Mathematics Home / Joint Research Seminars: Algebraic Geometry and Geometric Topology Seminar
Time & Location: All talks are on Monday in Gibson Hall 310 at 3:00 PM unless otherwise noted.
Organizer: Komendarczyk, Rafal; Co-organizer: Mahir Can, Tai Ha, Kalina Mincheva
January 30
Title: Spherical Tropicalization and Berkovich Analytification
Desmond Coles | University of Texas, Austin
Abstract:
Tropicalization is the process by which algebraic varieties are assigned a "combinatorial shadow". I will review the notion of the 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.
Location: Temporary location: GI 126
Time: 3:00
February 13
Title: Convexity Defect Functions and Reconstruction with Cech complexes
William Tran | Tulane University
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?
Location: DW-103 (special time and location)
Time: 2:00
February 20
Mardi Gras Holiday
February 27
Title: Equivariant enumerative geometry
Thomas Brazelton | UPenn
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.
Location: Dinwiddie 103
Time: 3:00
March 6
Title: TBA
Speaker | TBA
Abstract:
TBA
Location: Gibson Hall 310
Time: 3:00
March 13
Title: Persistent Homology, Merge Trees and Reeb Graphs
Tom Needham | Florida State University
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.
Location: Over Zoom
Time: 3:00
March20
Title: Polyak-Viro type formulas for high dimensional knots and links
Neeti Gauniyal | Kansas State University
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 R^3. We obtained such formulas for invariants of 2- and 3-component links of dimension (2m-1) in R^{3m}. At the end, I will give a conjectural formula for embeddings of R^3 in R^6.
Location: Over Zoom
Time: 3:00
March 27
Title: Vector bundles for data alignment and dimensionality reduction
Jose Perea | Northwestern University
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.
Location: Over Zoom
Time: 4:00 Time change
April 3
Spring Break
April 10
Title: Graphing, homotopy groups of spheres, and spaces of links and knots
Robin Koytcheff | University of Louisiana, Lafayette
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.
Location: Gibson Hall 310; May be over Zoom
Time: 3:00
April 17
Title: Convexity Defect Functions and Reconstruction with Cech complexes (part 2)
William Tran | Tulane
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.
Location: DW-103 (special time and location)
Time: 2:00 (special time and location)
April 24
Title: TBA
Speaker | TBA
Abstract:
TBA
Location: Gibson Hall 310
Time: 3:00
May 1
Title: TBA
Speaker | TBA
Abstract:
TBA
Location: Gibson Hall 310
Time: 3:00