk, we will be following Hitchin's 1982 paper, Monopoles and Geodesics, which defines spectral curves and discusses their relationship with SU(2) monopoles. This talk will focus on the creation of a holomorphic vector bundle on the space of oriented geodesics of R^3 as well as some of its properties.

21. February 17

    Anton Iliashenko - Curvature of almost Hermitian manifolds (Abstract)

    Let (M,g,J) be a Riemannian manifold with g-orthogonal almost complex structure J. The Riemann curvature tensor satisfies certain identities involving the torsion ∇J. We will focus on the Kahler case (when ∇J = 0) and if time permits we will make some remarks about the general case.

22. February 10

    Spencer Whitehead - Integrality theorems for symmetric instantons (Abstract)

    A symmetric instanton is a solution to the finite-energy anti-self-dual instanton equation on R^4 in which the connection commutes with some perscribed group of symmetries. This talk introduces the symmetric ADHM equations for structure group SU(N), and how index-theoretic methods can be used to derive integrality theorems for different symmetry groups. In the case of N=2 and a discrete subgroup of SU(2), we derive the prime charge theorem, which imposes a canonical choice of representation for the equations at prime number charges. In the case of N=2 for direct product subgroups of Spin(4), we use the index theorem to resolve a problem of Allen and Sutcliffe, showing that the minimal charge of a non-trivial instanton with the symmetries of the 600-cell is 119. Time permitting, we will describe the notion of "quasi-irreducibility" for these objects, and its use in a path to a complete classification of symmetric instantons.

23. February 3

    Spiro Karigiannis - Intro to calibrated geometry (Abstract)

    I will give a brief introduction to the ideas of calibrated geometry, as introduced in 1982 by Harvey-Lawson. In particular I will focus on the Kahler, special Lagrangian, and associative calibrations.

24. January 27

    Daren Cheng - Incompressible minimal surfaces and topological consequences of positive scalar curvature (Part 3) (Abstract)

    I will continue talking about the paper by Schoen and Yau (Annals, 1979) on the topological consequences of positive scalar curvature on 3-manifolds, focusing on the following special case of their main theorem: a closed 3-manifold admits no positive scalar curvature (PSC) metric if its fundamental group contains a subgroup isomorphic to ℤ ⊕ ℤ. I will begin with a quick review of my two previous talks, in which I explained how the PSC condition forbids the existence of immersed stable minimal tori, outlined how to use the assumption on the fundamental group to construct such a minimal torus, and described the initial steps of this latter construction. I will then attempt to explain the remainder of the construction, which involves minimizing the Dirichlet energy in two stages. Some classical regularity and compactness results for energy-minimizing harmonic maps will be taken for granted.

25. November 17

    Daren Cheng - Incompressible minimal surfaces and topological consequences of positive scalar curvature (Part 2) (Abstract)

    I'll continue talking about the paper by Schoen-Yau (Annals, '79) on the topology of 3-manifolds admitting positive scalar curvature (PSC) metrics. In Part 1 I explained how the PSC condition is incompatible with the existence of immersed stable minimal tori. In this talk, I'll explain how to produce such a minimal torus when the fundamental group of the 3-manifold contains a copy of Z \oplus Z. The idea is to first construct a \pi_1-injective map from the torus into the 3-manifold, and then find a stable minimal immersion by minimizing the Dirichlet energy first among maps inducing the same \pi_1-action, and then among conformal classes of metrics on the torus.

26. November 10

    Recording and Discussion: Jean-Pierre Bourguignon (President, European Research Council) - What is a spinor? (Abstract) (Note)

    This was the title of the lecture Sir Michael gave in September 2013 at IHES on the occasion of the farewell conference for my retirement as Director. This was most appropriate as I learned a lot from him about this subject. It is true that mathematicians struggled for a long time to get acquainted with spinors. It is in sharp contrast with the fact that physicists adopted them without hesitation as soon as Paul-Adrien Maurice Dirac showed they were essential to formulate a quantum equation invariant under the Poincaré group. Indeed spinors have a number of features that make them both subtle and powerful to deal with mathematical problems. Of great importance are of course the natural differential operators universally defined on spinor fields, namely the Dirac and the Penrose operators. The purpose of the lecture is to revisit historical steps taken to master these objects, explore their remarkable geometric content and present some mathematical problems on which they shed light.
    This is a *pre-recorded lecture* which was given as part of the Maxwell Institute's "Atiyah Lecture Series" in Edinburgh on January 11, 2021. We plan to watch the recorded lecture together (in-person and/or on Zoom, as usual this term) and we can pause the video at any moment to ask each other questions and hopefully get meaningful answers from each other. The video is available here.

27. November 3

    Anton Iliashenko - Riemannian Submersions, Part II (Abstract)

    This is the second talk on Riemannian Submersions. In the first talk we considered the local viewpoint by introducing the O’Neil tensors and used them to get identities for the curvature of the total space in terms of the base and the fibers. In this talk we switch to the global theory. In particular, assuming completeness of the total space we will prove that a Riemannian submersion is a locally trivial fiber bundle. Next, we will focus on the case with totally geodesic fibers, which has a very interesting family of examples.

28. October 27

    Daren Cheng - Incompressible minimal surfaces and topological consequences of positive scalar curvature (Abstract)

    I'll talk about one of the main theorems from the classical paper by Schoen and Yau (Annals, '79) where they find topological obstructions on a 3-manifold to the existence of metrics with positive scalar curvature (PSC). Specifically, the result says that if the fundamental group of M contains a finitely generated, non-cyclic abelian subgroup, then M does not carry any PSC metric. The method of proof resembles the Bochner technique, but uses minimal surfaces instead of harmonic forms. I'll start from the end of the story by explaining how the PSC condition is incompatible with the existence of immersed stable minimal tori. I'll then outline how they use harmonic maps to prove that the above condition on the fundamental group implies the existence of such a minimal torus. The remaining details of this last step will be described in another talk.

29. October 20

    Spiro Karigiannis - Generalized superminimal surfaces and holomorphic representation (Part 2) (Abstract)

    I will continue from where we left off last time. We have related conformal minimal immersions of a domain in the complex plane into R^n to harmonic maps, and packaged them in terms of holomorphic data. We have also discussed complex maps and superminimal maps as special cases. This time we will fit these special cases into a hierarchy using complex analysis.

30. October 4

    Anton Iliashenko - Riemannian submersions and the O’Neill tensors (Part I) (Abstract)

    This talk will be one of the two talks on Riemannian submersions. Both of them are based on Chapter 9 from the “Einstein Manifolds” book by Arthur L. Besse. First, we will consider things locally by introducing the O’Neill’s tensors A and T. We will use them to calculate the curvature of the total space in terms of the curvatures of the base and the fibers. After that, we will consider the global theory under some assumptions about the total space.

31. September 29

    Spiro Karigiannis - Generalized superminimal surfaces and the Weierstrass representation (Abstract)

    A superminimal surface in 4 dimensional Euclidean space is a special type of minimal surface, related to complex geometry. The Weierstrass representation of a minimal surfaces in n-dimensional Euclidean space is a way of describing the minimal surface using holomorphic data. I will discuss some old work from 2011 with my former USRA Li Chen, which I will be finally writing up this fall, where we generalize the notion of superminimal to higher codimension by characterizing (and then generalizing) the superminimal condition in terms of the Weierstrass representation and complex analysis.