# Research Projects

Our mathematics research involves carrying out mathematical modeling of brain networks, and developing and validating novel mathematical tools used for information processing and analysis.

Specific research includes:

**Affine Reflection Groups for Tiling Applications**

Knot Theory and Models of Genetic Molecules: In our research we examine applications of affine Weyl groups of rank 2, the symmetry group of the tiling/lattice, and develop the tools for applications requiring tilings of the Euclidean plane to elucidate the equivalence of these tilings as projections of knots.

The resulting mathematical structure produces a framework which utilizes knot theory for modeling the structure and function of genetics molecules – specifically the action of enzymes altering the topology of DNA in site-specific recombination, [BPP].

The tilings of the plane have a one-to-one correspondence with graphs of degree 2. These graphs in turn have a correspondence with 2-dimensional projections of knots, and when converted to signed graphs, a correspondence with 3-dimensional knots. Reflections of the Weyl group carried out in succession on the basic tiles defined by the simple roots of the 2-dimensional Lie groups produce sequences of specific surgeries (cutting and pasting of strands) on the corresponding knots which give a trajectory in the underlying knot/link space. These sequences of knots produce knots which are known to occur in the action of enzymes in altering the geometry and topology of DNA. Specifically, the reflection operations carried out on tilings corresponding to the base tiles from the 2-dimensional Lie groups are equivalent to the performance of specific surgeries which introduce new tangles into a knot which changes its topology and geometry to that of a new knot. These operations reveal the specific methods by which different enzymes may operate on DNA to enable them to rapidly change their geometry and topology to carry out necessary functions.

The tiling of the plane produced by the possible reflections of the tiles corresponding to simple Lie groups have been investigated up to 6 tiles, which empirically has been determined to produce all prime knots up to 10 crossings. In our research we attempt to characterize the knots produced by all of the 2-dimensional Lie groups tilling under reflections up to an arbitrary order, determine if the structure produces all prime knots up to 17 crossings, and attempt to generalize this to knots consisting of an arbitrary number of crossings. We examine the array of possible knots produced by this formalism. In addition, we attempt to determine the trajectories in knot/link space which have the relevance to the modeling of the structure and function of genetic molecules. We generalize this to non-crystallographic planar groups or quasi-crystals and determine whether these structures (along with the 2-dimensional simple Lie groups) span the set of all prime knots.

In addition, the affine groups allow one to develop the families of orthogonal polynomials. It is known that in the knot theory a number of knot invariants arise as polynomials in 2 variables (i.e. the HOMFLY polynomial and the Johnson Polynomial). We propose to investigate these invariant polynomials in terms of the affine Weyl group. We propose to completely characterize all the cases in 2 variables which may help to determine if there exist previously undetermined symmetries or relationships between the knots and the polynomials of the Weyl groups.

**Cubature Formula for Polynomials of Rank 2 Simple Lie Groups**

The cubature relation is the exact equality of an integral of a given function to the sum of values of that function sampled at precisely chosen (interpolation) points. It can be viewed either as a prototype of quantum mechanical relation between continuous and discrete description of physical phenomena, or traditionally as the best approximation of data.

The multidimensional cubature formula is a generalization of cubature formula of Chebyshev polynomials of the second kind in one variable. It includes the optimal approximation property generalized to polynomials of n variables based on the root lattices of compact simple Lie groups G of any type and of rank n.

**Generalization of the Nyquist-Shannon Sampling Theorem**

Nyquist-Shannon sampling theorem is the fundamental result frequently used in signal processing, communications, information theory, and generally in all fields in which 1-dimensional digital data are involved. The theorem can be stated as follows:

If a function of time x(t) contains no frequencies higher than B Hertz, it is completely determined by sampling its values at a series of points spaced 1/2B seconds apart.

**Hybrid Boundary Value Problem**

The classical boundary value problem (Dirichlet and Neumann) has been known for a century. The way to provide the correct boundary behavior is to use the expansion functions which satisfy required boundary behavior. In the Dirichlet case, they are equal to zero at the boundary, while in the Neumann case their normal derivative at the boundary is equal to zero.

In our research we construct efficient ways of how to ensure boundary behavior of Fourier expansion that can be understood as hybrid: partially Dirichlet and partially Neumann.

**Pattern and Pattern Recognition Mathematical Theory Development**

This research involves defining the concept of pattern and pattern recognition through a theory based on dynamical systems.