### Flip Graphs of Flag Triangulations of Surfaces

**Faculty Mentor:**

Allan Edmonds**Description:**Given one triangulation of a surface S one can produce a new triangulation with the same vertices by replacing an edge BC common to two adjacent triangles ABC and BCD with the the edge AD. This replaces the two original triangles with the triangles ABD and ACD. There is a graph whose vertices are the triangulations of S and whose edges correspond to such FLIPS. This is a well-studied procedure in discrete geometry and geometric or topological graph theory. We propose to examine the flip graph in the case of a special class of triangulations called FLAG triangulations. Such triangulations have the property that if there is a 3-cycle AB, BC, CA of edges then the triangulation includes the triangle ABC. Flag triangulations are triangulations completely determined by their 1-skeletons or graphs. Flag triangulations also come up in certain applications of geometry to topology. We will study the flip graph restricted to flag triangulations, and attempt to use it to classify small flag triangulations. We may also study the analogous, but more subtle, procedure for quadrangulations of surfaces (using quadrilaterals in place of triangles). There is the possibility of modifying an existing computer program to automate some of the exploration.

**Prerequisites:**

Linear algebra expected. Graph theory and/or some computer programming would be a plus. Background in topology could also be helpful.### Surfaces tiled by squares

**Faculty Mentor:**

Christopher M. Judge**Description:**One can construct surfaces that approximate any conformal type by gluing together n squares whose sides have length equal to one. In particular, one need only consider gluings where the top of a given square is identified with the bottom of one other, and the left side of a given square is identified with the right side of one other. Labeling the squares 1 through n, we see that the gluing pattern corresponds to a pair of permutations of {1,2,3, ..., n}, one corresponding to the vertical identifications and one corresponding to the horizontal identifications. How does this pair of permutations reflect the topology and geometry of the constructed surface? For example, given the pair of transpositions, how many connected components does the surface have? What is the genus of the surface? What are the symmetries? If pairs of permutations are multiplied pairwise, how does the geometry and topology change? These are the first of many questions that will be explored.

**Prerequisites:**

Basic group theory and linear algebra.### Mathematical theory in population genetics and molecular evolution

**Faculty Mentor:**

Michael Lynch, Department of Biology**Description:**Topics of investigation will involve the development of mathematical theory in population genetics / molecular evolution, involving the stochastic processes of random genetic drift and mutation; cf. [LA]. Specific subjects to be investigated include the evolution of complex functions that require modifications at more than one genetic locus to achieve the final adaptation, the evolution of duplicate genes, and the evolution of surveillance mechanisms for eliminating internal cellular damage. We are also developing maximum-likelihood approaches to estimate population-genetic parameters from high-throughput genomic data derived from multiple individuals in a population.

**Prerequisites:**

multivariable calculus, linear algebra, probability. Programming experience and familiarity with Markov chains helpful but not essential.### Evolution of phenotypes

**Faculty Mentor:**

Emilia Martins, Department of Biology**Description:**We consider modeling evolution of phenotypes along a phylogeny. Most phenotypes are complex traits that are the result of evolution in complex selective regimes. We currently have fairly simple models for considering the evolution of species means along a phylogeny, and could use some help in expanding models to deal with multiple, interacting traits that have evolved subject to pressures from many, sometimes competing, forces.

**Prerequisites:**

multivariable calculus, linear algebra, probability. Programming experience and familiarity with Markov chains helpful but not essential.### Lizard color signals

**Faculty Mentor:**

Emilia Martins, Department of Biology**Description:**We are also currently studying lizard color signals measured by a spectrophotometer as complex reflectance curves. Our analysis involves weighting those curves by the visual system of a lizard receiver or bird predator and the light environment in which it is seen. We then compare reflectance curves for an animal signal with that of various backgrounds. This yields complex data requiring statistical analysis.

**Prerequisites:**

programming experience and statistics.### Graphical partitions and complex dynamics

**Faculty Mentor:**

Kevin M. Pilgrim**Description:**For an integer d >=3 consider a partitition of the integer 2d-2 given by k_1 + k_2 + ... + k_n = 2d-2 where the k_i are positive integers. Consider planar graphs with no loops but with possibly multiple edges joining a pair of vertices. Question: suppose (i) n <= d, and (ii) k_i <= d-1 for all i=1, ... n. Does there exist a connected planar graph with n vertices v_1, ..., v_n with the valence at v_i equal to k_i? Related questions related to the characterization of ``graphical'' partitions have been investigated before. The question arises when trying to characterize the set of degree d rational functions f all of whose critical points z_i, i=1, ..., n are fixed points of f. The analogous question for polynomials has been investigated already, and the generalization is of recent interest to those in complex analytic and arithmetic dynamics.

**Prerequisites:**

none, but combinatorics and complex analysis would be helpful.### Polymer organization and cell function

**Faculty Mentor:**

Sidney Shaw**Description:**The Shaw laboratory investigates the mechanisms controlling polymer organization and function within cells. The microtubule cytoskeleton provides structure to a cell and vectoral information for intracellular transport. Microtubules are formed through the head-to-tail association of tubulin protein dimers into protofilaments, 13 of which create a hollow tube with important physical properties. Using genetics and live-cell imaging techniques, the Shaw laboratory has amassed substantial data on microtubule polymerization dynamics within living plant cells. Through analytical modeling and large-scale Monte Carlo simulations, we are attempting to reconstitute aspects of the measured biological system in order to determine how the polymer mass comes to steady state and how the microtubules form specific patterns within cells.

**Prerequisites:**

differential equations and numerical analysis or programming experience; Fourier series helpful but not essential.### Deformation spaces of polygonal tilings in the plane and space

**Faculty Mentor:**

Matthias Weber**Description:**The regular and uniform tilings of the plane are rigid if you restrict yourself to applying the usual rules: use only regular polygons, tile edge-to-edge, and stay in the plane. Relaxing these rules allows for more and sometimes surprising flexibility. We will begin by investigating examples and then develop a theory of such tilings.

**Prerequisites:**- Abstract Linear Algebra (i.e. familiarity with the notions of vector spaces, linear maps, bilinear forms)
- some Analytic Geometry (i.e. able to compute a line perpendicular to two skew lines, for instance)
- enthusiasm
- skills in Mathematica will be a plus