# Math Calendar

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A live feed of seminars and special events in the upcoming week.

May 3rd, 2013 (03:35pm - 04:25pm)
Seminar: Computational and Applied Mathematics Colloquium
Title: One-dimensional pressureless Euler/Euler-Poisson systems with or without viscosity
Speaker: Truyen Va Nguyen, Akron
Location: MB106

We study the initial value problem for one-dimensional pressureless Euler/Euler-Poisson systems with or without viscosity. A general global existence result is established by employing the sticky particles'' model and letting the number of particles go to infinity. We first construct entropy solutions for some appropriate scalar conservation laws, then we show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. Furthermore, an explicit rate of convergence of the sticky particle solutions to the solution for the continuous model is obtained via a contraction principle in the Wasserstein metric. Using this Wasserstein distance, we also study the vanishing viscosity limit for the systems. This is a joint work with Adrian Tudorascu.

May 6th, 2013 (10:00am - 12:00pm)
Seminar: Ph.D. Oral Comprehensive Examination
Title: "Poisson limit theorem for Gibbs-Markov systems"
Speaker: Xuan Zhang, Adviser: Manfred Denker, Penn State
Location: MB106

Sinai studied Poisson limit distribution behaviour for eigenvalues in the quantum kicked-rotator model. Inspired by Sinai's work, Pitskel in- vestigated the Poisson limit distribution for return times of dynamical systems. He proved results for Markov chains with nite states and for hyperbolic toral automorphisms. Independently Hirata proved his result for subshifts of nite types with Gibbs measures and for Axiom A sys- tems. Many results have been proved for di enrent kinds of systems in recent years. In this talk, we will try to explain that Poisson limit dis- tribution also exists for Gibbs-Markov systems. Certain Markov chains with countable states will be special cases. We basically follow Hirata's method with a few revisions.

May 7th, 2013 (03:00pm - 03:50pm)
Seminar: CCMA PDEs and Numerical Methods Seminar Series
Title: Title: Analyticity of the Stokes semigroup in space of bounded functions
Speaker: Yoshikazu Giga, University of Tokyo
Location: MB106
Abstract: http://

Abstract: The Stokes system is a linearized system of the Navier-Stokes equations describing the motion of incompressible viscous fluids. It is believed that the nonstationary problem is very close to the heat equation. (In fact, if one considers the Stokes system in a whole space R^n, the problem is reduced to the heat equation.) The solution operator S(t) of the Stokes system is called the Stokes semigroup. It is well-known that S(t) is analytic in the L^p setting for a large class of domains including bounded and exterior domains with smooth boundaries provided that p is finite and larger than 1. This property is the same as the heat semigroup. Moreover, for the heat semigroup it is analytic even when p equals the infinity. The corresponding (p=infinity) result for the Stokes semigroup S(t) has been open for more than thirty years even if the domain is bounded. Using a blowup-argument, we have now solved this long-standing problem for a large class of domains, including bounded and exterior domains. A key step is to derive a harmonic pressure gradient estimate by a velocity gradient. We give a sketch of the proof as well as a few possible applications to the Navier-Stokes equations. This is a joint work of my student Ken Abe and the main paper is going to appear in Acta Mathematica.

May 8th, 2013 (03:35pm - 04:35pm)
Seminar: Center for Dynamics and Geometry Seminars
Title: Singularly beautiful algebraic curves
Speaker: Joel Langer, Case Western Reserve University
Location: MB114

The theorems of Gauss on constructible n-gons and Abel on uniform subdivision of the Bernoulli lemniscate place the circle and lemniscate among only a handful of algebraic curves known to possess such nice subdivision properties. For these curves, unit speed parameterization (or its norm) extends to meromorphic (elementary or elliptic) functions on the complex plane. Such parameterizations are already rare, as may be seen from the polyhedral geometry on a (complex) curve C; this is de ned via the quadratic di erential on C induced by dx^2 +dy^2. The required behavior of this quadratic di erential forces rather special singularities of C and it follows, e.g., that Bernoulli lemniscates are the only curves of degree at most four with compact polyhedral geometry. In this talk, such results and related examples will be illustrated via a graphical technique for visualizing the (real) foci and polyhedral geometry of an algebraic curve.

