A powerful approach to understanding sequences of numbers is to study them p-adically. For instance, one can reduce a sequence modulo a prime p and consider the resulting values. For an interesting class of sequences, work of Furstenberg, Deligne, Denef and Lipshitz implies that, for fixed p, these values can be produced by a finite state automaton. These are very basic computers that can be represented visually as a graph. Interesting properties of our sequence can then be read off from these automata. On the other hand, recent work of Rowland and Yassawi demonstrates that these automata can obtained automatically (meaning that a computer can do the work for us). This allows us to experimentally go through lists of interesting sequences, compute the automata for some small primes, and deduce p-adic properties of the sequences (which, in the past, had to be proved by hand). We aim to find automatic proofs of known results and, possibly, interesting new congruences. Depending on the background and taste of participants, we will spend variable amounts of time on familiarizing ourselves with using Mathematica, refreshing our modular arithmetic, getting to know finite state automata and, above all, computing with lots of examples. It is hoped that, at the end of this project, you are inspired and ready to use computer algebra, and that you have seen a glimpse of the experimental nature of mathematical research.
Any five points in the plane (no four on a line) determine a unique conic. A set S is elliptesque (hyperbolesque) if the conic determined by any five points in S is an ellipse or circle (a hyperbola). Nontrivial examples exist! The set $\{(x,x^3):x \ge 0\}$ is provably hyperbolesque and the set $\{(x,x^{3/2}):x \in [1,1.38]\}$ is provably elliptesque. Other sets are probably -esque, but the evidence is numerical. The purpose of this project is to find more such example. See http://www.math.uiuc.edu/~reznick/8213ff-am.pdf for beamer slides of a talk on this subject which give an explanation of the numerical examples.
This is the third semester of an ongoing project aimed at implementing mathematical models for the processing of sensory input by the first layer (V1) of the human visual cortex. Last year, the team produced working versions of several models which recognize image contours by taking advantage of a sub-Riemannian geometric structure inherent to V1. In addition, the team began to implement diffusion algorithms for image completion in disoccluded images. The goals for the fall project are to continue the above research activities, to implement a second sub-Riemannian model for V1 which provides an alternate approach to disocclusion and to compare the efficiency and effectiveness of the models. A three-dimensional display module illustrating the mechanism via a cubical array of computer-controlled LEDs was previously initiated. Depending on the interest and skill set of the participants, the team may choose to continue this additional, `hands-on', activity.
This project will be particularly appropriate for students from applied disciplines, especially physics or engineering. It will involve a significant programming component: familiarity with Mathematica is a must, other software packages (e.g. Java) would be helpful. Prior background in neurobiology, signal processing, etc. is not necessary, but would be welcome.
We will build on last year's project on Random Forests which has lead to several very interesting conjectures on gap distribution and pair correlation properties of random subsets of sequences which have interesting gap distributions themselves, like the Farey sequence. This semester we will study other sequences of geometric and number theoretic interest.
Many properties of the natural numbers can be encoded as sequences of 1's and -1's. On the surface, such sequences often show no obvious pattern and indeed seem to behave much like sequences generated by true random experiments such as coin tosses. In this project we seek to obtain a deeper understanding of the behavior of such sequences via certain "random walks" in the plane formed with these sequences. These random walks provide a natural way to visualize the degree of randomness inherent in a sequence and to detect, and possibly explain, hidden patterns, but they can also open up new mysteries that defy explanation. In past semesters, we focused on random walks associated with quadratic residues and with the Moebius functions, a well-known number-theoretic functions with values +1, -1, and 0, and a random-like behavior that is closely connected to the Prime Number Theorem and the Riemann Hypothesis. In the course of these investigations we came across another, broader class of number-theoretic random walks with mysterious fractal-like patterns. These random walks will be main focus of the current project. For further details see http://www.math.illinois.edu/~hildebr/ugresearch/.
This project is part of an ongoing program to seek out and explore interesting problems in n-dimensional calculus and geometry that are accessible at the calculus level, but rarely covered in standard calculus courses. These problems typically arise as natural generalizations of familiar problems in 2 or 3 dimensions, and they are often motivated by applications to probability and statistics or other areas. In past projects we considered the volume of intersecting cylinders in n dimensions, the Broken Stick Problem (If a stick is broken up randomly into n pieces, what is the probability that the pieces can form a triangle?), and the Random Triangle Problem (If a triangle is chosen at random, what is the probability that it is acute?). For further details and reports on past projects, see http:// www.math.illinois.edu/~hildebr/ugresearch/
Same as before. We study particles whose interactions are governed by potentials such as those of electrodynamics.
