Georgia Institute of Technology College of Sciences

Filamentous materials may exhibit structure-dependent material properties and function that depend on their entanglement. Even though intuitively entanglement is often understood in terms of knotting or linking, many of the filamentous systems in the natural world are not mathematical knots or links. In this talk, we will introduce a novel and general framework in knot theory that can characterize the complexity of open curves in 3-space. This leads to new metrics of entanglement of open curves in 3-space that generalize classical topological invariants, like for example, the Jones polynomial and Vassiliev invariants. For open curves, these are continuous functions of the curve coordinates and converge to topological invariants of classical knots and links when the endpoints of the curves tend to coincide. These methods provide an innovative approach to advance important questions in knot theory. As an example, we will see how the theory of linkoids enables the first, to our knowledge, parallel algorithm for computing the Jones polynomial.

Importantly, this approach opens exciting applications to systems that can be modeled as open curves in 3-space, such as polymers and proteins, for which new quantitative relationships between their structure and material properties become evident. As an example, we apply our methods to proteins to understand the interplay between their structures and functions. By analyzing almost all protein structures in the Protein Data Bank, we derive for the first time a quantitative representation of the topology/geometry of the Topological Landscape of proteins. We show that 3 topological and geometrical parameters alone can predict the biological classifications of proteins with high accuracy. Moreover, preliminary results show that our proposed topological metrics based on static protein structures alone correlate with protein dynamics and protein function. The methods and results represent a new framework for advancing knot theory, as well as its applications to filamentous materials, which can be validated by experimental data and integrated into machine-learning algorithms.

Hypergraph Turán Problems became more approachable due to flag algebras. In this talk we will first focus on tight cycles without an edge. A tight $k$-cycle minus an edge $C_k^-$ is the 3-graph on the vertex set $[k]$, where any three consecutive vertices in the string $123...k1$ form an edge. We show that for every $k \geq 5$, k not divisible by $3$, the extremal density is $1/4$. Moreover, we determine the extremal graph up to $O(n)$ edge edits. The proof is based on flag algebra calculations.

Then we describe new developments in solving large semidefinite programs that allows for improving several other bounds on Turán densities.

This talk is based on joint work with Connor Mattes, Florian Pfender and Jan Volec.

MacPhersonians are a combinatorial analog of real Grassmannians defined by oriented matroids.  A long standing conjecture says that each MacPhersonian is homotopy equivalent to the corresponding Grassmannian.  Pseudolinear Grassmannians are spaces of topological representations of oriented matroids, and these are each homotopy equivalent to the corresponding Grassmannian in rank 3.  I will present a good cover of the rank 3 pseudolinear Grassmannian with nerve complex isomorphic to the order complex of the corresponding MacPhersonian, confirming the conjecture in rank 3.

Consider a sequence of independent and identically distributed SL(2, R) matrices. There are several classical results by Le Page, Tutubalin, Benoist, Quint, and others that establish various forms of the central limit theorem for the products of such matrices. I will talk about a recent joint work with Anton Gorodetski and Victor Kleptsyn, where we generalize these results to the non-stationary case. Specifically, we prove that the properly shifted and normalized logarithm of the norm of a product of independent (but not necessarily identically distributed) SL(2, R) matrices converges to the standard normal distribution under natural assumptions. A key component of our proof is the regularity of the distribution of the unstable vector associated with these products.