Ever wonder what math was like in the years BC? That is: Before Calculators, Before Computers, Before Calculus, Before Chalk...well I can't back that one up... In the first graduate student seminar of the semester, I'll discuss some of the highlights of ancient Greek mathematics, including a demonstration/proof of Hippocrates' famous quadrature of the lune. This result describes how to "transform" a crescent into a triangle of equal area, using only a compass and ruler, which were the only tools of the ancients. Time permitting, I will prove some other results from some other mathematcians of the time. The talk will be both historical and mathematical, but nothing too advanced for sure.

When first studying differential geometry, one encounters theorems (although not necessarily their proofs...) that discuss the possibility and limits of embedding real manifolds in affine space. While a comparable result is not true in general, for compact complex manifolds there is a roughly analagous results. In the case of a Riemann surface, we have the Kodaira embedding theorem. I will show a rough proof of this theorem, and in the process give a whirlwind tour of the study of Riemann surfaces, discussing holomorphic line bundles, their spaces of holomorphic sections, and their classification, and the Riemann-Roch theorem. No prior knowledge of anything very high tech will be required, I will give intuitive descriptions of any objects that are new to the audience. If folks enjoy the talk, I can continue it later on in the semester by going into more detail in the subject, which is very rich, interesting, and strange (to me).

1. Riemann Surfaces by Miranda
2. Riemann Surfaces by Forster
3. Principles of Algebraic Geometry by Griffiths and Harris (not for the faint of heart)
4. Compact Riemann Surfaces by Raghavan Narasimhan
5. C-Seminar Notes by Franz Pedit (available in the 15th floor GANG Lab)

Electrical fires and their impact on the campus economy will be closely examined. Faculty, staff, and graduate students of Lederle Graduate Research Tower were observed outside of their natural environment; our primary model will involve the influx of graduate students at the Union Billiards.

Gödel showed that in any useful theory in mathematics, there are true theorems that are not provable. I will give a comprehensive outline of the proof and discuss its implications.

1. Gödel, Escher, Bach by Douglas Hofstatder
2. An Introduction to Mathematical Logic by Elliot Mendelson
3. Logic Semantics and Mathematics by Tarski
4. Introduction to Infinitesimals by Luxemborg and Storyan

An arithmetic progression is a finite evenly spaced sequence of natural numbers. I will discuss arithmetic progressions in the context of van der Waerden's theorem on arithmetic progressions. This says that given any k-coloring of the natural numbers there exist monochromatic arithmetic progressions of arbitrary length. A special case of the theorem is one of the central results of Ramsey Theory a branch of Discrete Mathematics. I will focus on sketching a proof due to M. A. Lukomskaya and comment on the interesting aesthetics of this proof. This is a discrete mathematics talk so it should be more than appropriate for all graduate students and advanced undergrads as well.

1. Three Pearls of Number Theory by A. Y. Khinchin
2. Computer-Generated van der Waerden Partitions by R. S. Stevens and R. Shantaram, (Mathematics of Computation, Apr. 1978 635-636)
3. Ramsey Theory by Graham, Rothschild and Spencer

As we've heard in TWIGS talks, colloquia, etc. the absolute Galois group of Q is an object of great importance in number theory (there are people who say that number theory is the study of this group). One way to study this group is via its representations. Certain points on elliptic curves give rise to two-dimensional representations, which will be the focus of the talk. I'll start (essentially) from first principles, so don't be scared away. The prerequisites are linear algebra, Math 611 (rings and modules) and the definition of a Galois group.

1. The best reference for this material (in my opinion) is the book Abelian 1-adic Representations and Elliptic Curves by Serre, epecially the exercise on page IV-6.
2. The Arithmetic of Elliptic Curves by Joe Silverman, Chapter III
3. The Arithmetic of Elliptic Curves by John Tate (article)
4. Number Fields by Daniel Marcus (for more information on Galois theory, e.g. Frobenius substitution, Cebotarev density theorem, etc.)

The Great red spot (GRS) is a giant anti-cyclonic storm on Jupiter; in fact, it is the clearest marking on its surface. The intrigue about the GRS is its huge size, longevity (it has been there for at least 300 years), and that it has no known terrestrial analogue. The GRS seems to violate the standard stability criteria for coherent structures which is genuinely puzzling and requires some mathematical exploration. In this talk I will give a brief outline of the Statistical equilibrium theory that can predict GRS on the active weather layer of Jupiter. The talk will be non technical and everybody is welcome.

