# Poster Session and Reception

Tuesday, June 25, 2019 - 3:30pm - 5:00pm

Lind 400

**A Distance for Circular Heegaard Splittings**

Kevin Lamb (University of the Pacific)

For a knot $K\subset S^3$, its exterior $E(K) = S^3\backslash\eta(K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f\colon E(K)\to S^1$. When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose $E(K)$ into a pair of compression bodies. We call such a decomposition a \textit{circular Heegaard splitting} of $E(K)$. We define the notion of \textit{circular distance} (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circularHeegaard splitting is too large: (1) $E(K)$ can't contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for $K$ is unique up to isotopy.**Alternating knot types are exponentially rare**

Harrison Chapman (Colorado State University)

As a result of the Tait flyping conjecture (Menasco and Thistlethwaite '91), alternating links exhibit a very convenient

combinatorial structure. Sundberg and Thistlethwaite in '98 exploited this structure to develop and solve generating

functions leading to an enumeration of alternating links (see also Zinn-Justin and Zuber '99, Fusy, Kunz-Jacques, Schaeffer c'00). Thistlethwaite made further use of both this structure and precise enumeration to prove that all but exponentially few links are alternating.

The enumeration of alternating links depends strongly on connections between link diagrams and 4-valent planar maps,

a combinatorial class which is itself enumerated and well-studied. Alternating knots however are related to

a subset of these planar maps for which surprisingly little is known, and hence lack any enumeration or nice generating

function expression. As a consequence the common belief that alternating knots are rare among all knots

lacks any firm proof. I'll discuss a way to prove this result by using a pattern theorem for alternating knots to avoid

the necessity of precise enumeration.**Properties of codimension-two curve shortening flow**

Jiri Minarcik (Czech Technical University in Prague)

This contribution deals with problems associated with generalization of the curve shortening flow into higher dimensional space. The long term behavior of evolving closed curves embedded in $\mathbb{R}^3$ is analyzed and compared to the standard planar case. Although the motion of space curves in normal direction is similar to that in the plane, new phenomena may be observed in $\mathbb{R}^3$. Namely, embedded curves may develop new types of singularities, stop being simple or lose their convexity during the motion. These differences are illustrated on specific examples and results about motion of some invariant classes of space curves are presented.**Dynamics of Knotted DNA in Ionized Fluid**

Abby Pekoske Fulton (University of Pittsburgh)

Knotted DNA occurs as a result of the process of cellular replication. The goal of this work is to model the dynamics of knotted DNA in ionized fluid. We examine equilibrium configurations of knotted DNA plasmids and the dynamics of knot formation.

DNA is a double-helical polymer consisting of two strands. It has a negatively charged phosphate backbone linked by a sequence of nucleotide base pairs: Adenine-Thymine and Guanine-Cytosine. The phosphate backbone causes bending and torsional stiffness. During cellular replication, topoisomerase enzymes change the topology of the DNA segments.

We study these topological dynamics by modeling the double helical structure of DNA as an elastic rod with a central axis and a frame with one normal component in the direction of the phosphate backbone.**Curvature in Solvation Experiments**

Rhoslyn Coles (TU Berlin)

A solvation experiment is one in which a substance is added to a fluid and dissolves or mixes, eventually adopting a thermodynamically favourable configuration within the fluid environment. There is an interesting connection between thermodynamics and morphology making it possible to measure the change in thermodynamic potential via geometric measures of a surface associated to the solvation experiment. This makes it perhaps possible to investigate the role of entanglement in the solvation process. The poster presents ongoing work in this direction.**Variations of the Integral Menger Curvature**

Jan Knappmann (RWTH Aachen University)

Choosing three distinct points on a given knot defines a unique circle passing through all of these points. Integrating along the curve for each of these points over the reciprocal value of the aforementioned circle's radius gives a knot energy called the Integral Menger Curvature.

It is commonly conjectured that the round circle is the global minimizer of this functional, but so far it has only been verified by Hermes in 2014 that the round circle is critical.

