This seminar is for the Chancellors Faculty Excellence Program: Quantitative and Computational Developmental Biology Cluster Hire.
In most real life applications Mathematics not only face the challenge of modeling (typically by means of ODE and/or PDE), analysis and computer simulations but also the need control and design. And the successful development of the needed computational tools for control and design cannot be achieved by simply superposing the state of the art…
One way of drawing a fractal shape uses a random process called the Chaos Game to produce a predictable (non-random) picture. We need to explore an important theorem called the Contraction Mapping Theorem. We’ll only scratch the surface, but we’ll try to get a sense of the interesting ideas involved. No prior knowledge of fractal…
Many applications in the life sciences involve unknown system parameters that must be estimated using little to no prior information. In addition, these parameters may be time-varying and possibly subject to structural characteristics such as periodicity. We show how nonlinear Bayesian filtering techniques can be employed in this setting to estimate unknown, time-varying parameters, while…
Subalgebras of Lie algebra of Lorentz group will be discussed as the basic examples at the beginning of the report. The corresponding analysis is performed by canonical bases for subspaces of a vector space. All canonical bases for 5-dimensional and 4-dimensional subspaces of a 6-dimensional vector space are found, and they are utilized to find…
Radial glial progenitors follow patterns of proliferation and differentiation to produce neurons and glia in the developing brain. The cells first undergo neurogenesis, producing neurons in a fairly deterministic manner, and then a portion of the cell population switches to producing glia. This neurogenesis-to-gliogenesis switch (NGS) and the patterns of cell division in gliogenesis are…
Please join us for our weekly brown bag lunch! You bring your lunch, and we will bring a delicious treat. Everyone (not just women) is welcome to join or stop by for as long as they can!
We will discuss whether it is possible to reconstruct a phylogenetic tree from sequence data, given a specific model of evolution. To answer this question for a coalescent-based model, we study ideals associated to the model. By observing some facts about the geometry of the model, we can demonstrate that these ideals contain polynomials which may…
CUR decomposition is a low-rank matrix decomposition that is formed using a small number of actual columns and actual rows of the data matrix. Since this decomposition is explicitly constructed using entries from the actual data set, CUR decomposition may be interpretable by practitioners; for this reason, it is sometimes preferable to the Singular Value…
How many people need to be in a room to guarantee that there is a large probability that a pair of them have the same birthday? The number of people is surprisingly small, a fact that is sometimes called the ”birthday paradox” because it contradicts our intuition. The simple mathematics behind the birthday paradox leads to ”birthday attacks” on cryptographic protocols…
Advised by Ernie Stitzinger.
Please join us for our weekly brown bag lunch! You bring your lunch, and we will bring a delicious treat. Everyone (not just women) is welcome to join or stop by for as long as they can!
Given a determinantal representation by means of a cyclic weighted shift matrix, one can show the resulting polynomial is hyperbolic and invariant under the action of the dihedral group. Chien and Nakazato (2015) asked the converse question. By properly modifying a determinantal representation construction of Dixon (1902), we show for every hyperbolic polynomial with dihedral…
Discovered in the 1730s, the classical Euler-Maclaurin formula may be viewed as a formula for summing the values of a function over the lattice points in a one-dimensional polytope. Several years ago, Berline and Vergne generalized this formula to polytopes of arbitrary dimension, obtaining a formula for the sum of a polynomial function over the…
Advised by Hien Tran.
The Kuramoto model is used to describe synchronization behavior of a large set of oscillators. The equilibria of this model can be computed by solving a system of polynomial equations using algebraic geometry. Typical methods for solving such polynomial systems compute all complex equilibrium points when only the real equilibrium points are of physical interest,…