In this project we will consider the mean-field classical XY model, which is a twodimensional vector spin model with circular symmetry. The spins are unit vectors in lattice plane. This model shows a phase transition, which can be seen, experimentally in the LCD displays or in films of superconductors etc. We are interested in studying the behavior of the total spin as related to the inverse temperature. We will use simulations in order to study the phase transition and to study the energy maximization free energy function and rate function. The XY model, in absence of external field, can be represented as: $$ H = -J \sum_{i,j = 1}^n \sigma_i \cdot \sigma_j = - \frac{1}{2n} \sum_{i,j = 1}^n Cos(\theta_i - \theta_j), $$ where $J$ is the interaction variable which equals $\frac{1}{2n}$ for the mean field model, and $\sigma_i = (Cos \theta_i, Sin \theta_i)$. In theory at low temperature, the abundance of soft fluctuations results in non-existence of long-range order. In supercritical temperature regime the 2D XY model have short- range correlation due to propagation of vortices. Therefore vortices plays a main role in studying the phase transition in 2D XY model. There is a transition from high-temperature disordered phase with exponential correlation to low-temperature quasi-order phase, which is known as Kosterlitz-Thouless phase transition. The main goal of this project is to study the total spin behavior in different regions of inverse temperature (critical phase, subcritical phase and supercritical phase) using computer simulations. We will be interested in studying the large deviations principles for total spin with precise rate function. Some of the interesting simulation questions for this model are:
Depending on the time constraint and interest of the students, we may also study the numerical simulation results for the case where the external field is present. Some other directions include numerical comparison with other similar class of models namely the Ising model and the Heisenberg model.
Every closed curve on a surface determines (the homotopy class of) a homeomorphism (i.e. self-map) of that surface via the process of ``point pushing''. It is thus natural to wonder how various properties of the closed curve are reflected in the homeomorphism, and vice versa. The goal of this project is to tease out some of these relationships by running computer experiments. In the case of a punctured surface of genus $g$, closed curves (up to homotopy) can be easily and systematically encoded as finite words in $2g$ letters. Using this encoding, it is straightforward to algorithmically sample large sets of closed curves and calculate various quantities. On the ``topology of the curve'' side of the picture, we will primarily focus on the \emph{self-intersection number} of the curve (that is, the minimal number of self-crossings the curve has when drawn efficiently on the surface). There are effective algorithms to calculate self-intersection number in this setting, and certain things are known about it (e.g., its distribution is Gaussian). There are other interesting quantities to consider---such as hyperbolic length and other combinatorial measures of complexity--- and we will look for algorithms to calculate these as well. On the ``homeomorphism'' side of the picture, one expects that typical curves will produce \emph{pseudo-Anosovs}, which is just a fancy terminology for a surface self-map with a certain type of interesting dynamics. In this case the \emph{topological entropy} gives a good measure of the complexity of the homeomorphism. What's more, there is an efficient algorithm (implemented in Brinkmann's program {\tt xtrain}) to calculate the entropy. There are theoretical bounds relating selfintersection number to entropy for point-pushing homeomorphisms, but these computer experiments will hopefully reveal a more nuanced relationship and give empirical evidence for tighter bounds.
The mentor has shown that the four topics of the title have beautiful and fascinating interrelated realizations for a circle in the plane. The general definitions for each and their relations on a circle will be introduced and exemplified. The participants will try for extensions of the results to ellipses, where it is excepted that lovely new phenomena can be discovered using graphical methods.
The goal of this project is to visualize 3-dimensional hyperbolic space using the Oculus Rift, allowing the user to look around and walk through hyperbolic space as if they lived in it. One visualization of hyperbolic geometry "from the inside" was created in the 90's at the Geometry Center at the University of Minnesota as part of the video Not Knot (which can be found on YouTube). The aim of this project is to create a visualization similar to the one in this video, with two major differences: it will be a real-time animation controlled by the user, and it will use the Oculus Rift instead of an ordinary display.
Completion of Calculus 3, some programming experience. The challenges in this project are twofold: we have to understand the math of hyperbolic geometry, and there will be some serious computer graphics programming involved. The project will be particularly appropriate for people who have studied hyperbolic space before, or who have experience programming on multiple platforms, or who are familiar with the concepts of computer graphics.
Soil moisture is a major contributor to agricultural productivity. So far we have developed a computational model that deals with two of the fluxes of water at or near the soil surface: movement over the surface due to gravity and absorption into the soil surface. This model is written in Python and has been calibrated to (limited) experimental data. This project would deal with the next flux of water: water movement within the soil profile, especially heterogenous soils (can be thought of as movement of a liquid through porous medium).
PDEs, Numerical solutions/programming/computation, Python