We present the techniques used by Hermes including the first variation and follow up with an outline of the ongoing work dedicated to show that the second variation at this point is positive, i.e. trying to proof that the circle is in fact a local minimum of the energy.**Weaving Geodesic Foliations**

Josh Vekhter (The University of Texas at Austin)

We study discrete geodesic foliations of surfaces—foliations whose leaves are all approximately geodesic curves—and develop several new variational algorithms for computing such foliations. Our key insight is a relaxation of vector field integrability in the discrete setting, which allows us to optimize for curl-free unit vector fields that remain well-defined near

singularities and robustly recover a scalar function whose gradient is well aligned to these fields. We then connect the physics governing surfaces woven out of thin ribbons to the geometry of geodesic foliations, and present a design and fabrication pipeline for approximating surfaces of arbitrary geometry and topology by triaxially-woven structures, where the ribbon layout is determined by a geodesic foliation on a sixfold branched cover of the input surface.**Curvature Effects in Flocking Dynamics**

Franz Wilhelm Schlöder (Università di Milano - Bicocca)

We introduce a generalisation of the well-established Cucker-Smale model to complete Riemannian manifolds to study the influence of geometry and topology on the formation of flocks. The dynamics of the Cucker-Smale model facilitate the flocking of a group of particles in disordered motion into a coordinated one where all particles move parallelly with the center of mass. Despite their name, flocking models do not only illustrate the herding of animals but more generally the emergence of collective behaviour. The possible applications cover a broad spectrum of subjects such as linguistics, biology, opinion formation, sensor networks and robotics.

While the Cucker-Smale model already received much attention over the last decade, those efforts focused on particles moving in a Euclidean space. Because of the intimate relationship between the notion of parallelism on a Riemannian manifold and its curvature in our model the geometry constrains the ﬁnal ﬂocked state into speciﬁc patterns; a new feature that is absent in the Euclidean case. Next to this interesting phenomenology, our work also is a contribution towards the ﬂocking realizability problem raised by Chi, Choi and Ha: Given a manifold and a group of particles, construct a dynamical system that leads to a collective movement as a flock at least asymptotically. This project is joint work done with S.-Y. Ha (Seoul National U., Seoul)& D. Kim (Korea Institute for Advanced Study, Seoul).**Ribbonizing surfaces via rolling**

Matteo Raffaelli (University of Coimbra)

The method of approximating a surface in three-dimensional space by a net of planar triangles (triangulation) has a long history in both fundamental and applied mathematics. Here we present a novel rolling-based approach towards surface approximation and discretization. Such approach, called flat ribbonization, is based on the idea of using intrinsically flat (i.e., developable) strip-shaped surfaces, or ribbons, as the building blocks of the approximation. By construction, each ribbon has the same distribution of tangent planes (i.e., the same normal field) as the original surface along a given contact curve.**Artworks and Articles Meet Mapper and Persistent Homology**

Hongyuan Zhang (Grinnell College)

Since its recent birth, topological data analysis has proven to be a very useful tool when studying large and high dimensional data sets. We explored persistent homology tools and the Mapper algorithm with a data set describing artworks from Metropolitan Museum of Art (MET) and another data set describing a collection of arXiv papers. By using these tools, detailed insights are given in understanding the complexity of each data set. The effectiveness of the Mapper algorithm in guiding feature selection when building a logistic regression model will be illustrated.**Elastic Curves Confined to Balls**

Pascal Dolejsch (Albert-Ludwigs-Universität Freiburg)

Elastic rods that are inextensible and torsion-free can be modeled as embedded 1D curves. The relaxation of arbitrary initial shapes towards a minimally bent configuration, the elastica, can be calculated by a gradient-flow method. We therefore minimize the integrated curvature while preventing the curve from extending or contracting by additional constraints. This method, also including a mechanism to avoid self-intersections of the curve, has been

previously studied. Now, we examine penalty algorithms to confine the curves to spherical domains. This allows to describe possible elastica shapes and minimal energies.**A tree distinguishing polynomial and tree metrics**

Pengyu Liu (Simon Fraser University)

We present a bivariate polynomial for trees and show that the polynomial is a complete invariant for unlabeled trees, that is, two unlabeled trees are isomorphic if and only if they have the same polynomial. Thus, this polynomial can be applied to compare trees that arise in phylogenetics, linguistics and other sciences. We also present some results of classifying tree topologies using the metrics induced by the polynomial.**A Parametrization-Preserving Gradient Flow for Generalized Integral Menger Curvature: Long-Time Existence**

Daniel Steenebrügge (RWTH Aachen University)

Simon Blatt and Philipp Reiter introduced the generalized integral

Menger curvature, a knot energy. They also determined the space on which

said energy is finite, a Sobolev-Slobodeckij space. For appropriate

parameters, this space is a Hilbert space, enabling us to pose a

gradient flow equation.

