## GEOTOP-A SEMINAR SERIES

**Speaker:**De Witt L. Sumners

**Title:**DNA Topology

**Keywords:**DNA, topoisomerase, site-specific recombination, knots, links, tangle model, supercoiling, viral capsids

**Date:**August 17, 2018

Cellular DNA is a long, thread-like molecule with remarkably complex topology. Enzymes that manipulate the geometry
and topology of cellular DNA perform many vital cellular processes (including segregation of daughter chromosomes, gene
regulation, DNA repair, and generation of antibody diversity). Some enzymes pass DNA through itself via enzyme-bridged
transient breaks in the DNA; other enzymes break the DNA apart and reconnect it to different ends. In the topological
approach to enzymology, circular DNA is incubated with an enzyme, producing an enzyme signature in the form of DNA knots
and links. By observing the changes in DNA geometry (supercoiling) and topology (knotting and linking) due to enzyme action,
the enzyme binding and mechanism can often be characterized. This talk will discuss topological models for DNA strand passage
and exchange, including the analysis of site-specific recombination experiments on circular DNA and the analysis of packing
geometry of DNA in viral capsids.

**Speaker:**Rafael Herrera

**Title:**A brief introduction to Discrete Exterior Calculus

**Keywords:**differential forms, Hodge star operator, simplicial complex, boundary operator

**Date:**September 7, 2018

Discrete Exterior Calculus (DEC) is a relatively new method for solving partial differential equations
numerically. The central idea of the method is to develop a discrete version of Exterior Differential
Calculus. In this talk, we will briefly review the various operators of Exterior Differential Calculus
and their discrete versions. We will then focus our attention in 2D DEC, presenting a geometric
interpretation of DEC as well as numerical comparisons with the Finite Element Method in order to show
DEC’s competitive performance.

**Speaker:**Cristian Micheletti

**Title:**Knotted DNA: conformational, dynamical and pore-translocation properties

**Keywords:**pore translocation, DNA knots, DNA supercoiling

**Date:**September 21, 2018

Knots and supercoiling are both introduced in bacterial plasmids by catalytic processes involving
DNA strand passages. I will report on a recent study where we used molecular dynamics simulations and a mesoscopic
DNA model, to study the simultaneous presence of knots and supercoiling in DNA rings and the kinetic and metric
implications which may be relevant for the simplyfing action of topoisomerases. Finally, I will discuss how the same
modelling and simulation approach can be used shed light on the complex experimental phenomenology of knotted DNA
translocating through solid state nanopores.

**Speaker:**Vidit Nanda

**Title:**Discrete Morse Theory

**Keywords:**discrete Morse theory, persistent homology, cellular sheaf theory, flow category

**Date:**October 19, 2018

Large-scale homology computations are rendered tractable in practice by
a combinatorial version of Morse theory. Assuming that my internet connection survives, in
this talk I will introduce discrete Morse theory, describe how it helps with computing
various types of homology, and outline a powerful new higher-categorical extension. The
audience will not require prior knowledge of any of the technical terms mentioned above.

**Speaker:**Jesus Gonzalez

**Title:**Simplicial complexity: piecewise linear motion planning in robotics.

**Keywords:**topological complexity, simplicial topology, simplicial aproximation, contiguity of maps

**Date:**November 9, 2018

Using the notion of contiguity of simplicial maps, and its relation (via iterated
subdivisions) to the notion of homotopy between continuous maps, we adapt Farber's topological complexity
to the realm of simplicial complexes. We show that, for a finite simplicial complex K, our discretized
concept recovers the topological complexity of the realization of K.

**Speaker:**Renzo. L Ricca

**Title:**Geometric devils in topological dynamics

**Keywords:**inflexion, singularity, self-linking, twist, knots, braids, topological change, Möbius band, 2-sided disk

**Date:**November 23, 2018Geometric and topological features play an important role in many physical systems, especially in fluid dynamics where forces drive continuous deformation. The benign influence of geometry, which is the muscular actor in topological dynamics, has sometimes a more devilish nature. Here we show how two different types of singularities that may emerge during the evolution of magnetic fields and soap films are responsible for the spontaneous reorganization and energy change of these systems.

