Georgia Institute of Technology College of Sciences

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.

We will look at various instances of how working with skeins (diagrams) provides a way to describe the existence of various topological quantum invariants that were originally described using representation theory. This provides a very simple description of these invariants. Along the way we will look at how to describe the algebraic data (ribbon categories) topologically and also how one could observe instances of certain dualities that exist between certain categories using these diagrams. 

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.

Abstract TBA

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.

The emergence of large connected structures in networks has been a central topic in random graph theory since its inception, forming a foundation for understanding fundamental processes such as the spread of influence or epidemics, and the robustness of networked systems. The field witnessed significant growth from the early 2000s, fueled by a surge in experimental work from statistical physics that introduced fascinating concepts such as universality. Broadly speaking, universality suggests that the formation of a giant component in random graphs often depends primarily on macroscopic statistical properties like the degree distribution. In the theoretical literature, two universality classes have emerged, both closely related to Aldous’ seminal work on critical random graphs and the theory of multiplicative coalescents. In this talk, I will present a third universality class that emerges in the setting of percolation on random graphs with infinite-variance degree distributions. The new universality class exhibits fundamentally different behavior compared to multiplicative coalescents and reveals surprising phenomena concerning the width of the critical window—phenomena that were unforeseen in the substantial physics literature on this topic. Based on joint work with Shankar Bhamidi and Remco van der Hofstad.

We rigorously prove the filamentation phenomenon for a class of weak solutions to the Euler equations known as vortex caps. Vortex caps are characteristic functions representing time-evolving sets of Lagrangian type, with energy preserved at all times. The filamentation of vortex caps is characterized by L^1 -stability alongside unbounded growth of the perimeter of their interfaces. We recall the existence and stability results for vortex caps on the sphere, based on Yudovich theory. Using L^1 -stability, we derive a lower bound for the growth of the perimeter of vortex caps over time.

The regularity method for graphs has been well studied for dense graphs, i.e., graphs on $n$ vertices with $\Omega(n^2)$ edges. However, applying it to sparse graphs, i.e., those with $o(n^2)$ edges seems to be a harder problem. In the mid 2010s, the regularity method was extended to dense subgraphs of random graphs thus resolving the KŁR conjecture. Later, in another direction, Conlon, Fox, Sudakov and Zhao proved a removal lemma for $C_5$ in graphs that do not contain any $C_4$ (such graphs on $n$ vertices can contain at most $n^{3/2}$ edges). In this talk, we will consider a similar problem for sparse $3$-uniform hypergraphs. In particular, we consider an application of the regularity method to $3$-uniform hypergraphs whose vertices do not contain $C_4$ in their links and satisfy an additional boundedness condition. This is joint work with Vojtěch Rödl and Mathias Schacht.