1/24/2023 – Jacob Zoromski: The History of Calculus
The real number line, functions, limits, derivatives, integrals. When we teach a standard calculus course, these are the main concepts we cover. But calculus wasn’t always understood in these terms. How did these ideas develop? We will trace the history of calculus, from its roots in the area-finding of ancient Greeks, to its modern logical foundation(s). Along the way we will see how our mathematical predecessors viewed geometry, number, and infinity.
1/30/2023 – James Schmidt: What (else) can you do with partitions of unity?
Partitions of unity are a tool for making global constructions from local data, often introduced in the context of integration on manifolds. We consider two particular applications in continuous-time dynamical systems. In the first part, we recall stability as a notion of local robustness to disturbance in initial condition (simply: a continuity notion), and argue that under certain conditions maps of dynamical systems map stable points to stable points. In the second part, we present a general method for design of waypoint guidance which guarantees smoothness of global dynamics.
2/7/2023 – Karim Boustany: A Panoramic View of Symplectic Topology
Symplectic topology is a very active area of research which serves as a meeting ground for many different branches of mathematics. It also exhibits many curious features which result from an interesting hard/soft dichotomy inherent in the subject. In this talk, which is the first in a series of two, I will aim to introduce the basic notions of symplectic topology and survery some interesting examples and results, as well as maybe its connection to other fields. The talk is accessible to all graduate students familiar with the elementary machinery of smooth manifolds, as taught in Topology I for example. The sequel talk will focus on the theory of pseudoholomorphic curves in symplectic manifolds, bringing us closer to some more modern aspects of the subject.
2/14/2023 – Karim Boustany: Pseudoholomorphic Curves in Symplectic Topology
Pseudoholomorphic curve methods have been prevalent in symplectic topology ever since their introduction in Gromov’s seminal paper in the eighties. In this sequel to last week’s talk, I will give an introduction to the moduli space of these curves, discussing such topics as transversality, compactness, and bubbling. I will also survey the use of these moduli spaces in defining invariants of symplectic and contact manifolds. Examples include Gromov-Witten invariants as well as various flavors of homology, of which I will try to sketch at least one.
2/21/2023 – Connor Malin: The homotopy theory of configuration spaces
Understanding the homeomorphism types of manifolds is an ongoing problem in topology. It is difficult to apply the techniques of homotopy theory to such a problem because homotopy equivalence is a very coarse invariant. One proposed approach to study the homeomorphism class of a manifold M is to study the homotopy types of configuration spaces of M. We discuss what is known about the homotopy types of configuration spaces of manifolds and prove the counterintuitive result that after enough suspensions, the homotopy types of the configuration spaces of M are homotopy invariants of M.
2/28/2023 – Traci Warner: Computations in the Stable Homotopy Category
Using the classification of invertible Topological Field Theories (TFTs) as a motivating example, we examine some of the machinery at our disposal in the homotopy category of spectra. Along the way, we carry out a classic construction of an analogue of the Steenrod algebra for integer coefficients: the algebra of stable integral cohomology operations.
3/7/2023 – Xiyan Zhong: The Teichmüller Space of a Surface
The Teichmüller space \(\mathrm{Teich}(S_g)\) parametrizes all hyperbolic structures on the closed surface \(S_g\) of genus \(g\), up to isotopy. I will follow the book “A Primer on Mapping Class Groups” to give a brief introduction to two significant theorems on Teichmüller spaces. One is that \(\mathrm{Teich}(S_g)\) is homeomorphic to \(\mathbb{R}^{6g-6}\). The other is the \(9g-9\) theorem which shows that the hyperbolic structure on \(S_g\) is completely determined by the lengths of \(9g-9\) isotopy classes of assigned simple closed curves.
3/28/2023 – Jason Mitrovich: An Intro to the Yamabe Problem
The Yamabe problem is concerned with searching for conformal metrics from a prescribed scalar curvature. In this talk, I will first discuss some background regarding conformal metrics and scalar curvature. Then I will discuss Yamabe’s original motivation and how this problem is connected to Poincaré’s conjecture. With this, I will sketch out a proof for the Yamabe problem when we consider a restricted class of Riemannian manifolds. With the remaining time, I will discuss some progress made in other cases of interest.
4/11/2023 – Emanuela Marangone: Weak Lefschetz property and non-Lefschetz locus
A finite length graded R-module M has the Weak Lefschetz Property if there is a linear element \(l\) in R such that the multiplication map \(\times l : M_i \to M_{i+1}\) has maximal rank. The set of linear forms with this property forms a Zariski-open set and its complement is called the non-Lefschetz locus. In the first part of this presentation, I will first focus on the case of Artinian Complete intersections. An important result from Harima, Migliore, Nagel, Watanabe proved that any height three complete intersection has the Weak Lefschetz property. Moreover, Boij, Migliore, Miró-Roig, Nagel proved that for a general Artinian complete intersection the non-Lefschetz locus has the expected codimension and the expected degree. In the second part I will generalize the results for height tree complete intersection to the first cohomology module of a locally free sheaf \(E\) of rank 2 over \(\mathbb{P}^2\). It has been proved by Failla, Flores, Peterson, that these modules have the Weak Lefschetz property. Finally I will show that their non-Lefschetz locus has the expected codimension under the assumption that \(E\) is general.
4/25/2023 – Jui-Yun Hung: Sobolev Spaces and Smooth Vector Bundles
Sobolev spaces are important function spaces in the study of PDEs. In Euclidean space, we have the notion of weak derivatives and a Sobolev function is a measurable function that has weak derivatives up to certain order and with some control on the integral. In this talk, I will discuss one way to think about “weak derivatives” on smooth manifolds and more generally on smooth vector bundles, and use it to define Sobolev spaces on manifolds and vector bundles.
5/2/2023 – Khoi Nguyen: Scalar curvature under the collapse of metric
We prove a formula for the scalar curvature of a Riemannian manifold in terms of an orthonormal frame adapted to a distribution. Using the formula, we then investigate the effect of collapsing the metric along the distribution on the scalar curvature. This work is towards an effort of better understanding the scalar curvature of a manifold and its effect under scaling the underlying Riemannian metric.