UIUC Dept. of Mathematics Seminar Calendar

      April 2012              May 2012              June 2012      
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Tuesday, May 1, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Tuesday, May 1, 2012

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Submitted by ford.

Alina Cojocaru (Univ. Illinois Chicago)
Frobenius fields for elliptic curves
Abstract: Let E be an elliptic curve defined over $\mathbb{Q}$. For a prime p of good reduction for E, let $\pi_p$ be the $p$-Weil root of E and $\mathbb{Q}(\pi_p)$ the associated imaginary quadratic field generated by $\pi_p$. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes $p < x$ for which $\mathbb{Q}(\pi_p)$ is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. This is joint work with Henryk Iwaniec and Nathan Jones.

Harmonic Analysis and Differential Equations
1:00 pm in 347 Altgeld Hall, Tuesday, May 1, 2012

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Submitted by bronski.

Mary Pugh [email]Thin Liquid Films with Driving
Abstract: We present two thin liquid film problems with driving. The first problem is experimentally motivated and considers questions such as steady states and the existence of dynamic solutions. The second problem is more PDE-motivated and considers questions such as the presence (or absence) of finite-time blow-up. In the first problem, we consider a horizontal cylinder, rotating about its center. A viscous fluid is on the outside of the cylinder, coating the cylinder as it rotates. We consider a lubrication approximation of the Navier Stokes equations for the regime in which the fluid film is relatively thin and the surface tension is relatively large. The resulting lubrication model may have no steady state, a unique steady state, or more than one steady state. Using both numerics and analysis, we consider the dynamics of this flow, including whether or not solutions can become singular in finite time. In the second problem, we consider a long-wave unstable thin film problem $u_t = - (u^n u_{xxx})_x - B (u^m u_x)_x. The dynamics are strongly affected by the balance between the exponents n and m. We discuss the subcritical, critical, and supercritical regimes for the equation and present new results for finite-time blow-up for the problem on the line. This is joint work with Marina Chugunova (University of Toronto) and Roman Taranets (Nottingham).

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, May 1, 2012

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Laurentiu Leustean (Institute of Mathematics of the Romanian Academy)
Proof mining in nonlinear analysis
Abstract: The program of proof mining is concerned with the extraction of hidden finitary and combinatorial content from proofs that make use of highly infinitary principles. This new information can be both of quantitative nature, such as algorithms and effective bounds, as well as of qualitative nature, such as uniformities in the bounds or weakening the premises. Thus, even if one is not particularly interested in the numerical details of the bounds themselves, in many cases such explicit bounds immediately show the independence of the quantity in question from certain input data. This line of research, developed by Ulrich Kohlenbach in the 90's, has its roots in Georg Kreisel's program on unwinding of proofs, put forward in the 50's. In this talk I will present applications of proof mining to the asymptotic behavior of nonexpansive iterations and nonlinear generalizations of ergodic averages.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, May 1, 2012

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Christian Altomare (The Ohio State University)
Antimatroid minors: The Graph Minor Theorem, Laver's Theorem, and Forbidden Antimatroid Minor Theorems
Abstract: This talk is designed to be accessible and (hopefully) of interest to graph theorists and logicians. The Graph Minor Theorem says that every minor closed property of finite graphs has a finite forbidden minor description. (One version of) Laver's Theorem states the same is true for suborder closed properties of countable total orders. A notion of minor for antimatroids, thought of as "combinatorial proof systems", specializes to graph minor and total order embeddability for those respective objects. This allows a conjectural unification of these two seemingly quite distinct theorems. We discuss this unification and antimatroid minor theorems as well.