Modular forms are holomorphic functions on the complex upper half-plane that transform nicely under the action of elements of SL2(Z). Since the 1950’s they have occupied a central role in number theory. For example, they played a prominent part in the proof of Fermat’s last theorem.
L-functions are constructed from modular forms, and are of interest for the number-theoretic information they encode. Dirichlet was the first to work seriously with L-functions in the 1830’s, and their investigation has proved quite fruitful, although much of the work remains conjectural.
This defense will define modular forms and L-functions, and then prove the analytic continuation and functional equations of L-functions of cusp forms. We will then examine twisting L-functions by Dirichlet characters, and indicate one of the reasons for doing so. From there we will construct an adelic twisting operator and realize both twisted and untwisted L-functions as adelic integrals. Finally, we will indicate one advantage of the adelic approach by expressing the L-function as an Euler product.
Thursday, April 14, 2011
Yuri E. Gliklikh, Professor of Mathematics, Voronezh State University, Russia
“Investigation of Problems of Mathematical Physics by the Methods of Global and Stochastic Analysis. Mean derivatives.
3:30 – 4:20 (Refreshments after the talk).
208 Neville Hall.
The purpose of the talk is to show the importance of combination of methods of Global Analysis and Stochastic Analysis for investigation of problems of Mathematical Physics on the example of using the machinery of so-called Mean Derivatives both on linear spaces and on manifolds. We give the introduction to the Theory of Mean Derivatives and a survey of results obtained by the author within last years in investigation of equations and inclusions with such derivatives. Applications to the Mathematical Physics are contained in a special section (the last in the talk). The detailed description of this material can be found in the recent monograph by the author: Gliklikh Yu.E., Global and Stochastic Analysis with Applications to Mathematical Physics.- London: Springer-Verlag, 2011.
Biological populations live in a spatially structured heterogeneous world, where habitat quality varies from place to place. Organisms have evolved various movement/dispersal strategies to deal with this heterogeneity and to alleviate the overcrowding that would arise if no one ever moved.
The Internet is a type of heterogeneous environment as well. Early generations of computer worms used simple random strategies to try and spread from one host to another. For the past decade, such worms now regularly use biologically-inspired mixed dispersal strategies. My group has built an epidemiological simulation model of worms spreading through the entire Internet (4.29 billion hosts), efficient enough to run on an ordinary desktop computer. We’ve incorporated measurements of heterogeneity from the actual Internet into the model, to begin exploring various dispersal strategies of worms.
Kuratowski’s Complement Closure Problem asks: Given a topological space, is there a maximum number of different sets that are obtained by starting with a set and successively taking complements and closures? We will see different results that arise in specific situations, but that also there is a maximum value that holds generally.
In modern epidemiology there is a single number which is said to govern whether or not an infectious disease will successfully invade a susceptible population, causing an epidemic. This single number is called R_0 or the basic reproduction number; it denotes the expected number of secondary infections caused by a single infectious individual introduced into a completely susceptible population. This simple idea is a fundamental tool in modern epidemiology and is used to evaluate protective actions taken by public health officials. This talk will present the history and mathematical derivation of this important number.
I will share information about Concept Inventory assessments and how they have been used in education. In particular, I will give a little history of the development of Concept Inventories in physics and then share items from the Calculus Concept Inventory (CCI). Then we will look at some data we have on student performance from the CCI administered at a variety of schools and in a variety of types of classes.
Euler products play a key role in modern number theory. In this talk, I will describe them, and then describe a generalization, twisted Euler products, that is a recent construction. Twisted Euler products hint at new connections of number theory to quantum groups and to statistical mechanics.
Fermat’s Last Theorem is perhaps the most widely known and popularly celebrated mathematical result of the last century. Famously formulated in the margin of a book by Pierre de Fermat in the 1630’s, it states that for any nonzero integers a,b,c,
an+bn ≠ cn
when n is any integer greater than 2. This statement is simple enough that any high school student familiar with the Pythagorean Theorem could understand it, but its proof eluded mathematicians for centuries. Finally, in 1995 Andrew Wiles published a proof involving very sophisticated mathematical machinery. This talk will cover the most basic mathematical elements used in his proof. In particular, we will define and discuss elliptic curves, modular forms, the famous Taniyama-Shimura Conjecture connecting these two, and Serre’s so-called ε-conjecture. We will conclude the talk by sketching how these elements are combined to prove the theorem.
The talk will be preceded first by cookies, and then an excerpt from the NOVA special on the subject, entitled The Proof.