A first example is given by the inflexional instability of magnetic knots that under conservation of self-linking number brings inflexional magnetic knots to inflexion-free braids through an integrable singularity of torsion. A second example is given by the continuous deformation of a soap film Möbius band to a 2-sided disk by a drastic change of the soap film topology through a twisted fold catastrophe. These are examples of generic behaviors that trigger energy relaxation in a wide variety of different physical contexts.

[1] Moffatt, H.K. & Ricca, R.L. (1992) Helicity and the Călugăreanu invariant. *Proc. R. Soc. A* **439**, 411-429.

[2] Ricca, R.L. (2005) Inflexional disequilibrium of magnetic flux tubes. *Fluid Dyn. Res.* **36**, 319-332.

[3] Goldstein, R.E., Moffatt, H.K., Pesci, A.I. & Ricca, R.L. (2010) A soap film Möbius strip changes topology with a twist
singularity. *Proc. Natnl. Acad. Sci. USA* **107**, 21979-21984.

**Speaker:**Sergio Rajsbaum

**Title:**A very elementary introduction to the combinatorial topology approach to distributed computing

**Keywords:**distributed computing, combinatorial topology

**Date:**November 30, 2018An introductory lecture following the approach of the book by Herlihy, Kozlov, Rajsbaum "Distributed Computing through Combinatorial Topology", Elsevier-Morgan Kaufmann, based on the notions of indistinguishability and perspectives. Illustrating the use of simplicial complexes via simple examples. No preliminary knowledge either about topology or about distributed computing.

**Speaker:**Mark Dennis

**Title:**Polarization geometry of crystals and the blue sky

**Keywords:**polarization, optical axis, multiple scattering, birefringence

**Date:**December 7, 2018The world would look very different to us if, like many insects and other animals, our eyes could see the polarization of light. In particular, we would see a distinctive “fingerprint” pattern in the blue sky overhead, as well as many crystalline materials that surround us. In this talk I will describe the geometry of optical polarization patterns from Maxwell’s equations, and in doing so reveal the deep and surprising relationship between the polarization pattern of birefringent crystals and the blue sky, based on a geometric and topological understanding of the tensors involved in the propagation and scattering of light.

**Speaker:**Louis H Kauffman

**Title:**Unitary Braiding and Topological Quantum Computing

**Keywords:**quantum computation, Hilbert space, complex space, state vector, unitary transformation, measurement, quantum Hall effect, Fibonacci model, Majorana Fermion, Artin braid group, unitary braiding, Clifford algebra

**Date:**January 25, 2019Quantum computation can be regarded as a methodical study of the structure preparation, evolution, and measurement of quantum systems in a computational framework. In a quantum system, a physical state corresponds to a unit length vector in a complex space (Hilbert space or finite dimensional complex space depending on the problem). A physical process corresponds to the application of a unitary transformation to a state. A measurement of a state projects that state to one of a number of orthogonal possibilities represented by an orthonormal basis for the complex space. The probability of a measurement is the absolute square of its coefficient in the state vector. A quantum computer consists in a unitary transformation and the capability to prepare states, apply the unitary transformation and measure the results. The statistics of repeated measurement gives the information for the computation. Thus the key to quantum computing is the design of unitary transformations that can accomplish a particular task. There are a number of designs that are very interesting such as Shor’s method of factoring integers on a quantum computer. The design of actual quantum computers is very difficult because it is hard to control measurement and keep the system from decohering in its interaction with the environment. It is thought that topological physical situations, such as the behaviour of anyons in the quantum Hall effect or the behaviour of Majorana Fermions in nano wires, may hold the key to the desired fault tolerant computation. In both of these instances one can formulate the unitary transformations in terms of representations of the Artin Braid group that are related to the physics of the situation. It is thought that such topological phases of physical systems will be resistant to perturbation and will make scalable quantum computing possible. It is the purpose of this talk to describe braiding related to both the quantum Hall effect (the Fibonacci model) and unitary braiding related to Majorana Fermions (braiding from Clifford algebras). The talk will be self-contained and as pictorial as possible.

