In 1932 von Neumann proposed classifying the statistical behavior of physical systems. The idea was to take a diffeomorphism of a compact manifold and describe what one might observe as random (as in coin flipping) or predictable (as in a translation on a compact group), or even better have a dictionary in which one could look up the precise behavior.
Remarkable progress was made on this problem; benchmarks include the Halmos-von Neumann theorem on discrete spectrum and the work of Kolmogorov on Entropy that culminated in the Ornstein classification of Bernoulli shifts. One genre of applications of this theory were the results of Furstenberg on Szemeredi’s theorem and eventually the work of Green and Tao.
Still the problem resisted a complete solution. Strange examples of completely determined systems that showed completely random statistical behavior began to surface. Starting in the 1990’s anti-classification theorems began to appear. These results showed, in a rigorous way, that complete invariants for measure preserving systems cannot exist. Moreover the isomorphism relation itself is completely intractable. Very recently these results were extended to measure preserving diffeomorphisms of the 2-torus.
What is the maximum number of rational points on a curve of genus g over a finite field of size q? What is the distribution of rational point counts for degree d plane curves over a fixed finite field? We discuss these and several related questions and show how to use curves over finite fields to construct interesting error-correcting codes.
This week's graduate seminar will feature a question-and-answer session with experienced TAs. Come prepared with your own questions, and try to make them as specific as possible. For example, "How do you teach effectively?" vs "How do you encourage participation?" vs "Have you ever had a class where students will not speak, no matter how hard you try?" Do you agree that the last one is so much easier to answer?
Special room! MSTB 226! We'll show you some of our favorite ways to use technology in the classroom. In some classes it's more natural than others, but even for a class like Abstract Algebra, there are lots of possibilities! We might talk about Graphmatica, Doceri, Screencastomatic, Canvas, Wolfram Demonstrations Project. We are always looking to learn about new resources, so please let us know about any favorites you have (or resources you've heard about but never tried).
Do you have fond memories of your favorite college professor? What made their teaching memorable? What do you want your students to remember about your own way of teaching? In this talk, we will share inspirations for good teaching in mathematics.
This is the second meeting of our sequence of teaching seminars. We will continue to illustrate (and practice) best strategies to bring active learning into our classrooms (especially in the calculus discussion sessions). In addition, we will focus on understanding and preventing academic dishonesty with the help of a special guest, Don Williams, Director of PS Student Affairs.
Have you ever regretted asking a certain quiz or exam question as soon as you started grading it? Maybe the question was too easy and everyone got it right? Maybe nobody got it? Maybe the question itself was fine but grading it was super painful? We'll talk about some tips for writing good quiz and exam questions, and then continue with the overall seminar theme of active learning.
This is the first of a sequence of six seminars dedicated to teaching. All grad students are welcome; participation from 1st and 2nd year grad students is required!
One of the main themes of the graduate seminar this quarter will be active learning, especially with reference to Math 2A and Math 2B. Adrienne Williams, the new campus Teaching and Digital Strategy Consultant <http://sites.uci.edu/awilliams/welcome/> in the UCI Division of Teaching and Learning, will talk to us about the benefits of active learning.
If you are teaching 2A/2B, you are especially encouraged to attend! You will walk out with concrete suggestions for active learning activities that can be implemented in Math 2A and Math 2B discussions as early as Week 1!
