In statistical physics, systems like percolation and Ising models are of particular interest at their critical points. Critical systems have long-range correlations that typically decay like inverse powers. Their continuum scaling limits, in which the lattice spacing shrinks to zero, are believed to have universal dimension-dependent properties. In recent years critical two-dimensional scaling limits have been studied by Schramm, Lawler, Werner, Smirnov and others with a focus on the boundaries of large clusters. In the scaling limit these can be described by Schramm-Loewner Evolution (SLE) curves.

In this talk, I'll discuss a different but related approach, which focuses on cluster area measures. In the case of the two-dimensional Ising model, this leads to a representation of the continuum Ising magnetization field in terms of sums of certain measure ensembles with random signs. This is based on joint work with F. Camia (PNAS 106 (2009) 5457-5463) and on work in progress with F. Camia and C. Garban.

Alexandrov geometry reflects the geometry of Riemannian manifolds when stripped from everything but their structure as metric spaces with a (local) lower curvature bound. In this talk I will define Alexandrov spaces and discuss basic properties, constructions and examples. By now there are numerous applications of Alexandrov geometry, including Perelman's spectacular solution of the geometrization conjecture for 3-manifolds.

The utility of Alexandrov geometry to Riemannian geometry is due to a large extend by the fact that there are several geometrically natural constructions that are closed in Alexandrov geometry but not in Riemannian geometry. These include, but are not limited to (1) Taking Gromov-Hausdorff limits, (2) Taking quotients, and (3) forming cones, jones etc of positively curved spaces. In the talk I will give examples of applications of each of these and one additional new construction.

In the late 1980's I began my efforts to increase the success rate of minorities in first semester calculus. The interventions that I devised were very time consuming and as the number of minority students increased, I could not manage that kind of effort. I developed my Calculus Minority Advising Program in an effort to meet with scores of minority students each semester. This program consists of a twenty-minute meeting with each student at the beginning of each semester. These meetings with students eventually transformed my own attitude about the importance of mathematics in their undergraduate curriculum.

I took over the position of Associate Head for Undergraduate Affairs in the department in 2003. I set a very modest goal for myself: to double the number of mathematics majors. With almost 600 mathematics majors I have reached that goal. I think the next doubling is going to be much harder to achieve. My work with minority students provided me with the tools to accept this new challenge of working with all students.

This talk will describe my own efforts to encourage ALL of our students that a mathematics major, or adding mathematics as a second major, is a great career choice. I will also describe the support that I have from the university and the department that enables me to carry out these tasks.

Theory of knowledge and learning spaces is used to assess readiness and de-
termine course placement for mathematics students at or below introductory calculus at
the University of Illinois. Readiness assessment is determined by the articially intelligent
system ALEKS. The ALEKS-based mechanism used at the University of Illinois eectively
reduces overplacement and is more eective than the previously used ACT-based mech-
anism. Signicant enrollment distribution changes occured as a result of the mechanism
implementation. ALEKS assessments provide more specic skill information than the ACT.
Correlations of ALEKS subscores with student maturity and performance meets explecta-
tions in many cases, and revels interesting characteristics of the student population in other
(systematic weakness in exponentials and logarithms). ALEKS revels skill bimodality in the
population not captured by the previous placement mechanism.
The data shows that preparation, as measured by ALEKS, correlates positively with
course performance, and more strongly than the ACT in general. The trending indicates
that while a student may pass a course with a lower percentage of prerequisite concepts
known, students receiving grades of A or B generally show greater preparedness. Longi-
tudinal comparison of students taking Precalculus shows that ALEKS assessments are an
eective measure of knowledge increase. Calculus students with weaker skills can be brought
to the skill level of their peers, as measured by ALEKS, by taking a preparatory course.
Interestingly, the data provided by ALEKS provides a measure of course eectiveness when
students preformance is aggregated and tracked longitudinally. The data is also used to
measure course eectiveness and visualize the aggregate skills of student populations.

Quantum resonances describe metastable states created by phenomena such as tunnelling, radiation, or trapping of classical orbits. Mathematically they are elegantly defined as poles of meromorphically continued operators such as the resolvent or the scattering matrix: the real part of the pole gives the rest energy or frequency, and the imaginary part, the rate of decay. With that interpretation they appear in expansions of linear and non-linear waves. And they can be found in other branches of mathematics and science: as poles of Eisenstein series and zeta functions in geometric analysis, scattering poles in acoustical and electromagnetic scattering, Ruelle resonances in dynamical systems, and quasinormal modes in the theory of black holes. In my talk I will present some basic concepts and illustrate recent mathematical advances with numerical and experimental examples.