**Speaker:**Sara Kalisnik

**Title:**Learning Algebraic Varieties from Samples

**Keywords:**persistent homology, dimension estimates, learning varieties

**Date:**February 8, 2019"I will discuss how to determine vanishing polynomials on a fixed finite sample of points and what to do with that information. For example, from the equations defining a variety one can learn the degree and the dimension of the variety. One can also construct ellipsoid complexes which, based on the experiments, strengthen the topological signal for persistent homology. All the algorithms needed are made available in a Julia package."

**Speaker:**Neza Mramor-Kosta

**Title:**Discrete Morse theory in action

**Keywords:**discrete Morse theory, feature tracking, topological complexity, robot motion planners

**Date:**February 22, 2019In this talk, the basics of discrete Morse theory will first be shortly summarized. We will then describe two generalizations which both have applications to mathematical models associated with motion. Parametric discrete Morse theory can be applied to tracking the motion of features in a sequence of images or scenes, and fiberwise discrete Morse theory to robot motion planning.

**Speaker:**Sophie E. Jackson

**Title:**Why are there knots in proteins?

**Keywords:**protein structure, protein stability, knotted proteins, methyltransferase, ubiquitin c-terminal hydrolase, knot-promoting loops, UCH-L1, YibK, YbeA

**Date:**March 8, 2019There are now more than 1700 protein chains that are known to contain some type of topological knot in their polypeptide chains in the protein structure databank. Although this number is small relative to the total number of protein structures solved, it is remarkably high given the fact that for decades it was thought impossible for a protein chain to fold and thread in such a way as to create a knotted structure. There are four different types of knotted protein structures that contain 31, 41, 52 and 61 knots and over the past 15 years there has been an increasing number of experimental and computational studies on these systems. The folding pathways of knotted proteins have been studied in some detail, however, the focus of this talk is to address the fundamental question “Why are there knots in proteins?” It is known that once formed, knotted protein structures tend to be conserved by nature. This, in addition to the fact that, at least for some deeply knotted proteins, their folding rates are slow compared with many unknotted proteins, has led to the hypothesis that there are some properties of knotted proteins that are different from unknotted ones, and that this had resulted in some evolutionary advantage over faster folding unknotted structures. In this talk, I will review the evidence for and against this theory. In particular, how a knot within a protein chain may affect the thermodynamic, kinetic, mechanical and cellular (resistance to degradation) stability of the protein will be discussed.

**Speaker:**Jorge Urrutia

**Title:**Local solutions for global problems in wireless networks

**Keywords:**routing, algorithms, network, nodes

**Date:**March 29, 2019In this paper we study on-line local routing algorithms for plane geometric networks. Our algorithms take advantage of the geometric properties of planar to route messages using only local information available at the nodes of a network. We pay special attention to on-line local routing algorithms which guarantee that a message reaches its destination. A message consists of data that have to be sent to a destination node, i.e. the message itself plus a finite amount of space that keeps only a constant amount of data to aid a message to reach its destination, e.g. the address of the starting and destination nodes, a constant number of nodes visited, etc. Local means that at each site we have at our disposal only local information regarding a node and its neighbors, i.e. no global knowledge of the network is available at any time, other that the network is planar and connected. We then develop location aided local routing algorithms for wireless communication networks, in particularly cellular telephone networks.

**Speaker:**Joanna Sulkowska

**Title:**Mysteries of entanglement – proteins, life and physics Biological role of knotting

**Keywords:**proteins, knots, lassos, links, theta curves

**Date:**April 12, 2019 Knotted proteins are believed to be functionally advantageous and to provide extra
stability to protein chains [1]. Twenty years of their investigation suggests that they fold via a
slipknot conformation, across the native twisted loop. During the talk I will show that diversity of
identified folds of knotted proteins and their locations in cells is still growing and surprising us.

