Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS200, part 2: From algebraic geometry to geometric topology: Crossroads on applications
Time:
Wednesday, 10/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F007
30 seats, 59m^2

Presentations
3:00pm - 5:00pm

From algebraic geometry to geometric topology: crossroads on applications

Chair(s): Jose Carlos Gomez Larrañaga (CIMAT), Renzo Ricca (University of Milano-Bicocca), De Witt Sumners (Florida State University)

The purpose of the Minisymposium "From Algebraic Geometry to Geometric Topology: Crossroads on Applications" is to bring together researchers who use algebraic, combinatorial and geometric topology in industrial and applied mathematics. These methods have already seen applications in: biology, physics, chemistry, fluid dynamics, distributed computing, robotics, neural networks and data analysis.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Privileged topologies of self-assembling molecular knots

Cristian Micheletti
SISSA

The self-assembly of objects with a set of desired properties is a major goal of material science and physics. A particularly challenging problem is that of self-assembling structures with a target topology. Here we show by computer simulation that one may design the geometry of string-like rigid patchy templates to promote their efficient and reproducible self-assembly into a selected repertoire of non-planar closed folds including several knots. In particular, by controlling the template geometry, we can direct the assembly process so as to strongly favour the formation of constructs tied in trefoil or pentafoil, or even of more exotic knot types. A systematic survey reveals that these "privileged", addresable topologies are rare, as they account for only a minute fraction of the simplest knot types. This knot discovery strategy has recently allowed for predicting complex target topologies [1,2,3], some of which have been realized experimentally [4,5].
References
1) G. Polles et al. "Self-assembling knots of controlled topology by designing the geometry of patchy templates", Nature Communications, 2015
self-assembly video demonstration available at this link
2) G. Polles et al. "Optimal self-assembly of linked constructs and catenanes via spatial confinement", Macro Letters (2016)
3) M. Marenda, et al. “Discovering privileged topologies of molecular knots with self-assembling models", Nature Communications, 2018
4) J. Danon et al. "Braiding a molecular knot with eight crossings.", Science (2017)
5) Kim et al. "Coordination-driven self-assembly of a molecular knot comprising sixteen crossings", Angew. Chem. Int. Ed. (2018)
 

Why are there knots in proteins?

Sophie Jackson
University Of Cambridge

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.

 

The study of 2-stratifolds as models for applications (Part 1)

Jose Carlos Gomez Larrañaga1, Wofgang Heil2, Francisco Gonzalez Acuña3
1CIMAT, 2FSU, 3UNAM and CIMAT

In physics, the morphological structure of granular samples in mechanical equilibrium has been modeled by graphs and analyzed by using the first Betti number (S. Ardanza-Trevijano et al.). in TDA, persistent homology is used to study graphs arising from sampling point clouds. Graphs can be viewed as 1-dimensional stratified spaces and possibly more information could be obtained by modeling with 2-dimensional stratified spaces, since these provide more topological invariants. For example, in Physics, the study of singularities of soap films (E. Goldstein et al.) and in Chemistry and Biology, the study of cyclo-octane energy landscapes (S. Martin et al.) led to 2-dimensional complexes that consist of unions of 2-manifolds intersecting along a curve. These 2-complexes are special cases of 2-dimensional stratified spaces. In TDA, techniques have been developed (Bendich et al.) for organizing, visualizing and analyzing point cloud data that has been sampled from or near a 2-dimensional stratified space. There is no topological classification of these spaces.

A systematic study of 2-dimensional stratifiedspaces without boundary curves or 0-dimensional singularities, the 2-stratifolds, was begun by W. Heil, F.J. González-Acuña and the speaker.

In this talk, we wil explore 2-stratifolds with trivlal fundamental group.

 

The study of 2-stratifolds as models for applications (Part 2)

Jose Carlos Gomez Larrañaga1, Wolfgang Heil2, Francisco Gonzalez Acuña3
1CIMAT, 2FSU, 3UNAM

In physics, the morphological structure of granular samples in mechanical equilibrium has been modeled by graphs and analyzed by using the first Betti number (S. Ardanza-Trevijano et al.). in TDA, persistent homology is used to study graphs arising from sampling point clouds. Graphs can be viewed as 1-dimensional stratified spaces and possibly more information could be obtained by modeling with 2-dimensional stratified spaces, since these provide more topological invariants. For example, in Physics, the study of singularities of soap films (E. Goldstein et al.) and in Chemistry and Biology, the study of cyclo-octane energy landscapes (S. Martin et al.) led to 2-dimensional complexes that consist of unions of 2-manifolds intersecting along a curve. These 2-complexes are special cases of 2-dimensional stratified spaces. In TDA, techniques have been developed (Bendich et al.) for organizing, visualizing and analyzing point cloud data that has been sampled from or near a 2-dimensional stratified space. There is no topological classification of these spaces.

A systematic study of 2-dimensional stratified spaces without boundary curves or 0-dimensional singularities, the 2-stratifolds, was begun by J.C. Gómez-Larrañaga, F.J. González-Acuña and the speaker.

In this talk, we will describe an efficient algorithm on the labeled graph of a 2-stratifold that determines its homotopy type and an efficient algorithm that determines if its fundamental group is infinite cyclic. Also, we will discuss embeddings of 2-stratifolds as 3-manifold spines and talk about the solvability of the word problem for 2-stratifold groups.