Robin Koytcheff, UL Lafayette (Zoom). Integrals, trees, and spaces of pure braids and string links
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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 $k$.
September 27
Henry Adams, Colorado State University (Zoom). Vietoris-Rips thickenings of spheres
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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.
October 4
Daniel Bernstein, Tulane University (GI-325). Rigidity theory for Gaussian graphical models,
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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.
October 11
Mahir Can, Tulane University (GI-325). Vector bundles and affine Nash groups,
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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.
October 18
Michał Marcinkowski, University of Wrocław (Poland) (Zoom). Quasimorphisms, diffeomorphism groups of surfaces and $L^p$-metrics.
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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.
October 25
Lara Bossinger, National Autonomous University of Mexico (UNAM) (Zoom). On toric degenerations.
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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.
November 1
Baris Coskunuzer, University of Dallas Geometric Approaches on Persistent Homology
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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.
November 8
Mahir Can, Tulane University (GI-325). A Classification of One-dimensional Nash Supergroups.
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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.
November 15
TBA TBA
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TBA
November 29
Claudia Yun, Brown University (GI-325). TBA
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TBA
December 6
Elise Walker, Texas A&M (Zoom) TBA
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TBA
January 31
TBA TBA
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TBA
February 7
TBA TBA
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TBA
February 14
TBA TBA
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TBA
February 21
TBA TBA
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TBA
March 7
TBA TBA
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TBA
March 14
TBA TBA
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TBA
March 21
Clayton Shonkwiler, Colorado State University TBA
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