Hongkun Zhang, Assistant Professor in the Department of Mathematics and Statistics at the University of Massachusetts Amherst, has received a Faculty Early Career Development (CAREER) Award from the National Science Foundation. Her proposal, titled “CAREER: The Nature of SRB Measures for Nonequilibrium Hyperbolic Systems,” was funded for a five-year award in the amount of $400,001 with an effective date of August 1, 2012.

The theory of billiards has recently been developed to the point where it provides mathematical tools to address open problems in physics. Hongkun’s long-term research goal is to use the new tools developed in the theory of billiards to solve challenging problems arising in equilibrium and nonequilibrium statistical mechanics. In pursuing this goal, her focus is on the nature of Sinai-Ruelle-Bowen (SRB) measures and their modifications for nonequilibrium phenomena, on the statistics of relevant observations and on significant changes in dynamical landscapes using innovative approaches.

Many sophisticated techniques have been developed and beautiful results obtained for equilibrium systems. However, the research topics of random and nonequilibrium systems have remained wide open as chaotic phenomena in these settings have a different flavor. Recently developed mathematical tools in chaotic billiards offer hope for significant progress. The PI seeks both theoretical understanding and new ways to connect mathematical ideas to a variety of complex phenomena by addressing the following two challenging questions, using spectral analysis, coupling methods, and other innovative approaches.

1. Statistical properties for nonequilibrium billiards. The goal is to prove the existence of SRB measures that characterize the steady states, to study the time correlation functions that relate to the diffusion matrices in the transport processes, and to verify other limit theorems for perturbed billiard systems. Specific models include Lorentz gases under general forces, ideal gases with slow-moving scatterers, and random billiards with microstructure.
2. Properties of SRB measures and related physical laws. The goal is to understand the nature of SRB measures including their Kolmogorov-Sinai entropy and Hausdorff dimensions and to derive obtain Ohm's law, the Einstein relation, and other physical laws for certain nonequilibrium billiards originating in physics.

The proposed research will use mathematical tools to address applied problems in physics, chemical engineering, and other sciences. The theory of billiard systems has provided excellent models or paradigms for understanding chaos and diffusion processes in nonequilibrium systems. Because the study of SRB measures and their asymptotic properties provides new insight into the nature of nonequilibrium steady states and transport phenomena, the projected research will greatly contribute to modern statistical physics. Thus this project will have a broad impact outside of mathematics in the physical sciences and in other scientific communities. The proposed education plan contributes to and draws upon the research projects, whose impact will be broadened by presenting simulations of chaotic systems. The PI will be active in mentoring VAPs and graduate students. The proposed research will be integrated into both graduate and undergraduate research projects as well as into curriculum development, including a new topics course for graduate students on Stochastic Differential Equations and a new undergraduate course that will focus on chaos and fractals.

Professor Farshid Hajir is a recipient of the university’s 2011 Distinguished Teaching Award (DTA), the highest honor given on campus for excellence in teaching. One of three tenure-system faculty winners this year, Farshid joins the ranks of extraordinary faculty who have received the award since it was instituted in 1962. Those ranks include from our department the late Clair Naylor, who won the award in 1973.

At an April 29 Faculty Honors dinner, Farshid was presented with a plaque acknowledging the honor. The award carries a monetary prize and recognition at the May Undergraduate and Graduate Commencement ceremonies. And Farshid’s name will be added to the list of previous winners that is inscribed on a permanent display on the concourse wall of the Murray D. Lincoln Campus Center.

Students play a key role in selecting DTA recipients: only current students and alumni may make nominations. A committee of undergraduate and graduate students, joined by faculty, review teaching evaluations and letters of recommendation in order to select the winners.

Farshid is passionate about his teaching, and his students enthusiastically appreciate his efforts. As we reported in the 2006 issue of this
ewsletter, when Farshid won the Outstanding Teaching Award from the College of Natural Sciences and Mathematics (as our college was then known), students described him as “awesome,” “great,” “fabulous,” and “amazing.” No wonder that Farshid continues to garner some of the very highest course evaluation ratings in our department, across a wide range of courses large and small, from beginning undergraduate to advanced graduate.

Asked about what most influenced his teaching, aside from the role models provided by his own teachers, Farshid particularly cites the constructive criticism he received from now-Emeritus Professor Arthur Mattuck of MIT, who supervised beginning teaching assistants. Mattuck, he says, had a good eye for detecting what a novice teacher was doing wrong and challenged him at times to hold back from sharing his own view of the material so as to allow students to come to their own understanding. Farshid, in turn, has helped initiate our department’s own new TAs by presenting a hilarious model minilecture in which he deliberately commits the typical mistakes a beginner might make!