Jan Knappmann was able to show that said gradient is locally Lipschitz

continuous and even gave some estimates which yield lower bounds on the

existence time of the flow.

The bounds depend on the norm and the bi-Lipschitz constant of the

initial curve, so if we can control these, we can show long time

existence. We are able to do the former via means provided by Blatt and

Reiter, but only if the parametrization does not change along the flow.

To ensure this, the gradient is projected as done in chapter 6 of J.W.

Neuberger's Sobolev gradients and differential equations.

The constraint used for the projection is due to Henrik Schumacher,

Sebastian Scholtes and Max Wardetzky.**Biomedical Image Analysis via Persistent Homology**

Ülgen Kilic (University at Buffalo (SUNY))

Even though the whole is not the sum of its parts, examining the neurobiological systems at the cellular level entails so much information about the whole system. Cell locations, neuron’s in particular, is extremely useful in the imaging data for further analysis to take place. Here, we are borrowing a tool from algebraic topology whose applications have gained a lot of attention by a wide range of research fields recently. We apply persistent homology to grayscale imaging data in which regions of interests (ROI) appear either as filled regions which are in contrast with the background using 0-dimensional homology, or as loopy, donut-like regions using 1-dimensional homology. More importantly, this method is robust to the frequent occurrence of white noise in this type of data up to some amount. We localize cells on the image by finding the maxima of the landscape that we constructed. We verify the performance of our technique for the images where ROIs occur as filled regions in 2018 KAGGLE Data Bowl: Nuclei Detection Competition data set in which ground truth segmentation is also provided and obtained high accuracy. Images with loopy ROIs are even harder to deal with especially in the images with multiple ROIs because of the non-uniform distribution of the saddles. Nevertheless, we get great results in the calcium images as well as in the myelin imaging data with a few ROIs. Cell segmentation information obtained by persistent homology can be combined with other image analysis techniques. Finally, we highlight the non-uniqueness of the homology generators in the 1-dimensional case which is a problem in its own right in this area. However, we can address this issue by turning it into an optimization problem which can be dealt with linear programming to find optimal cycles.**Topological surgery and cosmology**

Stathis Antoniou (National Technical University, Zografou Campus)

We connect topological changes that can occur in 3-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena, including relationships between quantum entanglement and wormhole formation. By considering the initial manifold as the 3-dimensional spatial section of spacetime, we describe the changes of topology occurring in these processes by determining the resulting 3-manifold and its fundamental group. As these global changes are induced by local processes, we use the local form of Morse functions to provide an algebraic formulation of their temporal evolution and propose a potential energy function which, in some cases, could give rise to the local forces related to surgery. We further show how this topological perspective gives new insight for natural phenomena exhibiting surgery, in all dimensions, while emphasizing the 3-dimensional case, which describes cosmic phenomena. This work makes new bridges between topology and natural sciences and creates a platform for exploring geometrical physics.**Some stochastic SICA epidemic models for HIV transmission**

Jasmina Djordjevic (University of Nis)

We propose a stochastic SICA epidemic model for HIV transmission. The model is improved with a stochastic model for HIV/AIDS transmission where pre-exposure prophylaxis (PrEP) is considered as a prevention measure for new HIV infections. A white noise is introduced into both model, representing fluctuations in the environment that manifest themselves on the transmission coefficient rate. We prove existence and uniqueness of a global positive solution of the stochastic models, and establish conditions under which extinction and persistence in mean hold. Numerical simulations are provided which illustrate the theoretical results and conclusions are derived on the impact of the fluctuations in the environment.