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 Schubertâ€™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