Moreover recently we went one step further and found that proteins can be even more entangled than
knots – they also form lassos and links, which consist of several components. Based on the search
through the entire Protein Data Bank we identified several sequentially nonhomologous chains that
form a Hopf link, a Solomon link [2], and various types of lassos. I will show that topological
properties of these proteins are related to their function and stability. Finally I will explain how
the presence of links affects folding pathways of proteins and present new reaction coordinate to
study entangled proteins.

During my talk I will also present knotted TrmD which is the leading antimicrobial drug target
owing to its essentiality for bacterial growth, its broad conservation across bacterial species,
and its substantial differences from the human and archaeal counterpart Trm5. I will also present
our achievements in designing selective inhibitor for TrmD, based on combining theoretical and
experimental methods [3].

All entangled proteins – 7% of proteins deposited in the PDB – are collected in databases: KnotProt,
LassoProt and LinkProt.

[1] JI Sulkowska, E Rawdon, KC Millett, J Onuchic, A Stasiak, PNAS (2012) 109: E1715-23

[2] P Dabrowski-Tumanski, JI Sulkowska, PNAS (2017) 114, 3415–3420

[3] T Christian, et al., Nature S&MB (2016) 23, 941-948

**Speaker:**Eric Goubault

**Title:**Directed topological complexity

**Keywords:**directed algebraic topology, topological complexity, directed invariants, homotopy theory, differential inclusions, control theory

**Date:**Mayo 10, 2019 I will introduce in this talk a variant of topological complexity, that can be applied to help classify directed spaces, and has applications to the study of dynamical
systems in the large, such as differential inclusions. Directed topological complexity looks for specific partitions {F1,…,Fn,…} of the set of reachable states of some directed space X,
such that there is a continuous section from each of the Fi to the space of directed paths which is right inverse to the end points map. As in the classical case, this sheds an interesting
light on a number of directed invariants, and we discuss in particular dicontractibility. We will also show some examples of calculations on directed graphs, non-positively curved cubical
complexes, and directed spheres.

This ongoing work with Aurélien Sagnier and Michael Farber.

**Speaker:**Gareth Alexander

**Title:**Geometric Topology of Liquid Crystal Textures

**Keywords:**Crystal, textures, cholesteric, chirality, distortions, nematic

**Date:**May 24, 2019The textures and phases of liquid crystals are replete with geometric motifs, and the geometric approach to elasticity underpins a large portion of nonlinear theories. Despite this, the basic characterisation of topology comes from the homotopy theory without particular attention to geometric features. I will describe our recent work developing geometric approaches to liquid crystal topology, describing cholesteric point defects and topological chirality, and the geometric features of bend distortions, illustrated by applications to the twist-bend nematic phase.

**Speaker:**Joel Hass

**Title:**Some New Geometric Methods to Distinguish Shapes and Applications in Biology

**Keywords:**shape, metric, space, genus-zero, surfaces

**Date:**August 23, 2019Almost everything we encounter in our 3-dimensional world is a surface - the outside of a solid object. Comparing the shapes of surfaces is, not surprisingly, a fundamental problem in both theoretical and applied mathematics. Deep mathematical results are now being used to study objects such as bones, brain cortices, proteins and biomolecules. This talk will discuss joint work with Patrice Koehl that introduces a new metric on the space of genus-zero surfaces and applies it in this context.

**Speaker:**Jose Perea

**Title:**Topological Dimensionality Reduction

**Keywords:**Dimensionality reduction, topological data analysis, persistent homology, Eilenberg-MacLane coordinates

**Date:**September 6, 2019When dealing with complex high-dimensional data sets, several machine learning tasks rely on having appropriate low-dimensional representations. These reductions are often phrased in terms of preserving statistical or metric information and may fail to recover important topological features. We will describe in this talk several schemes to take advantage of the underlying topology of a data set, in order to produce informative low-dimensional coordinates.

