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T.W.I.G.S.
The "What Is ...?" Graduate Seminar

Room 1634, Lederle Graduate Research Tower
University of Massachusetts, Amherst 		ComplexCubicUnitAction
The Seminar meets Wed 3:00-4:00 in 1634 LGRT.    
Driving Directions   and   Campus Maps
Overview of the Seminar as well as Unofficial mottos and sponsors of TWIGS
Maintained by Farshid Hajir.

NOTE Here are the schedules for Fall 2002, Spring 2003, Fall 2003, Spring 2004, Fall 2004, Spring 2005, Fall 2005, and Spring 2006.
Wednesday September 20, 2006 Farshid Hajir, UMass Amherst. "What is Bertrand's Postulate?"
Abstract: This is one of those results whose statement is so fabulous, it continues to bear the name of its hypothecater (yes, that's a real word!). It states that for every positive integer n, the interval (n,2n] contains a prime number. To check that it's true for all n up to 4000, it's enough to note that 2,3,5,7,13,23,43,83,163,317,631,1259,2503,4001 are all prime; why is this enough? Joseph Bertrand checked it for all n up to 3 million. It was first demonstrated by Pafnuty Chebyshev in 1850. There are many proofs: I'll present one of them, and time permitting, discuss generalizations and further conjectures about distributions of primes.
Wednesday September 27, Farshid Hajir , UMass Amherst. "What is Turan's Graph Theorem?"
Abstract: First I'll remind everyone what a "graph" is and then give several proofs of a theorem of Turan about how many edges a p-clique-free graph on n vertices can have.
Wednesday October 4, Tom Weston , UMass Amherst. "Why is exp(Pi*sqrt(163)) so close to an integer?"
Abstract: exp(Pi*sqrt(163)) is *really* close to being an integer, even though it is a transcendental number. So what? There are lot of transcendal numbers that are really close to integers. Ah, but exp(Pi*sqrt(163)) + 744 is really close to 640320^3! Not impressed yet? To find out what any of this has to do with elliptic curves, modular forms, the class number of binary quadratic forms, .... come to Tom's talk.
Wednesday October 11, Farshid Hajir , UMass Amherst. "What is a binary quadratic form?"
Abstract: A binary quadratic form over a ring R is just a map from R^2 to R given by (x,y) goes to ax^2+bxy+cy^2 with a,b,c in R. There is a natural notion of equivalence of bqf, under which the discriminant D=b^2-4ac remains unchanged. The natural question then is: How many equivalence classes of bqfs of fixed discriminant D are there? For R=Z, this number h(D), called "the class number" has been studied for hundreds of years. I'll give you some theorems and conjectures dating from 1801.
We interrupt this schedule to bring you the latest news. TWIGS is being hi-jacked for the next six weeks by a speaker who wishes to inflict upon the TWIGS audience his own musings on five types of objects that have nothing to do with each other, which will then be followed by a sixth talk on a kind of object that has nothing to do with the previous five objects either. The TWIGS authorities apologize for our inability to block this brazen individual from usurping the seminar in this cavalier fashion. -- The Management
Wednesday October 18, Farshid Hajir , UMass Amherst. "What is a tower of number fields?"
Abstract: A number field is a field K containing Q as a subfield and having finite dimension as a Q-vector space. This dimension is called its degree. Another important invariant of K is its discriminant. A major problem in number theory is to understand how the discriminant grows with the degree. I'll give an overview of what is known and conjectured. I'll describe a construction of towers of number fields that are "good" in the sense that they have slowly growing discriminants.
Wednesday October 25, Farshid Hajir , UMass Amherst. "What is a good linear code?"
Abstract: Let F be a finite field, with q elements. An [n,k,d] linear code C over F is a k-dimensional subspace of F^n such that any two distinct elements of C differ in at least d-1 positions but such that there is a pair of elements of C that differ in exactly d positions. We us codes (in cell phones, CDs, hard drives, ...) to transmit information reliably and efficiently. Reliability comes from having a large d/n; efficiency comes from having a large k/n. What is the "best" linear code possible, assuming very large n? I'll describe what is known and conjectured. The audience will be able to see for itself that linear codes do not resemble in any way any of the following objects: number fields, lattices, function fields of curves.

Note that the Oct. 25th and Nov. 8th talks switched places.
Wednesday November 1, Farshid Hajir , UMass Amherst. "What is the function field of a curve?"
Abstract: It was one of the greatest discoveries of the 19th century that there is an "equivalence of categories" between certain kinds of nice curves and certain kinds of nice fields. So, a Riemann surface, for instance, which is a 1-dimensional complex manifold (a complex curve) can be understood as an algebraic object, viz. pairs (x,y) satisfying a certain polynomial equation f(x,y)=0. One object, two points of view: the geometric and the algebraic; this was a spark for the birth of algebraic geometry. In the twentieth century, the fact that this dictionary works over base fields which are finite sparked "arithmetic geometry." I'll stick mostly to the fields point of view; the invariants I'll concentrate on are the genus and the number of "degree 1" points on the curve. We'll try to find curves that have a lot of points (with respect to the genus). We will emphasize the fact that function fields look totally different from number fields and of course they have absolutely nothing to do with lattices.
Wednesday November 8, Farshid Hajir , UMass Amherst. "What is a lattice sphere packing?"
Abstract: Imagine a bunch of regularly spaced dots in n-dimensional space. Imagine solid spheres centered at these dots having maximal radius (given that the spheres are not allowed to encroach on each other's space; they can "kiss" but that's it). What percentage of space do the balls hog? How can we space the lattice points so as to maximize the percentage of space taken up by the balls, i.e. so that we can squeeze in the most balls in a given volume? These are weighty questions. I'll give some of the basic results from Conway and Sloane's monumental tome on this subject. We will see that lattices have utterly nothing to do with number fields; they are completely different objects.
Wednesday November 15, Farshid Hajir , UMass Amherst. "What is an expander graph?"
Abstract: I'll define graphs, their adjacency matrices, Cheeger constants, "second Eigenvalues", connectedness, girth, a lot of the standard invariants. Then I'll introduce the Dodziuk/Alon-Millman Theorem. Ramanujan graphs will make their stately entrance. Brief overview of what all this has to do with communication networks and modular forms will be touched on. Emphasis will be placed on the truly outstanding lack of connections to the theory of number fields, lattices, codes and curves.
Wednesday November 29, Farshid Hajir , UMass Amherst. "What is an AGF?"
Abstract: An AGF is a formal object that has nothing to do with lattices, codes, curves, number fields or graphs. An AGF has an associated function called the "asymptotic envelope." The asymptotic envelope has a lower and an upper bound. Actual real-world AGFs have upper bounds that come from zeta funtions with a Riemann Hypothesis. Their lower bounds are even more mysterious and ad hoc. An AGF isn't happy until its lower and upper bounds meet. There are simply not enough happy AGFs, or at least we haven't been able to prove that many happy AGFs exist. When you interrogate any happy AGF you come across, sooner or later you realize that your happy AGF originally comes from a land called "modular forms."



The picture above was created by Paul Gunnells. It visualizes the natural action of the group of units of a complex cubic field on 3-space. Consult Paul for more details.

Last modified: Feb 2006 by Farshid Hajir