Algebraic Geometry and Geometric Topology Seminar
We meet weekly on Mondays at 3:00 PM (Central time), Gibson 325
Organizers: Mahir Can, Tài Huy Hà, Rafał Komendarczyk, Kalina Mincheva.
Schedule: Fall 2021 and Spring 2022
September 20-
Robin Koytcheff, UL Lafayette (Zoom).
Integrals, trees, and spaces of pure braids and string links .The based loop space of configurations in a Euclidean space $\mathbb{R}^n$ can be viewed as the space of pure braids in $\mathbb{R}^{n+1}$. In joint work with Komendarczyk and Volic, we studied its real cohomology using an integration map from a certain graph complex and recovered a result of Cohen and Gitler. Specifically, the map we studied is a composition of Kontsevich’s formality integrals and Chen’s iterated integrals. We showed that it is compatible with Bott-Taubes integrals for spaces of 1-dimensional string links in $\mathbb{R}^{n+1}$. As a corollary, the inclusion of pure braids into string links in $\mathbb{R}^{n+1}$ induces a surjection in cohomology for any $n>2$. More recently, we showed that the dual to the integration map embeds the homotopy groups of the space of pure braids into a space of trivalent trees. We also showed that a certain subspace of these homotopy groups injects into the homotopy groups of spaces of k-dimensional string links in $\mathbb{R}^{n+k}$ for many values of $n$ and $k$.
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Henry Adams, Colorado State University (Zoom).
Vietoris-Rips thickenings of spheres .If a dataset is sampled from a manifold, then as more and more samples are drawn, the persistent homology of the Vietoris-Rips complexes of the dataset converges to the persistent homology of the Vietoris-Rips complexes of the manifold. But little is known about Vietoris-Rips complexes of manifolds. An exception is the case of the circle: as the scale parameter increases, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, ..., until finally they are contractible. The Vietoris-Rips thickenings of the n-sphere first obtain the homotopy type of the n-sphere, and then next the $(n+1)$-fold suspension of a (topological) quotient of the special orthogonal group $SO(n+1)$ by an alternating group $A_{n+2}$. Not much is known at later scales, even though (as we will explain) these homotopy types have applications for generalizations of the Borsuk-Ulam theorem, for projective codes (packings in projective space), and (conjecturally) for Gromov-Hausdorff distances between spheres. This is joint work with Michal Adamaszek, Johnathan Bush, and Florian Frick.
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Daniel Bernstein, Tulane University (GI-325).
Rigidity theory for Gaussian graphical models, .Many modern biological applications require one to fit a statistical model with many parameters to a dataset with relatively few points. This begs the question: for a given model, what is the fewest number of data points needed in order to fit? In this talk, I will discuss this question for the class of Gaussian graphical models, highlighting connections to discrete geometry, convex geometry, classical combinatorics, and rigidity theory.
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Mahir Can, Tulane University (GI-325).
Vector bundles and affine Nash groups, .In this talk we will make a gentle introduction to the theory of (the vector bundles on affine) Nash manifolds. We will introduce a special family of affine Nash groups. Then we will announce a classification theorem related to these new Nash groups.
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Michał Marcinkowski, University of Wrocław (Poland) (Zoom).
Quasimorphisms, diffeomorphism groups of surfaces and $L^p$-metrics. .Quasi-(homo)morphisms are real functions on a group that pretend to be homomorphisms. On many groups there is plenty of interesting quasimorphisms. I will ilustrate this notion with simple geometric and combinatoric examples. In particular I will describe how using braids one can construct quasimorphisms on $\text{Diff}_{0}(S,\omega)$, the group of area preserving diffeomorphis of surface $S$. These quasimorphisms are generalisations of the Calabi invariant.
In our recent work with M. Brandenbursky and E. Shelukhin we showed that there exist many quasimorphisms on $\text{Diff}_{0}(S,\omega)$ that are Lipschitz with respect to the $L^p$-norm, $p \geq 1$. The proof uses the compactification of the configuration space of $S$. This allows to show e.g., that right angled Artin groups can be embedded quasi-isometrically into $\text{Diff}_{0}(S,\omega)$ with the $L^p$ norm. I will explain these notions and show the idea of the proof.
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Lara Bossinger, National Autonomous University of Mexico (UNAM) (Zoom).
