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Department of Mathematics Newsletter { Winter 2011 {MMathath NewsNews A publication of the Department of Mathematics at the University of Nebraska–Lincoln VIEW FROM THE Curto wins noted Sloan Research Fellowship CHAIR arina Curto, assistant profes- tion, which Csor of mathematics, has been announced its John Meakin selected for a Sloan Research Fel- newest recipi- lowship for her research in the fi eld ents on Feb. 15, he year 2011 of mathematical neuroscience. This 2011, awards Thas been one two-year fellowship awards Curto 118 Sloan of opportuni- $50,000 to put toward her research. Research Fel- ties, challenges “I was thrilled to receive the lowships each and successes for news,” Curto said. “This award will year, bringing the Department benefi t my research signifi cantly, es- total grants in of Mathematics pecially because of its fl exible nature. the program to $5.9 million annu- and indeed for I greatly appreciate all those who ally. The fellowships seek to stimulate the University of supported me in my nomination, as fundamental research by early-career Nebraska-Lin- well as my close collaborators.” scientists and scholars of outstand- coln. In what seems destined to be one The Alfred P. Sloan Founda- CURTO of the most signifi cant events in the See on Page 11 history of the university, UNL became one of the 12 members of the Big Ten (as opposed to one of the 10 remain- ing members of the Big 12). A tribute to Meakin’s leadership This move is much more than a As this edition’s View from the Chair addresses, John Meakin, after eight years of change in athletic conference for the serving as Department Chair, will be handing over the reins as chair to Judy Walker. university. UNL joins an elite group of In honor of his dedicated service, we wish to highlight some of the contributions institutions with a seat on the Com- Meakin has made to the department and thank him for a job well done. mittee on Institutional Cooperation, the academic arm of the Big Ten. uch like the mark of a good “Administration is a huge job. John Our university and our department Mpilot is an uneventful fl ight, the takes care of that, and it insulates the enjoy a new set of colleagues, oppor- mark of a good department chair is rest of us. While the work is a burden tunities, and expectations. There is a when faculty members remain oblivi- on the chair, done effectively it helps great deal of energy, optimism and ous to the work that it entails. The the entire department.” confi dence on campus, and expecta- most important aspect of the job is to As a result, much of the work is tions of growth and additional success ensure that faculty members feel en- behind the scenes, making Meakin’s throughout the university are evi- couraged and well supported, enabling hard-working and selfl ess nature dent. Our department and indeed the them to perform at their very best. among his greatest attributes as chair. CHAIR Professor of Mathematics David This became apparent almost im- See on Page 3 Pitts, who came to UNL in 1986, said, See MEAKIN on Page 3 { { INSIDE: DEPARTMENT NEWS: Progress FACULTY NEWS: Grant provides ALUMNI NEWS: Rolle fi nds in math education Page 4 math scholarships Page 9 math gives edge in law Page 12 www.math.unl.edu/friends Winter 2011 1 {DDepartmentepartment NewsNews Leavitt’s work builds foundation for exciting new area of research he Winter 2010 Math News In 1962, UNL Professor Tcontained an article describing William G. Leavitt wrote the outstanding contributions and an article for the presti- accomplishments of UNL’s Commuta- gious journal Transac- tive Algebra group. But don’t get the tions of the American wrong impression –– UNL’s non- Mathematical Society, Commutative Algebra community is in which he showed that having quite a worldwide impact too. there is a plethora of Here’s some mathematical back- rings which do not have ground. If you think back to your Intro the “Invariant Basis to Modern Algebra Math 310 course, Number” property. In the most likely all of the rings you saw as past seven years, his work basic examples have what’s known as has received much atten- the “Invariant Basis Number” prop- tion, and these algebras erty. This means that if you have a are now called Leavitt fi nitely generated free module M over path algebras in honor of R (the moral equivalent of a fi nite his foundational work in dimensional vector space over R), then the area. any two bases for M must have the COURTESY PHOTO same number of elements. Thinking back even further, you probably saw a i j R-modules RR and RR are not iso- the 