**Speaker:**Jean-Luc Thiffeault

**Title:**Random entanglements: Winding of planar Brownian motions

**Keywords:**planar Brownian motion, entanglement, winding number

**Date:**October 4, 2019There exists a wide class of winding problems, where one or more planar Brownian motions get entangled around each other and obstacles in the plane. These can be seen as models of entanglement of long molecules, or as a general framework for understanding dynamical topological complexity. I will review some classical results, such as Spitzer's law, which gives the angular distribution for a planar Brownian motion around the origin. I will discuss a recent variation (with Huanyu Wen) where we add a drift that promotes entanglement. Finally, I will present ongoing work (with Gage Bonner and Benedek Valko) on the much more challenging problem of winding around several obstacles. In some limits, this can be reduced to computing the escape rate for a random walk on a so-called regular languages.

**Speaker:**Mariel Vázquez

**Title:**Topological modeling of DNA recombination

**Keywords:**DNA knots, reconnection, link nomenclature, DNA topology

**Date:**October 18, 2019DNA replication in bacteria yields two interlinked circular chromosomes. Returning the chromosomes to an unlinked monomeric state is essential to cell survival. Simplification of DNA topology is mediated by enzymes, such as recombinases and topoisomerases. We here focus on site-specific recombinases that recognize two short segments of DNA (the recombination sites), introduce two double stranded breaks and recombine the ends. The local action of site-specific recombinases is a reconnection event which is modeled mathematically as a band surgery. The banding is coherent or non-coherent, depending on the relative orientation of the recombination sites. We use tools from low dimensional topology to investigate the mechanism of action of these enzymes, and we complement the analytical work with computer simulations. The numerical work provides a quantitative measure to distinguish among pathways of topology simplification by recombination, and also informs the search for bandings between specific pairs of link types.

**Speaker:**Jason Cantarella

**Title:**Modeling topological polymers with Gaussian random graph embeddings

**Keywords:**topological polymers, network polymers, random polygons, random graph embeddings, Kirchhoff index, biopolymers

**Date:**November 8, 2019 Very recently, chemists have succeeding in synthesizing purified samples of polymers which are more topologically
complicated than the linear or ring polymers which have previously been created. The new polymers have different graph types, including
theta-curves, multithetas, K_{3,3}, and K_4, with new types arriving at a rapid rate. In this talk, we develop a theory of Gaussian
random graph embeddings in order to model the shapes of these polymers in solution. The theory allows us to connect the expected radius
of gyration of the polymers with the Kirchhoff index of the graph. We will prove a theorem that describes the asymptotic behavior of the
expected radius of gyration for large graphs of fixed topology in terms of the degree-Kirchhoff index of the graph. Last, we show that
our theory provides a good fit to data from the Tezuka lab's experimental measurements of the relative sizes of topological polymers with
different graph types.

This talk is joint work with Tetsuo Deguchi and Erica Uehara (Ochanomizu University) and Clayton Shokwiler (Colorado State University),
with support from the Simons Foundation (#354225 and #52412) and
the Japan Science and Technology Agency (CREST Grant Number JPMJCR19T4, Deguchi Lab) and JSPS KAKENHI Grant Number JP17H06463.

**Speaker:**Roy Meshulam

**Title:**Aspects of High Dimensional Expansion

**Keywords:**High dimensional expansion, simplicial cohomology, random complexes

**Date:**November 15, 2019 The Cheeger constant is a measure of the edge expansion of a graph, and as such plays a key role in diverse mathematical
areas, in particular in combinatorics and theoretical computer science. In recent years there is an interest in higher dimensional notions
of expansion that likewise provide a quantitative measure of cohomological acyclicity of a complex in dimension *k*. In this talk we will
focus on a *k*-dimensional version of the graphical Cheeger constant called *coboundary expansion*, and on a related parameter called
*cosystolic expansion*. After introducing these notions and outlining their relevance to random complexes and to topological overlap
theorems, we will discuss two results. The first concerns an application of cosystolic expansion to a stability version of the
cohomological classification of *G*-covering spaces (joint work with Irit Dinur). The second is a lower bound on coboundary expansion under
certain symmetry assumptions (joint work with Dmitry Kozlov).

