Geometry & Topology RTG

Previously this week (week of August 3rd) I was able to attend the Geometry & Topology RTG workshop at the University of Notre Dame. The workshop was a week long event consisting of two parts, I and II, the first being an introduction into geometry & topology, and the latter being lectures on more advanced topics including student presentations. I’ve included their website link here. I attended part II and thought I’d speak about my experiences.

Closed Geodesics

This talk was given by post-doc Gabor Szekelyhidi. One starts with a closed Riemann surface $M \subset \mathbb{R}^{3}$ . We call a smooth non-trivial loop $r:[0,1] \to M$ such that $r'(0) = r'(1)$ a closed geodesic if it is a map of minimal length or equivalently a critical point of the length functional (think a non-squiggly curve at small scales). The goal is to deduce if there are closed geodesics in $M$ . Szekelyhidi then broke the argument into two cases, either $M$ is simply-connected or it is not. If not, then we have the following well-known theorem:

Theorem: If $\pi_{1}(M) \ne \{1\}$ , then every non-trivial homotopy class contains a closed geodesic.

The proof is essentially finding the shortest loop in each homotopy class and we know such a loop exists because on $M$ there always exists some length $L$ such that if the length of a loop $r$ is less than $L$ , written $\ell(r) < L$ , then $r$ must be homotopic to the identity (for example, any loop less than length $2\pi$ in the torus cannot go around length-wise or width-wise and so must be trivial). To find the shortest loop we apply a process called curve shortening flow, which is a process that minimizes the curvature everywhere on the loop in a continuous manner without affecting the homotopy class of the loop.

On the other hand, if $M$ is simply-connected then all loops are homotopic to the identity and we need to work a little harder. Instead of considering classes of loops we consider loops of loops (commonly called “sweepouts”). Formally, a sweepout is a loop $\Phi:[0,1] \to \Omega M$ into the space of all loops on $M$ (not homotopy classes of loops) such that the induced map $S^{2} \to M$ viewing $S^{2} \cong [0,1] \times S^{1}/\sim$ is not trivial. We can think of a sweepout as taking a non-trivial loop in $M$ and pulling it in $M$ so it traces out $M$ in terms of the loop (think about pulling a rubber band around the sphere from top to bottom). We then define $W(\Phi) = \max_{s \in [0,1]}\ell(\Phi(s))$ and try to find the sweepout minimizing width because this sweepout will contain the geodesic. In fact, Birkhoff proved this in 1917.

Theorem (Birkhoff 1917): There exists closed geodesics in $M$ .

Current research aims to generalize this idea to higher dimensional manifolds where we are now looks for surfaces of minimal area, $3$ -manifolds of minimal volume, etc. In fact, in 2016 and again in 2018 it was proved (by Sang and Marques-Neves respectively) that if $M$ is a closed Riemann $3$ -manifold, then it has infinitely many minimal surfaces.

Relating Topology and Geometry of Manifolds

This talk was given by Notre Dame professor Stephan Stolz. He started with a short introduction about the differences between topology and geometry. His view, was that topologists are interested in qualitative aspects of spaces such as the number of connected components, holes, punctures, and twists. While on the other hand, geometers are more interested in quantitative aspects of spaces such as the curvature, area, length, and area. This viewpoint is quite nice because it also alludes to the fact that most homeomorphisms (mapping that preserve topological structure) are often very far off from being a Riemannian isometries (a smooth homeomorphism which respects the Riemannian metric between manifolds), that is qualitative aspects being preserved doesn’t imply the quantitative ones are as well. However in this viewpoint the converse is true because every Riemannian isometry is in particular a homeomorphism.

The next piece of his talk introduced the Euler characteristic of a surface $M$ , which is a topological invariant. Its obtained, in a crude manner, by putting a pattern of of polygons $\Gamma$ on $M$ , defining $v(\Gamma)$ , $e(\Gamma)$ , and $f(\Gamma)$ to be the number of vertices, edges, and faces in $\Gamma$ respectively, and then defining the Euler characteristic $\chi(M,\Gamma)$ by the alternating sum $\chi(M,\Gamma) = v(\Gamma)-e(\Gamma)+f(\Gamma)$ . It turns out that $\chi(M,\Gamma)$ is independent of the type of patterning $\Gamma$ so that we may write $\chi(M)$ for the characteristic and that it is topologically invariant, so homeomorphic manifolds have the same Euler characteristic. This gives a nice method for determining if two surfaces are not homeomorphic, and if we assume the surfaces $M$ and $N$ are compact orientable then they are homeomorphic if and only if they have the same Euler characteristic.

