Title: Is polynomial interpolation really that bad?
Abstract: A myth in numerical analysis (according to a nice article of N. Trefethen) is the belief that polynomial interpolation has to be avoided in practice since it is not stable and converges in general badly to the interpolated function.
In this talk, we are going to shed light on this myth by considering different aspects of polynomial interpolation as numerical stability and convergence properties. We will discuss some ot the theories of Trefethen why polynomial interpolation has such a bad reputation. At the end of the talk, I will give some examples how Chebyshev polynomials can be used efficiently to interpolate data points on Lissajous curves.
Title: Locomotion and Rotation with three stiff
legs at Low Reynolds Number
For biological organisms the ability to turn and reorient in space is of vital importance to their evolutionary fitness. Motivated by the kinematics of swimming crustaceans, this paper analyzes the hydrodynamics of a theoretical tripodal organism whose legs extend radially from a spherical body with small radius. Each leg moves sinusoidally about a specified time-averaged angle relative to the swimmer’s orientation. Arguments of symmetry are presented to establish expectations about the swimmer’s kinematic dynamics; then, applying classical results from slender-body theory to the model we specify a resistance matrix and present numerical results to the equations of motion depending on the amplitude, phase, and average angle for each leg. As the prescribed phase shift of each leg is varied the model predicts that maximal turning effciency occurs when the phase
difference between adjacent legs is 2π/3 with maximal net translation occurring coincidentally.
Title: A MODULAR FORMS APPROACH TO ARITHMETIC CONVOLUTED IDENTITIES
In 2004, H. Farkas found a series of identities which relate the convolution of a certain arithmetic function with an analogue of the classical σ-function. In 2009, P. Guerzhoy and W. Raji interpreted series of identities of this kind using generating functions and modular forms. Their results pertain to primes p ≡ 3 mod 4. In this paper, we address the primes p ≡ 5 mod 8 and obtain four new series of similar identities. Our methods are close to those employed by Guerzhoy and Raji.
Title: Zeta polynomials for modular forms
Abstract: The speaker will discuss recent work on Manin’s theory of zeta polynomials for modular forms. He will describe recent results which confirm Manin’s speculation that there is such a theory which arises from periods of newforms. More precisely, for each even weight $k > 2$ and newform $f$, the speaker will describe a canonical polynomial $Z_f(s)$ which satisfies a functional equation of the form $Z_f(s) = Z_f(1-s)$, and also satisfies the Riemann Hypothesis: if $Z_f(\rho) = 0$, then $\Re(\rho) = 1/2$. This zeta function is arithmetic in nature in that it encodes the moments of the critical values of $L(f, s)$. This work builds on earlier results of many people on period polynomials of modular forms. This is joint work with Seokho Jin, Wenjun Ma, Larry Rolen, Kannan Soundararajan, and Florian Sprung.
Title: New theorems at the interface of number theory and representation theory
Abstract: Ramanujan’s first and last letters to Hardy have a breathtaking legacy. In representation theory alone they inspired the development of vertex operator algebras and the Fields medal winning work of Borcherds on Monstrous Moonshine. The speaker will recall this history, and then explain very recent developments which illustrate that these results are only glimpses of even larger theories.
Speaker: Plamen Iliev (Georgia Tech)
Title: Bispectrality and superintegrability
Abstract: The bispectral problem concerns the construction and the classification of operators possessing a symmetry between the space and spectral variables. Different versions of this problem can be solved using techniques from integrable systems, algebraic geometry, representation theory, classical orthogonal polynomials, etc. I will review the problem and some of these connections and then discuss new results related to the generic quantum superintegrable system on the sphere.