Algebraic statistics advocates polynomial algebra as a tool for addressing problems in statistics and its applications. This connection is based on the fact that most statistical models are defined either parametrically or implicitly via polynomial equations. The idea is summarized by the phrase "Statistical models are semialgebraic sets". I will try to illustrate this idea with two examples, the first coming from the analysis of contingency tables, and the second arising in computational...

Mathematical concepts are often difficult for students to acquire. This difficulty is evidenced by failure of knowledge to transfer from the learning situation to a novel isomorphic situation. What choice of instantiation most effectively facilitates successful transfer? One possibility is that grounding the concept through a concrete, contextualized instantiation may facilitate learning and in turn facilitate transfer. On the other hand, several cognitive factors influence the process of...

This presentation does not require previous knowledge of C*-algebras, labeled graphs, or group actions. A labeled graph over an alphabet consists of a directed graph together with a labeling map . One can associate a C*-algebra to a labeled graph in such a way that if the labeling is trivial then the resulting C*-algebra is the C*-algebra of the graph . In this presentation, I will discuss joint work with Teresa Bates and David Pask concerning (discrete) group actions on labeled graphs...

Current community models in the geosciences employ a variety of numerical methods from finite-difference, finite-volume, finite- or spectral elements, to pseudospectral methods. All have specialized strengths but also serious weaknesses. The first three methods are generally considered low-order and can involve high algorithmic complexity (as in triangular elements or unstructured meshes). Global spectral methods do not practically allow for local mesh refinement and often involve cumbersome...

I am an active mathematical physicist who has also engaged long-term with mathematics education; particularly, with research on mathematical learning and problem solving. This led, perhaps inevitably, to a focus not only on cognition, but also on the psychology of what is called “the affective domain” – i.e., emotional feelings, attitudes, beliefs, and values – in relation to mathematics. In this talk, I shall discuss some important affective constructs which relate directly to mathematical...

Approximating functions or data by polynomials is an everyday tool, starting with Taylor series. Approximating by rational functions can be much more powerful, but also much more troublesome. In different contexts rational approximations may fail to exist, fail to be unique, or depend discontinuously on the data. Some approximations show forests of seemingly meaningless pole-zero pairs or "Froissart doublets", and when these artifacts should not be there in theory, they often appear in...

We propose a new design for phase II oncology clinical trials based on two considerations. (1): Currently most phase II oncology trials use complete remission (CR) as the primary end point. The drugs having higher CR rates enter into subsequent phase III trials, which are usually required to demonstrate benefit on survival. Although achieving CR is necessary for prolonging survival, it is not sufficient because patients may relapse shortly after achieving CR. This discrepancy was one of the...

Hosted by Professor Kyeong Hah Roh Abstract Much of what we say and write in our mathematics classes assumes that our students understand linguistic and logical conventions that have never been made explicit to them. What problems result from this assumption, and how can we address them? Biography Susanna S. Epp (Ph.D., University of Chicago, 1968) is Vincent de Paul Professor of Mathematical Sciences at DePaul University. After initial research in commutative algebra, she became interested...

The School of Mathematical and Statistical Sciences presents Dr. Erica Flapan, Lingurn H. Burkhead Professor, Department of Mathematics, Pomona College presenting on the topic, Topological Symmetries of Molecules.

We will describe what a genus one curve is and then specialize to the case of elliptic curves. After discussing a simple looking problem through which elliptic curves become objects that we want to understand, we will summarize some known results about elliptic curves. To conclude, we will go back to the general genus one curve over the rationals and see how close this object is to being an elliptic curve. Mirela Çiperiani is an Assistant Professor in the Department of Mathematics at the...

In algebraic topology, we learn to associate groups Hn(T) to locally compact spaces which “count the n-dimensional holes in T". In this talk, I want to describe how to realize H3(T) as a set Br(T) of equivalence classes of certain well-behaved C* -algebras. The group structure imposed on Br(T) via its identification with H3(T) is very natural in its C* -setting. With this group structure, Br(T) is called the Brauer groupof T. Depending on your point of view, this result can be viewed either...

The increase of entropy was regarded as perhaps the most perfect and unassailable law in physics and it was even supposed to have philosophical import. Einstein, like most physicists of his time, regarded the second law of thermodynamics as one of the major achievements of the field, and it entered his work in several ways. The essence of the second law is the statement that all processes can be quantified by an entropy function whose increase is a necessary and sufficient condition for a...

I will focus on two issues of spatial population dynamics. The first will be on the dynamics and spread of populations in space, which is joint work with Brett Melbourne that begins with experimental work with the flour beetle Tribolium which is then coupled with analyses of stochastic population models incorporating different sources of variability to understand highly variable spread rates. The second part of the talk will cover questions related to control of spread of invasive species.

We propose a general framework to design asymptotic preserving schemes for the Boltzmann kinetic kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the...

Starting point of my talk are approximation algorithms for NP-hard problems in combinatorial optimization which are based on semidefinite programming (SDP), a recent and powerful method in convex optimization. One example is the theta number of Lovasz which provides an upper bound for the largest size of an independent set of finite graphs based on a solution of a semidefinite program. Many problems in extremal discrete geometry can be formulated in this way but for infinite geometric...

The hypothesis that the structure of high Reynolds number turbulence conists of thin shear layers, with thickness of the order of the Taylor micro scale, has been further confirmed by numerical studies by Ishihara and Kaneda of conditional statistics and local dynamics; by PIV measurements of lab experiments by Wirth and Nickels, and by further developments of the theory, especially the transport of energy into the layers leading to the generation of intense structures, on the scale ofthe...

“It is impossible to trisect an arbitrary angle.” We (the mathematical community) have been certain of this for the past 170 years. Missing in that statement is the qualifying phrase“... using a straightedge and compass.” But if we are clumsy enough to scratch our straightedge in two places, we can in fact trisect an arbitrary angle, a result that was known to Archimedes. We are equally confident in the much more modern assertion: “There is no algorithm to solve an arbitrary quintic,” where...

Fluid flows in the presence of free surfaces occur in a great many situations in nature; examples include waves on the ocean and the flow of groundwater. In this talk, I will discuss my contributions to the understanding of the systems of nonlinear partial differential equations which model such phenomena. The most important step in these results is making a suitable formulation of the problem. Influenced by the computational work of Hou, Lowengrub, and Shelley, we formulate the problems in...

This lecture is part of the Fall 2010 Seminar series and was recorded on September 3, 2010 in Physical Science Center A Wing, Room 107,