Stolz then shifted gears to talk a little about Riemannian manifolds: smooth manifolds $M$ with an inner product $g_{p}$ on the tangent space $T_{p}M$ such that the inner product varies smoothly with respect to $p$ . With this inner product we have a way to measure lengths of curves, distance between points, and areas in $M$ analogous to how such quantities are measured in calculus. More importantly, we have a notation of scalar curvature at $x \in M$ (up to normalization) which is defined to be $-3(n+2)$ times the second derivative with respect to $r$ evaluated at $r = 0$ of $\text{vol\,}B_{r}(x,M)/\text{vol\,}B_{r}(0,\mathbb{R}^{2})$ (this is a measure of how fast the manifold is curving away or towards $x \in M$ ). Varying $x$ over all of $M$ we get a function $sc:M \to \mathbb{R}$ called the scalar curvature function. Incredibly, this relates to the Euler characteristic of $M$ in the following manner:

Theorem (Gauss-Bonnet): If $M$ is a compact orientable surface, then $\int_{M}s(x)\,dx = 4\pi\chi(M)$ .

As an interesting corollary, it can be shown that the sphere is the only compact orientable surface admitting a positive Euler characteristic (namely $2$ ). Hence by Gauss-Bonnet it is the only compact orientable surface that can possibly admit an everywhere positive scalar curvature function (and in fact it does).

Directed Homology

One of the undergraduate presentations was on a new field of study: directed homology. Here one has a topological space with a sense of “time flow” on the manifold. Homological calculations can then be performed to gain some understanding of space. The presentation was primarily computational, so naturally I was curious about the homological properties these spaces have. The talk ended with a section on future research and some open questions were if a Mayer-Vietoris sequence and the standard homology axioms hold in this setting. I spoke with the presenter after about pursuing this topic and he agreed to get me in touch with his mentors.

For peace of mind, a directed space is a pair $(X,dX)$ consisting of a topological space $X$ with a subset $dX \subset C(I,X)$ of continuous paths from the interval into $X$ such that every constant path is in $dX$ , $dX$ is closed under composition of increasing maps $I \to I$ , and $dX$ is closed under path-concatenation. Notice that since $X$ is usually uncountable, $dX$ quite often a very large space. A morphism $f:X \to Y$ between directed spaces is a continuous map which preserves directed paths in the sense that if $\gamma \in dX$ , then $f \circ \gamma \in dY$ . With these definitions in mind, simplicial complexes $\Delta^{n}$ should become directed spaces by the induced ordering on their vertices (I’m not exactly sure how this should best be worked out), so the directed homological theory would begin by considering the chain groups $C_{n}(X)$ consisting of all singular directed morphisms $\sigma:\Delta^{n} \to X$ . I’m looking into this as time permits, so keep a look out for future posts on the topic.

Some Soft Comments

Overall I really enjoyed the level of mathematics the workshop was at. I understood a majority of the content, and for the the things I didn’t quite understand I either had a good sense of intuition for them or I had someone else give me more detail after the talk. There were quite a few more talks then the two presented here by the way (seven professor/post-doc talks and twelve student presentations). Between talks there was a $30$ minute break intended to be a time to discuss the talk in greater detail, ask questions, and chat with other members of the conference. However, there was really no external indication that a workshop was taking place (no direction signs, banner, etc.) and that gave a feeling of “if I was in the local area and didn’t get invited I could of just shown up and no one would of minded”. I would have also preferred if there was some type of lunch session where invitees could go grab food with some of the speakers and talk about their research and interests in a less formal setting. Perhaps these things occurred in Part I and not in Part II, but I would have preferred them nonetheless. Overall, it was a good experience and I’ll be applying again next summer.

By the way, this is a neat shot my roommate got of me after the conference when I was playing around with some Hagoromo chalk. If you’re wondering what the math on the blackboard is, it’s diagrams for operad multiplication of strings.

Closed Geodesics

Relating Topology and Geometry of Manifolds

Directed Homology

Some Soft Comments

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