# Abstracts of supporting lectures

- Yuliy Baryshnikov

*Applied Configuration Spaces*
I will discuss some versions of configurations spaces that are of relevance in applications: configuration spaces of hard disks in a box, or of particles that cannot occupy the same position with multiplicity greater than \(k>1\). - Vin de Silva

*Presistent cohomology and circle coordinates*

- John Etnyre

*A quantitative Darboux theorem and the three dimensional contact sphere theorem*
Since the work of Chern and Hamilton there has been a great deal of work studying Riemannian metrics adapted to contact structures, but most of this work has focused on properties of the Riemannian metric. There have been few results concerning properties of the contact structure in terms of the Riemannian metric. In this talk I will discuss a "quantitative Darboux theorem." More precisely, I will give estimates on the size of a geodesic ball in a contact metric manifold that is contactomorphic to the standard contact 3-ball. This is one of the first results relating the key contact geometric notion of tightness to adapted Riemannian metrics. This leads to a "contact sphere theorem": if you have a contact 3-manifold and a metric adapted to it with positive \(\frac{4}{9}\)-pinched curvature then the manifold is the sphere and the contact structure is standard. This is joint work with Rafal Komendarczyk and Patrick Massot. - Steve Ferry

*Taming Wild Homotopies*
We will discuss techniques for replacing arbitrary homotopies by
homotopies with nicer paths. - Vivian Ferry

*Quasicrystals and Spatial Correlations of Nanopatterns in Nanostructured Solar Cells*
Photovoltaic technology is currently enjoying substantial growth and investment. Effective light trapping, where nanostructures are incorporated into the solar cell to increase absorption while decreasing the volume of material, is essential for cost reduction. The enhanced light absorption is governed by both local resonances of the individual nanostructures and collective modes dependent on the patterning of the nanostructures. Experiments and simulations showing the effect of periodic, quasi-crystal, and pseudo-random patterns on the photocurrent will be discussed. Applications of quasicrystal patterns to other areas of nanophotonics such as chemical sensing will also be covered. - Erik Guentner

*Coarse geometry of graph coverings*

- Jerry Kaminker

*Noncommutative topology and tilings*
Noncommutative topology and geometry has played an
interesting and important role in the study of tilings of \(\mathbb{R}^d\). After
suggestions of Connes, many researchers have developed the theory-for
example, Jean Bellissard in the direction of solid state physics, and Ian
Putnam toward dynamical systems. In this talk we will explain the use
of noncommutative methods and why this is a natural approach. An
important step is deciding which noncommutative algebra to associate
to a tiling, and, once this is done, nding ways to extract information
from it. One application of this is the proof of Bellissard's Gap Labeling
Conjecture, which we will discuss. In fact, there are many possible ways
to extend these ideas further. We will present some of these and also
consider how one can adapt noncommutative methods to Weinberger's
very interesting generalizations of tilings and tiling spaces to the case
of complete Riemannian manifolds with bounded geometry. - Sanjeevi Krishnan

*Directed sheaf homology and network optimization*
The max-flow min-cut theorem, traditionally applied to
problems of maximizing the flow of commodities along a network (e.g.
oils in pipelines) and minimizing the costs of disrupting networks
(e.g. damn construction), has found recent applications in image
segmentation, communication, and other forms of information
processing. In this talk, I will recast and generalize max-flow
min-cut as a form of twisted Poincare Duality. Flows correspond to
the top-dimensional homology, taking local coefficients and values in
a sheaf of semigroups, on "directed" spaces. Cuts correspond to
certain distinguished sections of a dualizing sheaf. Thus max-flow
min-cut dualities extend to higher dimensional analogues of flows,
higher dimensional analogues of directed graphs (e.g. dynamical
systems), and constraints more complicated than upper bounds. I will
describe the formal result, including a construction of directed sheaf
homology, and some applications to network communication.
- Robert Kusner

*Mobius transformations, configuration spaces and canonical differential forms*

- Yevgeny Liokumovich

*Contracting the boundary of a Riemannian 2-disc*
Let \(D\) be a Riemannian 2-disc that has diameter \(d\) and length of the boundary \(L\).
M.Gromov asked the following question. Does there exist a universal constant \(C\),
say \(C=10^{10}\), such that for any disc \(D\) one can contract the boundary of \(D\) to a point through curves
of \( length \leq C \max \{L,d \}\ ?\)
S. Frankel and M. Katz gave a negative answer to this question. Then they asked
if there exists a homotopy of the boundary to a point through curves of length bounded in terms
of \(L\), \(d\) and the area of the the disc \(A\)?
We answer this question positively, showing that the boundary can be contracted
through curves of length \(\leq L+200 d \max \{ 1, \log ( \frac{\sqrt{A}}{d}) \}\).
This is a joint work with Alexander Nabutovsky and Regina Rotman. - Elizabeth Meckes

*Betti numbers of random simplicial complexes*
There has been much recent interest in the topology of random spaces. In this talk, we focus on three different models of random simplicial complexes: \( Erd\ddot{o}s-R\acute{e}nyi\) random complexes, random Vietoris-Rips complexes, and random Cech complexes. In the case of \( Erd\ddot{o}s-R\acute{e}nyi\) random complexes, a complex is constructed by beginning with an \( Erd\ddot{o}s-R\acute{e}nyi\) random graph and taking the maximal simplicial complex with this 1-skeleton. A random Vietoris-Rips complex is constructed similarly, but beginning with a random geometric graph as 1-skeleton. A random \(\check{C}ech\ complex\) is a fuller analog of a random geometric graph, in which faces are filled in or not based on the proximity of their randomly constructed vertices. We will review some known results on threshholds for the vanishing of homology and expectations of Betti numbers, and focus on the limiting distributions of the Betti numbers. In particular, we will show how to prove central limit theorems using Stein's method and, in the Vietoris-Rips and \(\check{C}ech\) cases, a Poissonization and de-Poissonization procedure. - Monica Nicolau

*Unraveling the biology of disease through data decomposition and topological data analysis*
The past decade has witnessed developments in the field of biology that have brought about profound
changes in understanding the dynamic of disease and of biological systems in general. New technology has
given biologists an unprecedented wealth of information, but it has generated data that is difficult to analyze
mathematically, thereby making its biological interpretation challenging. Essentially the data lives in
extremely high dimensional space, making it so sparse that traditional analysis methods have difficulty
identifying meaningful patterns.
These challenges have given rise to a myriad novel exciting mathematical challenges and have
provided an impetus to modify and adapt traditional mathematics tools, as well as develop novel techniques
to tackle the data analysis problems raised in biology.
I will discuss a general approach to address some of these computational challenges by way of a
combination of data decompositions and topological methods, to highlight specific biologically driven questions.
These methods have been applied in a wide range of settings, in particular for the study of the biology of
disease. I will discuss some concrete applications to these methods, including their use to discover a new
type of breast cancer and the associated biology that drives the disease, identify disease progression trends,
and highlighting the driving mechanisms in acute myeloid leukemia.