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.

Fall 2022

• Rafal Komendarczyk, Tulane University.
  From integrals to combinatorial formulas of finite type invariants - a case study. abstract.
  We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the argument presented here and the classical arguments is that the vanishing of integrals over hidden faces does not require the involution trick due to Kontsevich. The integral formula yields the well-known arrow diagram expression for the invariant, first presented in the work of Polyak and Viro. We also take the next step of extending the arrow diagram expression to multicrossing knot diagrams. The primary motivation is to better understand a connection between the classical configuration space integrals and arrow diagram formulas for finite type invariants. This is a joint work with Robyn Brooks, which builds on her thesis.

• Rafal Komendarczyk, Tulane University.
  From integrals to combinatorial formulas of finite type invariants (part 2). abstract.
  This is part 2 of last week's talk.

• Jaiung Jun, SUNY at New Paltz.
  Quiver representations over $\mathbb{F}_1$ and Euler characteristics of quiver Grassmannians. abstract.
  A quiver is a directed graph, and a representation of a quiver assigns a vector space to each vertex and a linear map to each arrow. Quiver representations arise naturally in many areas of mathematics. Quiver representations over $\mathbb{F}_1$, ``the field with one element'' can be considered as a (degenerated) combinatorial model of quiver representations over a field, where vector spaces and linear maps are replaced by $\mathbb{F}_1$-vector spaces and $\mathbb{F}_1$-linear maps. I will introduce several aspects of quiver representations over $\mathbb{F}_1$, and its potential applications. This is joint work with Alex Sistko.

• Louis Theran, University of St. Andrews.
  Rigidity for sticky disks. abstract.
  A packing of $n$ disks in the Euclidean plane is an arrangement of the disks so that their interiors are disjoint. Two related questions are: If the radii of the disks are generic, how many pairs can be tangent? If the radii of the disks are generic and m pairs are in contact, how many degrees of freedom to move preserving contacts and radii does the packing have? I’ll discuss the answers to both of these, and, along the way make a connection to combinatorial rigidity theory. This is joint work with Bob Connelly (Cornell) and Shlomo Gortler (Harvard).

• Scott McKinley, Tulane University.
  Toward Statistical Significance in Topological Data Analysis, Part 1. abstract.
  As TDA becomes more and more prominent in applications, the need for a rigorous statistical theory is becoming more urgent. The basic question is to say, I see this unusual topological feature (for example, a big hole in a 2d point cloud) and we would like to have a rigorous method for stating whether or not a feature of that size is likely to occur by randomness alone. A successful theory would require a clear articulation of a null model, and an understanding of the probability distribution that this null model induces on topological summaries of the data. In this first of two planned talks I will first show how this arose quite naturally in my mathematical biology work. I will show how my collaborator Veronica Ciocanel and I dealt with the question at the time. Then we will look at why exactly this problem is so hard, even in “trivial” topological settings. Finally I will survey some of the techniques that have arisen in the last five years, and then outline the results that Veronica and I submitted (still under review!) in a second paper this fall. Through extreme value theory in probability, we believe we know the correct probability distribution to use when conducting TDA hypothesis tests. At the level of computational statistics, our theory is quite successful, but we hope to “turn these results into math” in the coming years.

• Elizabeth Denne, WLU (over Zoom).
  Folded ribbon knots. abstract.
  We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. We give an overview of many of the bounds on the folded ribbonlength for knot families like torus, twist, 2-bridge, and pretzel knots. We also relate folded ribbonlength to the crossing number of a knot. In addition, the folded ribbon knot is a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We seek to find the minimum folded ribbonlength of a knot while respecting the ribbon linking number. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number $\pm n$ is bounded from above by $2n$.

• Michał Jabłonowski, University of Gdansk, Poland (over Zoom).
  Moves on immersions of surfaces into four-space. abstract.
  We consider smooth immersions of surfaces in the four-space, it is, on one hand, a generalization of classical knots (closed curves in three-space) and on the other hand generalization of immersions of curves in the plane. We derive a minimal generating set of spatial moves for banded singular diagrams of surfaces immersed in the four-space, which translates into a generating set of planar moves. We also discuss invariants coming from the immersed surface-link complement in the four-space.

• Cédric Oms (over Zoom).
  Existence and classification of $b$-contact structures. abstract.
  $b$-contact structures are a generalization of contact structures and can be seen as a Jacobi structure which is not transitive but transversally vanishing. Using the $h$-principle, we will give existence results for those structures in dimension 3. Furthermore, the overtwisted/tight dichotomy can be prescribed in each contact leaf of the Jacobi structure. Time permitting, we will classify these structures for some manifolds with a distinguished hypersurface. This is based on joint work with Robert Cardona.

• Scott McKinley, Tulane University.
  Toward Statistical Significance in Topological Data Analysis, Part 2. abstract.
  In this talk we will make the concepts of the first talk concrete by developing TDA on one-dimensional point clouds. We will look at three popular methods for defining filtrations, and for each consider what barcodes, persistence diagrams, and persistence landscapes emerge. (An interesting question for an inquisitive student is to look at the tropical geometry mapping defined by Sara Kalisnik and collaborators for this problem.) In 1-dimension there is an explicit formula for the distribution for the persistence lengths that emerge from a Poisson spatial point process. We will study this distribution in the context of extreme value theory and show why the Gumbel distribution is a natural candidate to use for critical values when conducting hypothesis tests.

• TBA.
  TBA. abstract.
  TBA