May 13th, 2013 (10:00am - 12:00pm)
Seminar: Ph.D. Thesis Defense
Title: "An Analytical Approach for Sustainable Transportation Network Design"
Speaker: Ke han, Adviser: Alberto Bressan, Penn State
Location: MB106
Abstract: http://

This dissertation work emphasizes the modeling and formulation of multi-agent, dynamic, interdependent, complex and competitive transportation systems, by invoking mathematically canonical and tractable forms. The design and management of a transportation network include not only problems of adding/removing capacity by changing nodes and arc sets, but also the determination of piecewise smooth decision variables like prices, tolls, traffic signals, information accessibility, and other control variables associated with Stackelberg mechanisms and so-called second-best strategies. The primary modeling paradigm employed by this dissertation is differential Nash-like game among travelers that captures several key aspects of modern traffic networks, including travel choices, travel modes, economic, social and environmental impacts. The well-known concept of dynamic network user equilibrium developed in the early 1990s has been extended in this dissertation to incorporate elastic travel demands and bounded rationality. Several network traffic flow models, including the Lighthill-Whitham-Richards hydrodynamic model featuring vehicle spillback, have been analyzed along with their qualitative properties and numerical results presented. As applications of the aforementioned theoretical work, several network design problems, such as congestion pricing and dynamic signal control, are presented. These problems explicitly address environmental sustainability in a mathematically tractable way. The results reveal various types of complexity inherent in the transportation networks, and provide insights into the management of such systems.

May 20th, 2013 (10:30am - 12:30pm)
Seminar: Ph.D. Thesis Defense
Title: "Rigidity of periodic cyclic homology under certain smooth deformations"
Speaker: Allan Yashinski, Adviser: Nigel Higson, Penn State
Location: MB106

Given a formal deformation of an algebra, Getzler defined a connection on the periodic cyclic homology of the deformation, which he called the Gauss-Manin connection. We define and study this connection for smooth one-parameter deformations. Our main example is the smooth noncommutative n-torus viewed as a deformation of the algebra of smooth functions on the n-torus. In this case, we use the Gauss-Manin connection to give a parallel translation argument that shows that the periodic cyclic homology groups of noncommutative tori are the same as in the commutative case. As a consequence, we obtain differentiation formulas relating various cyclic cocycles on noncommutative tori. By considering the properties leveraged in the case of noncommutative tori, we generalize to a larger class of deformations, including nontrivial crossed product algebras by the group of real numbers. The algebras of such a deformation extend naturally to differential graded algebras, and we show that they are fiberwise isomorphic as A_infinity-algebras. As a corollary, periodic cyclic homology is preserved under this type of deformation. In particular, this gives yet another calculation of the periodic cyclic homology of noncommutative tori and a proof of the Thom isomorphism in cyclic homology.

May 20th, 2013 (02:30pm - 04:30pm)
Seminar: Ph.D. Thesis Defense
Title: "A Model of Intuitionism Based on Turing Degress"
Speaker: Sankha Basu, Adviser: Stephen Simpson, Penn State
Location: MB106

Intuitionism is a constructive approach to mathematics introduced in the early part of the twetieth century by L. E. J. Brouwer and formalized by his student A. Heyting. A. N. Kolmogorov, in 1932, gave a natural but non-rigorous interpretation of intuitionism as a calculus of problems. In this document, we present a rigorous implementation of Kolmogorov's ideas to higher-order intuitionistic logic using sheaves over the poset of Turing degrees with the topology of upward closed sets. This model is aptly named as the Muchnik topos, since the lattice of upward closed subsets of Turing degrees is isomorphic to the lattice of Muchnik degrees which were introduced in 1963 by A. A. Muchnik in an attempt to formalize the notion of a problem in Kolmogorov's calculus of problems.