1. Dynamics of Jovian atmospheres by Dowling,T.E., Annual Rev. Fluid Mech.27:293-334,1995
2. Jupiter's Great red spot and other vortices by Philip S. Marcus, Ann. Rev. Astronomy & Astrophysics 1993.31:523-573
3. Statistical Equilibrium predictions of Jets and spots on jupiter by Bruce Turkington, Andrew Majda, Kyle Haven and Mark Dibattista, Proc.Nat.Acad.Sci. (2001)
4. Nonequivalent Statistical equilibrium ensembles and refined stability theorems for most probable flows by Richard S.Ellis, Kyle Haven and Bruce Turkington, Nonlinearity (2001)

Given k polynomials F_1, ... , F_k which each define a subvariety of a k-1 dimensional variety X, the zero set of each F_i "should cut down the dimension by one", so that the F's shouldn't have a common zero on X. In fact the condition that they have a common zero on X is a polynomial in the coefficients of the F's, called the Resultant of F_1, ..., F_k. Computing general resultants is hard, so I'll only talk about the one variable case (X = complex numbers) in any detail, then hopefully talk about using more general resultants to solve polynomial systems.

1. Ideals, Varieties and Algorithms, by Cox, Little and o'Shea
2. Using Algebraic Geometry, but Cox, Little and o'Shea
3. Discriminants, Resultants and Multidimensional Determinants, by Gel'fand, Kapranov, and Zelevinsky

The human kidney is composed of roughly one million nephrons, the basic functional unit of the kidney. A conservation equation describing the transport of chloride in one certain part of the nephron and its regulation will be described and the stability of it's evolution in certain scenarios will be characterised. Coupling of nephrons will be mentioned too, since nephrons tend to be hooked up either alone,in pairs or in triplets to a common afferent arteriole. Basic anatomy and physiology of the nephron will be presented as background. The math involved won't be too technical: chain rule, basic numerical analysis and ODEs and basic complex analysis.

1. Feedback-Mediated Dynamics in Two Coupled Nephrons by E. B. Pitman, R. M. Zaritski, K. J. Kesseler, L. C. Moore, and H. E. Layton. Bulletin of Mathematical Biology, February, 2004
2. Bifurcation Analysis of TGF-mediated oscillations in SNGFR by H.E. Layton, E. B. Pitman, L.C. Moore. American Journal of Physiology (Renal Fluid Electrolyte Physiol.), 1991
3. Renal Physiology, by A. J. Vander
4. Effect of Backleak on Nephron Dynamics by P.G. Kevrekidis and N. Whitaker. Physical Review E, 2003

One technique we use to solve constrained min/max problems is the method of Lagrange multipliers. In most cases the calculations are tedious. In this talk I will present an example that involves Lagrange multipliers method and then use convex analysis to provide a much simpler and elegant approach to more general problems. No background is necessary for the talk (except of course, I assume that you have taken multivariable calculus at some point in your life!).

1. Limit theorems for the empirical vector of the Curie-Weiss-Potts model by R.S.Ellis and K.Wang. Stochastic Process. Appl. (1990)
2. Limit theorems for maximum likelihood estimators in the Curie-Weiss-Potts model by R.S.Ellis and K.Wang. Stochastic Process. Appl. (1992)
3. Behavior in large dimensions of the Potts and Heisenberg models by H. Kesten and R.H.Schonmann. Rev. Math. Phys. (1990)
4. Convex analysis by R.T. Rockafellar (1970)

In this talk I will try to explain what solitons are and describe some of their properties. I will start with an overview from the theory of waves and then I will outline how solitons were first observed in nature and later in numerical simulations. I will conclude with the derivation of a soliton solution of the Korteweg-de Vries equation, which models the 1-dimensional behavior of small amplitude, shallow water waves. The talk will be non technical and accessible to everyone.

1. Solitons in Mathematics and Physics by Alan C. Newell (CBMS-NSF)
2. Interactions of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States by N.J. Zabusky and M.D. Krushkal. Phys. Rev. Letters Vol. 15 #6, p240-243 (1965).

In this talk I intend to describe the contribution of the theory of Fourier Series in the development of modern Analysis. I will discuss why Dirichlet,Riemann and Cantor are three names that represent important developments in the Analysis of the 19th century. Then I will talk about the main problem of the theory: Given a function f, determining the Fourier coefficients c_{n}(f) and having form the Fourier series,under what conditions does the series represent the function f. I will present without proofs the main results especially in dimension one.

1. A treatise on trigonometric series by N.Bary. Pergamon Press, NY 1946
2. Trigonometric seriesby A.Zygmund. Cambridge Univ. Press, 1959
3. Studies in harmonic Analysis by A.Zygmund. Vol 13, J.M.Ash editor,MAA,1976