In the 2006
Newsletter article, we described how Farshid makes his classroom such a comfortable place to learn by various techniques, which include sharing what he refers to as his “goofy sense of humor.” Asked how his approach to teaching has evolved since then, Farshid believes he pays even more attention now to mediating the sometimes competing demands of syllabus content and student learning — “not merely covering, but helping to uncover the material,” as he puts it. Moreover, he thinks he now more carefully tries to diagnose what students do not understand by listening, for example, to conversations among them in class as they work in groups to solve a problem.

For some years Farshid has been involved with K-12 math education, working with groups in the School of Education at UMass Amherst and, more recently, being engaged in public activities. At present he chairs the School Committee of the Town of Leverett, where he lives. In 2009–2010 he also chaired the Amherst-Pelham Regional School Committee, which oversees the secondary schools for the towns of Amherst, Pelham, Leverett, and Shutesbury. In that position, he had to contend with budgetary and administrative personnel crises that beset the schools.

Another manifestation of Farshid’s strong interest in school mathematics education has been his past teaching of our standard two-semester sequence for pre-service K-8 teachers. In recent years, mathematicians and educators have come to realize not only that two semesters are insufficient, but also that the traditional mathematical content for teacher preparation does not align well with what teachers actually need to understand about mathematics. Accordingly, Farshid has become involved in a multi-campus project to revamp the mathematical preparation of K-8 teachers. The past semester he taught Math 597T, “Mathematics Knowledge for Teaching I,” off-campus to a group of inservice teachers, including several math coaches from local schools.

Math 597T is a pilot run of the first course of a proposed three-semester sequence, the sort of preparation that Massachusetts wants all prospective K-8 teachers to have. While the teacher-preparation curriculum for Massachusetts is being developed by mathematicians and educators at the Amherst, Boston, Dartmouth, and Lowell UMass campuses, the purpose of the particular pilot course here, Farshid explains, is to see how to teach the material in a more interactive way.

The target school curriculum for which Farshid and his colleagues elsewhere want to better prepare teachers is based upon the Common Core State Standards Initiative, now endorsed by more than forty states including Massachusetts. Elsewhere in this
ewsletter, Farshid himself writes about the Common Core K-12 math subject matter— the “Standards for Mathematical Content” — and the accompanying ways that learners of mathematics should engage with the subject matter — the “Standards for Mathematical Practice.”

One reason that Farshid regards the teacher preparation effort in which he is involved to be so important, and greater funding for it to be so essential, is that “today’s kids will be tomorrow’s university students.” If students don’t learn math well enough in school, from teachers who have a sufficiently broad and deep understanding of math themselves, then those students will be unable to cope later with college-level math, science, and engineering work.

While Farshid’s mathematical knowledge and broad teaching experience at the university level have informed how he prepares K-8 teachers, his interactions with those teachers has, reciprocally, affected his approach to other courses. K-8 teachers, he explains, routinely expect to do group work in class and not just receive “direct instruction,” and they are accustomed to differentiating instruction according to their students’ abilities and needs. Farshid tries to does similar things in our math major offerings, for example, in Math 300, the introduction to rigor and proof required of all math majors. In that course he incorporates in-class group work and tries to keep all students engaged, among other ways, by posing optional, more challenging problems for them to solve.

During the past semester Farshid also directed an advanced Commonwealth Honors College seminar titled “Epistlary Vignettes in Mathematics.” In this seminar, students read the correspondence among 17th-century through 19th-century mathematicians, using the content and form of the letters to gain insight into the mathematical culture and wider intellectual milieu of those eras. As an example, Farshid offered the case of Fermat and Pascal engaged in analyzing the probabilities involved in “dividing the stakes” from an aborted gambling game; in effect, they were trying to predict the future — a radical thing to do. Such correspondence provides a rare opportunity for students to see mathematics “in the raw” as it was being developed. Farshid reports that students were amazed at how quickly, in those pre-electronic times, letters were delivered, read, and answered; were intrigued by how ostensibly polite yet sometimes contentious the exchanges could be; and were surprised at how things so obvious to us today often gave mathematicians such trouble then.

Teaching that undergraduate honors seminar has even impinged upon the graduate complex analysis course that Farshid taught during the spring semester of 2011. From his readings in Cauchy’s letters, he was able to relate to the graduate students Cauchy’s love for the idea of contour integration.

We congratulate Farshid Hajir upon the Distinguished Teaching Award, the latest recognition of his excellence in teaching and his significant broader contributions to mathematical education.

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