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, 2018

Geometric 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, 2018

An 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, 2018

The 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, 2019

Quantum 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, 2019

In 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, 2019

There 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, 2019

In 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, 2019

The 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, 2019

Almost 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, 2019

When 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.