Georgia Institute of Technology College of Sciences

The prediction of RNA secondary structures from sequence is a well developed task in computational RNA Biology. However, in a cellular environment RNA molecules are not isolated but rather interact with a multitude of proteins. RNA secondary structure affects those interactions with proteins and vice versa proteins binding the RNA affect its secondary structure.  We have extended the dynamic programming approaches traditionally used to quantify the ensemble of RNA secondary structures in solution to incorporate protein-RNA interactions and thus quantify these effects of protein-RNA interactions and RNA secondary structure on each other. Using this approach we demonstrate that taking into account RNA secondary structure improves predictions of protein affinities from RNA sequence, that RNA secondary structures mediate cooperativity between different proteins binding the same RNA molecule, and that sequence variations (such as Single Nucleotide Polymorphisms) can affect protein affinity at a distance mediated by RNA secondary structures.

https://gatech.bluejeans.com/348270750

Homotopy continuation methods are numerical methods for solving polynomial systems of equations in many unknowns. These methods assume a set of start solutions to some start system. The start system is deformed into a system of interest (the target system), and the associated solution paths are approximated by numerical integration (predictor/corrector) schemes.

The most classical homotopy method is the so-called total-degree homotopy. The number of start solutions is given by Bézout's theorem. When the target system has more structure than start system, many paths will diverge, This behavior may be understood by working with solutions in a compact projective space.

In joint work with Telen, Walker, and Yahl, we describe a generalization of the total degree homotopy which aims to track fewer paths by working in a compact toric variety analagous to projective space. This allows for a homotopy that may more closely mirror the structure of the target system. I will explain what this is all about and, time-permitting, touch on a few twists we discovered in this more general setting. The talk will be accessible to a general mathematical audience -- I won't assume any knowledge of algebraic geometry.

The Teichmüller space is the space of hyperbolic structures on surfaces, and there are different flavors depending on the class of surfaces. In this talk we consider the enhanced Teichmüller space which includes additional data at boundary components. The enhanced version can be parametrized by shear coordinates, and in these coordinates, the Weil-Peterson Poisson structure has a simple form. We will discuss a construction of the quantum Teichmüller space corresponding to this Poisson structure.

Bluejeans: https://bluejeans.com/872588027

The differential equation method is a powerful tool used to study the evolution of random combinatorial processes. By showing that the process is likely to follow the trajectory of an ODE, one can study the deterministic ODE rather than the random process directly. We extend this method to ODEs in infinite-dimensional Banach spaces.
We apply this tool to the classical n-queens problem: Let Q(n) be the number of placements of n non-attacking chess queens on an n x n board. Consider the following random process: Begin with an empty board. For as long as possible choose, uniformly at random, a space with no queens in its row, column, or either diagonal, and place on it a queen. We associate the process with an abstract ODE. By analyzing the ODE we conclude that the process almost succeeds in placing n queens on the board. Furthermore, we can obtain a complete n-queens placement by making only a few changes to the board. By counting the number of choices available at each step we conclude that Q(n) \geq (n/C)^n, for a constant C>0 associated with the ODE. This is optimal up to the value of C.

Based on joint work with Zur Luria.