**Speaker:**Massimo Ferri

**Title:**Non-topological persistence

**Keywords:**Topological persistence, concrete category, weighted graph, coherent sampling, ranging feature, steady feature

**Date:**November 22, 2019After recalling "classical" topological persistence, I introduce the generalization of the persistent Betti number functions to what we call persistence functions. These are defined axiomatically on a wide class of concrete categories. Then, three types of processes generating persistence functions are shown: coherent samplings and ranging or steady features. Examples of all of them are given on weighted graphs coming from real situations. This is based on a joint work with Mattia G. Bergomi, Pietro Vertechi and Lorenzo Zuffi.

**Speaker:**Kathryn Hess

**Title:**Topological exploration of neuronal network dynamics

**Keywords:**network dynamics, persistent homology

**Date:**December 6, 2019One of the paramount challenges in neuroscience is to understand the dynamics of networks of interconnected neurons. In this talk I will describe a novel approach to classifying network dynamics that employs tools from topological data analysis. The efficacy of our method was established by studying simulated activity in three small artificial neural networks in which we varied certain “biological” parameters, giving rise to dynamics that could be classified into four regimes, and showing that a machine learning classifier trained on features extracted from persistent homology could accurately predict the regime of the network it was trained on and also generalize to other networks not presented during training. I will also discuss the application of these methods to networks of actual neurons in mice. This is joint work with Jean-Baptiste Bardin and Gard Spreemann.

**Speaker:**Jean-Daniel Boissonnat

**Title:**Delaunay triangulation of manifolds

**Keywords:**computational geometry, computational topology, Delaunay triangulation, manifold

**Date:**January 24, 2020The Delaunay complex is a natural structure to consider when seeking to triangulate a space equipped with a metric. It plays a central role in the development of algorithms for meshing Euclidean domains. In applications where an anisotropic mesh is desired, a standard approach is to consider a Riemannian metric defined over the domain and to construct an approximate Delaunay triangulation with respect to this metric. In the case of surfaces, it has been shown that a Delaunay triangulation exists if the set of sample points is sufficiently dense but this is no longer true in higher dimensional manifolds: sample density alone is not sufficient to ensure that the Delaunay complex is a triangulation. To walk around this difficulty, we first introduce perturbation algorithms in Euclidean space that make the Delaunay triangulation of well-distributed point sets quantifiably stable with respect to perturbations of the metric or of the point positions. We then adapt this Euclidean perturbation algorithm to the context of compact manifolds equipped with a metric that can be locally approximated by a Euclidean metric. In particular, this includes Riemannian metrics, as well as the extrinsic metric on submanifolds variations of which have been exploited in algorithms for reconstructing submanifolds of Euclidean space from a finite set of sample points. The simplicial complex produced by our algorithm is naturally equipped with a piecewise flat metric that is a quantifiably good approximation to the metric on the original manifold.

**Speaker:**Heather A. Harrington

**Title:**Algebraic Systems Biology

**Keywords:**algebraic systems, biology, models, TDA

**Date:**February 7, 2020Signalling pathways can be modelled as a biochemical reaction network. When the kinetics follow mass-action kinetics, the resulting mathematical model is a polynomial dynamical system. I will overview approaches to analyse these models with steady-state data using computational algebraic geometry and statistics. Then I will present how to analyse such models with time-course data using differential algebra and geometry for model identifiability. Finally, I will present how topological data analysis can provide additional information to distinguish these models and experimental data from wild-type and mutant molecules. These case studies showcase how computational geometry, topology and dynamics can provide new insights in the biological systems, specifically how changes at the molecular scale (e.g. MEK WT and mutants) result in phenotypic changes (e.g. fruit fly mutations).