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Here are potential topic ideas for an independent REU project. The professor who generated the idea is listed. If you are interested in a certain topic, contact the professor. Topics are by no means limited to those listed below. Algebraic GeometryFinding equations of Tautological rings (Daniele Arcara):Tautological rings belong to some deep apsects of algebraic geometry and topology and finding equations in tautological rings used to employ very difficult mathematics. Recently, it has been proposed that thewre might be an algorithm which can be used to discover these equations. This algorithm has been put on test in a few cases and has so far been successful. The goal of this project is to develop a computer program implementing this algorithm. Linear Algebra and Calculus are the only mathematics required. Mathematical computing skills (Mathematica/Maple/Matlab) are desireable. MaterialsTransport properties of sea ice (Kenneth M. Golden):Sea ice is a composite material of pure ice with brine, air, and salt inclusions. The polar sea ice packs are important in global climate, and they host algal and bacterial communities which support the rich polar food webs. The transport of fluid, heat, and light through sea ice controls a broad range of geophysical and biological processes. The composite microstructure of the sea ice in turn controls transport. There are many possible projects here for students to work on, which revolve around modeling the fluid, thermal, and electromagnetic transport properties of sea ice and how they depend on the composite microstructure. Students generally should have completed the calculus sequence, and should have some exposure to differential equations, linear algebra, probability, and basic physics. Some students have traveled to the Arctic for sea ice field work in connection with their research. Math BiologyTime Delays in Highway Traffic Models (Kim Montgomery):Delay differential equations are differential equations in which the rate of change of a variable depends not only on the current value of the variable but also on its value at a previous time (e.g. dx(t)/dt = k x(t-a)). Interestingly, the solution to a differential equation can change dramatically when time delays are added. For instance, while the solution to dx(t)/dt = k x(t) is always exponential, the solution to dx(t)/dt = k x(t-a) can be oscillatory. In modeling the dynamics of highway traffic, time delays due to driver reaction time can play an important role. One possible project is to analyze the effect of adding time delays to a simple model of traffic flow. Mathematical Modeling in Biology (David Goulet): In sciences such as physics and chemistry, there are many well defined physical laws which lead to broadly applicable systems of governing equations. Examples of such laws and equations include the Newton-Einstein gravitational laws, Maxwell's equations of electromagnetism, the Dirac-Heisenberg formulation of quantum mechanics, and the Navier-Stokes description of fluid dynamics. These equations have historically served as the starting points and springboards for building models of complex physical and chemical phenomena. The field of biology is somewhat less fortunate. No such broadly applicable mathematical laws currently exist. A mathematical biologist, without mathematical laws to build from, is often required to draw upon the techniques of the other sciences and to develop their own techniques. This summer project will focus on mathematical modeling in biology. The student can choose any biological system of interest. Our goal to will be to develop several mathematical models of the phenomena and to solve and analyze these models. We will compare the strengths and weaknesses of our models and together try to decide what would constitute a "good" model of our biological system and decide whether we have succeeded in creating a "good" model. The minimum requirements for participation are: The completion of Calculus 2 or some other course which introduces differential and difference equations. The completion of a college level or AP course in biology, ecology, sociology, psychology, or physiology. Some knowledge of using the software package Matlab is also preferred, but not required. Math FinanceBlack-Scholes theory of option pricing (Jingyi Zhu):In an ingenious use of stochastic calculus and hedging principles, Black, Scholes and Merton started the theory of option pricing which paved the way for the wide spread use of financial derivatives. We will investigate certain aspects of the theory and applications in areas such as mortgage and municipal bonds. Mathematical modeling in credit risk (Jingyi Zhu): The goal of research in this area is to quantify the risk faced by investors in government and corporate liabilities. Intuitive but also sophisticated mathematical theories are developed to obtain quantitative information from the market, and then the development of credit derivatives helps investors to hedge against default risks. Numerical AnalysisMultilevel evaluation of radial basis function expansions (Grady Wright):A common problem that arises in many science and engineering disciplines is the approximation of a function f from a discrete set of samples. Radial basis functions (RBFs) are a particularly simple and powerful technique for solving this problem, especially when the samples are scattered in two and higher dimensions. Applications of this relatively new method include cartography, neural networks, geophysics, pattern recognition, graphics and imagining, and the numerical solution of partial differential equations. RBFs are, however, not without computational challenges. For example, the efficient evaluation of an n-center RBF expansion at m points. A direct summation requires O(nm) operations. Recently, a new multilevel method has been developed by O.E. Livne and G.B. Wright for reducing this cost to O(n + m) operations when the radial kernel is smooth. The algorithm has can be also applied beyond RBFs, to discrete integral transform evaluation, Gaussian filtering and de-blurring of images, and particle force summation. Some potential projects relating to this research include:
TopologyRigidity of Affine Patterns, Christopher CashenPrereqs: Linear Algebra, Euclidean Geometry Let V be a linear subspace of R^n. Let P_V, the affine pattern induced by V, be the collection of affine subspaces parallel to V. Let F={V_1,V_2,..} be a finite collection of subspaces, and let P_F be the collection of the P_V_i's. A map from R^n to R^n preserves P_F if for each i it takes P_V_i to P_V_i in an appropriate sense. A map from R^n to R^n is a quasi-isometry if it preserves distances up to bounded multiplicative and additive errors. Every pattern has some quasi-isometries which preserve it, we can just rescale R^n by some factor. We call a pattern "rigid" if these are the only quasi-isometries preserving it. Which patterns are rigid? We could probably enumerate the cases for small dimensions, but it would be better to come up with some rigidity criteria. A related question would be to describe the quasi-isometry groups preserving non-rigid patterns. For example, in R^2 let F consist of the x-axis and the y-axis. P_F is then the collection of horizontal and vertical lines in the plane. This is not a rigid pattern. We could rescale the horizontal direction by one factor and the vertical direction by a different factor, and this will take vertical lines to vertical lines and horizontal lines to horizontal lines. In fact we can do more than rescale the two directions, we could take any two quasi-isometries of the real line and let them act on the two coordinates, so the group of quasi-isometries preserving the pattern is a product of two copies of the quasi-isometries of the real line. On the other hand, if F has at least three distinct lines, then with some work we can see that the pattern is rigid. |
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