On toric degenerations. .In this talk I will give an overview on different constructions of toric degenerations, in particular from valuations and from Gröbner theory. I will show how they are related. By the end of the talk we will explore a possible construction to extend the notion of moment polytope to not-necessarily toric varieties via toric degenerations.
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Baris Coskunuzer, University of Dallas (Zoom).
Geometric Approaches on Persistent Homology .Persistent Homology is one of the most important techniques used in Topological Data Analysis. In this talk, after giving a short introduction to the subject, we study the persistent homology output via geometric topology tools. In particular, we give a geometric description of the term “persistence”. The talk will be non-technical, and accessible to graduate students. This is a joint work with Henry Adams.
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Mahir Can, Tulane University (GI-325).
A Classification of One-dimensional Nash Supergroups. .In this second part of our seminar on semialgebraic geometry, we will continue to explain our new categories related to Nash manifolds. In particular, in this talk we will present our classification theorem of one-dimensional Nash supergroups.
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Umberto Hryniewicz (RWTH Aachen University) (Zoom).
Quantitative conditions for right-handedness. .We present numeric conditions for a dynamically convex Reeb flow on the 3-sphere, in the sense of Hofer-Wysocki-Zehnder, to be right-handed, in the sense of Ghys. Once right-handedness is checked, the following interesting conclusions about the dynamics can be deduced: (a) every link of periodic orbits is a fibered link, and (b) every finite collection of periodic orbits spans a global surface of section for the flow. As an application, we find an explicit pinching constant $0 < d < 0.7225$ such that if a Riemannian metric on the 2-sphere is pinched by at least $d$ then its geodesic flow lifts to a right-handed flow on the 3-sphere. This is joint work with Anna Florio.
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Claudia Yun, Brown University (GI-325).
Homology representations of compactified configurations on graphs. .The $n$-th ordered configuration space of a graph parametrizes ways of placing $n$ distinct and labelled particles on that graph. The homology of the one-point compactification of such configuration space is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We give a cellular decomposition of these configuration spaces on which the actions are realized cellularly and thus construct an efficient free resolution for their homology representations. As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces $\mathcal{M}_{2,n}$, equivalently the rational homology of the tropical moduli spaces $\Delta_{2,n}$, as a representation of $S_n$ acting by permuting point labels for all $n\leq 10$. This is joint work with Christin Bibby, Melody Chan, and Nir Gadish. Our paper can be found on arXiv with ID 2109.03302.
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Elise Walker, Texas A&M (Zoom).
An optimal numerical algorithm for solving polynomial systems. .Numerical homotopy continuation is a useful numerical algorithm for computing the solutions of a system of polynomial equations. Such solution sets are sometimes known as varieties. Homotopies compute varieties by tracking paths from the solutions of a similar, pre-solved system. Generally, homotopies may track extraneous paths, which wastes computational resources. A homotopy is optimal if paths are smooth and there are no extraneous paths. Embedded toric degenerations are one source for optimal homotopy algorithms. In particular, if a variety has a toric degeneration, then there is an optimal homotopy for computing linear sections of that variety. There is a toric degeneration for any variety which has an associated finite Khovanskii basis. This work provides the appropriate embeddings for the Khovanskii toric degeneration and gives the corresponding optimal homotopy algorithm for computing a linear section of the variety. This is joint work with Michael Burr (Clemson University) and Frank Sottile (Texas A&M University).
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Erin Chambers, Saint Louis University (GI-310).
Reeb graph metrics from the ground up. .The Reeb graph has been utilized in various applications including the analysis of scalar fields. Recently, research has been focused on using topological signatures such as the Reeb graph to compare multiple scalar fields by defining distance metrics on the topological signatures themselves. In this talk, we will introduce and study five existing metrics that have been defined on Reeb graphs: the bottleneck distance, the interleaving distance, functional distortion distance, the Reeb graph edit distance, and the universal edit distance. This talk covers material from a recent survey paper, which has multiple contributions: (1) provide definitions and concrete examples of these distances in order to develop the intuition of the reader, (2) visit previously proven results of stability, universality, and discriminativity, (3) identify and complete any remaining properties which have only been proven (or disproven) for a subset of these metrics, (4) expand the taxonomy of the bottleneck distance to better distinguish between variations which have been commonly miscited, and (5) reconcile the various definitions and requirements on the underlying spaces for these metrics to be defined and properties to be proven.