1960s. proof of this fact in your introduction morphic; on the other hand, for each These newfangled algebras are to linear algebra course, in the case positive integer the free left called Leavitt path algebras in honor that R is the fi eld of real numbers. i j modules RR and RR are isomorphic of Bill Leavitt’s foundational work in More formally, the Invariant Basis precisely when (mod n – m). the area. (Yes, the L in L (E) stands Number property for a ring R means K Less formally, what Professor for ‘Leavitt’.) Many mathematicians that for every pair of positive integers throughout the world have been m Leavitt’s Theorem shows is that, in the m and n, if the free left R-modules RR context of isomorphisms between free focused on these LK(E) algebras for and Rn are isomorphic, then m = n. It R modules, anything that can happen, the past seven years. Interest in them is not too hard to show that not only does happen. comes not only from the noncom- does this IBN property hold for fi elds, Over the next three-plus decades mutative algebra community, but from it actually holds for all commutative this interesting work received some- the operator algebra community as rings, all (fi nite-sized) matrix rings what modest attention in the math- well, because there is an intimate con- over fi elds, and most, if not all, of the ematics community. But in the past nection between Leavitt path algebras rings you saw in Math 310. seven years, the attention level sur- with coeffi cients in the complex num- In 1962, UNL Professor William rounding these algebras has risen from bers and analytic structures G. Leavitt wrote an article for the modest to intense. Here’s why: In 2004 called prestigious journal Transactions of the a handful of algebraists came up with a During April 2011 a pair of talks American Mathematical Society, in construction which associates with any were delivered at UNL in which some which he showed that there is a pletho- fi nite directed graph E and fi eld K a K- of the current happenings in the fi eld ra of rings which do NOT have IBN. algebra, denoted L (E), whose defi nition of Leavitt path algebras were discussed. Specifi cally, Bill proved this: K is based on the confi guration of vertices How nice it was that Bill Leavitt, at age Theorem. For any pair of posi- and edges in E. As it turns out, if you 95 and still going strong, could attend. tive integers m, n with m < n and any Professor Emeritus Leavitt’s start with the directed graph Rn having fi eld K there exists a K-algebra LK(m, one vertex and loops based impact on mathematics research will n) with the following properties: for endure well into the future. at that vertex, then LK(Rn) is precisely each positive integer i < m, and each –– Gene Abrams, University of the algebra LK(1,n) that Bill Leavitt positive integer , the free left constructed and investigated back in Colorado-Colorado Springs 2 Winter 2011 www.math.unl.edu/friends DDepartmentepartment NNewsews { as UNL’s fi rst recipient of a Gates portunities to appoint new faculty in CHAIR From Page 1 Scholarship to Cambridge University. the department are expected for the whole university has been engaged In April we hosted the KUMUNU coming year, however, as the univer- in an examination of all aspects of commutative algebra conference and sity’s budget situation improves. This its mission, from curricular issues the Rowlee Lecture by David Eisen- will be needed if the department is to research expectations to the role bud, and in October we were pleased to maintain the momentum it has that we all play in society and in our to host two more major conferences, established in the past several years, discipline. the Fall Central Sectional Meeting of and if it is to take advantage of the The department is also in the the American Mathematical Society, opportunities afforded by its rising midst of a change in leadership. including the 2011 Erdos Lecture by profi le and the university’s move into The current semester (Fall 2011) Emmanuel Candes, and the Enacting the Big Ten. will be my last semester as depart- Standards for Mathematical Practices It has been a real privilege to ment chair: Professor Judy Walker Conference in mathematics education serve as department chair, working will assume this position effective honoring the many contributions of with a talented group of faculty, staff January 1, 2012. Jim Lewis to the profession. These and students, and I am grateful for Judy has had a events are featured in this edition of the support of the administration and huge impact on Math News.
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