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.