## 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.