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.

Speaker: Piper Harron (UH Manoa)

Title: Equidistribution of shapes of number fields of degree 3, 4, and 5.

Abstract: In her talk, Piper Harron will introduce the ideas that there are number fields, that number fields have shapes, and that these shapes are everywhere you want them to be. This result is joint work with Manjul Bhargava and uses his counting methods which currently we only have for cubic, quartic, and quintic fields. She will sketch the proof of this result and leave the rest as an exercise for the audience. (Check your work by downloading her thesis!).

Speaker: Gideon Zamba (U. Iowa)

Title: Applied Mathematics in Action through Biostatistics