Kalina Mincheva

Algebraic Geometry and Geometric Topology Seminar

We meet weekly on Mondays at 3:00 PM (Central time), Room: Dinwiddie 103

Organizers: Mahir Can, Tài Huy Hà, Rafał Komendarczyk, Kalina Mincheva.

Spring 2023

• Desmond Coles, Uiversity of Texas, Austin.
  Spherical Tropicalization and Berkovich Analytification. abstract.
  Tropicalization is the process by which algebraic varieties are assigned a "combinatorial shadow". I will review the notion of tropicalization of a toric variety and recent work on extending this construction to spherical varieties. I will then present how one can construct a deformation retraction from the Berkovich analytification of a spherical variety to its tropicalization.

• William Tran, Tulane University.
  This talk will be at 2pm, in room DW-103.
  Convexity Defect Functions and Reconstruction with Cech complexes. abstract.
  We discuss previous results from Attali, Lieutier, and Salinas. Given a set of points that sample a shape, can we give conditions -- in terms of convexity of the shape -- that guarantee that a Cech complex built from our sampled points is homotopy equivalent to our shape?

• Thomas Brazelton, UPenn.
  Equivariant enumerative geometry. abstract.
  Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the sum of regular representations of the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any symmetric cubic surface.

• Tom Needham, Florida State University.
  This talk will be on Zoom.
  Persistent Homology, Merge Trees and Reeb Graphs. abstract.
  Topological Data Analysis is an approach to data science where the main idea is to featurize a dataset via topological methods, such as associating a sequence of homology groups to it. This homological signature is known as the persistent homology of the dataset. In this talk, I will discuss some enriched summaries of persistence - namely, decorated Reeb graphs and decorated merge trees - which capture richer topological information than standard persistent homology. Spaces of such objects admit natural metrics, and I will describe stability properties of these metrics. I will also discuss computational issues and applications to analysis of complex data. This is joint work with Justin Curry, Haibin Hang, Washington Mio, Osman Okutan and Florian Russold.

• Neeti Gauniyal, Kansas State University.
  This talk will be on Zoom.
  Polyak-Viro type formulas for high dimensional knots and links. abstract.
  I will talk about the problem of finding a high dimensional analogue to Polyak-Viro type formulas given in the classical case of 1-dimensional knots in ${\mathbb{R}}^3$. We obtained such formulas for invariants of 2- and 3-component links of dimension $(2m-1)$ in ${\mathbb{R}}^{3m}$. At the end, I will give a conjectural formula for embeddings of ${\mathbb{R}}^3$ in ${\mathbb{R}}^6$.

• Jose Perea, Northwestern University
  This talk will be at 4pm, over Zoom.
  Vector bundles for data alignment and dimensionality reduction. abstract.
  Vector bundles have rich structure, and arise naturally when trying to solve synchronization problems in data science. I will show in this talk how the classical machinery (e.g., classifying maps, characteristic classes, etc) can be adapted to the world of algorithms and noisy data, as well as the insights one can gain. In particular, I will describe a class of topology-preserving dimensionality reduction problems, whose solution reduces to embedding the total space of a particular data bundle. Applications to computational chemistry and dynamical systems will also be presented.

• Robin Koytcheff, Uiversity of Louisiana, Lafayette.
  Graphing, homotopy groups of spheres, and spaces of links and knots. abstract.
  We show that the homotopy groups of spaces of 2-component long links, up to knotting, are given by homotopy groups of spheres in a range of degrees that depends on the dimensions of the source manifolds and target manifold. In one degree higher, joining components sends both a parametrized long Borromean rings class and a Hopf fibration to a generator of the first nontrivial homotopy group of the space of long knots. For spaces of two-component long links, we give generators of the homotopy group in this dimension in terms of this class from the Borromean rings and homotopy groups of spheres. A key ingredient in most of our results is a graphing map that increases source and target dimensions by one..

• William Tran, Tulane University.
  This talk will be at 2pm, in room DW-103.
  Convexity Defect Functions and Reconstruction with Cech complexes (part 2). abstract.
  We discuss previous results from Attali, Lieutier, Salinas. Given a set of points that sample a shape, can we give conditions -- in terms of convexity of the shape -- that guarantee that a Cech complex built from our sampled points is homotopy equivalent to our shape? We will review the topological results from the previous discussion, then focus on geometric results.

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