Speaker: Daniel Erman (University of Wisconsin)
Title: Ultraproducts, Hilbert’s Syzygy Theorem, and Stillman’s
When and where: 3-3:50pm, December 7, in Keller 403
Abstract: Hilbert’s Syzygy Theorem is a classic finiteness result about
a construction in algebra known as a free resolution. Stillman once
proposed an analogue of Hilbert’s result, which involved potentially
considering polynomials in infinitely many variables. Stillman’s
Conjecture was recently solved, and perhaps the simplest proof is based
upon a novel use of an ultraproduct. I’ll give an expository overview
of the history of Stillman’s Conjecture (very little algebraic
background will be assumed), and then explain how and why ultra products
came to play such a key role.
Title: Klingen Eisenstein series and symmetric square $L$-functions
Abstract: It is well-known in number theory that some of the deepest results come in connecting complex analysis in the form of $L$-functions with algebra/geometry in the form of Galois representations/motives. In this talk we will consider this for a particular case. Let $f$ be a newform of weight $k$ and full level. Associated to $f$ one has the adjoint Galois representation and the symmetric square $L$-function. The Bloch-Kato conjecture predicts a precise relationship between special values of the symmetric square $L$-function of $f$ with size of the Selmer groups of twists of the adjoint Galois representation. We will outline a result providing evidence for this conjecture by lifting $f$ to a Klingen Eisenstein series and producing a congruence between the Klingen Eisenstein series and a Siegel cusp form with irreducible Galois representation. time permitting, we will discuss a modularity result for a 4-dimensional Galois representation that arises from the congruence and studying a particular universal deformation ring. This is joint work with Kris Klosin.
Title: Hydrodynamics of Ciliary Swimming
Planktonic microorganisms are ubiquitous in water, and their population dynamics are essential for forecasting the behavior of global aquatic ecosystems. Their population dynamics are strongly affected by these organisms’ motility, which is generated by their hair-like organelles, called cilia or flagella. However, because of the complexity of ciliary dynamics, the precise role of ciliary flow in microbial life remains unclear.
In terms of fluid dynamics, ciliary swimming has been analyzed by using a squirmer model. A classical squirmer model propels itself by generating surface tangential and radial velocities. Recently, we developed a novel squirmer model in which, instead of a velocity being imposed on the cell surface, a shear stress is applied to the fluid on a stress shell placed slightly above the cell body. The shear stress expresses the thrust force generated by cilia, and the fluid must satisfy the no-slip condition on the cell body surface. The stress squirmer model has been successful in reproducing experimentally observed cell-cell interactions and cell-wall interactions.
In order to understand swimming energetics, we further developed a ciliate model incorporating the distinct ciliary apparatus. The hairy squirmer model revealed that over 90% of energy is dissipated inside the ciliary envelope. By using the hairy squirmer model, we found that there exists an optimal number density of cilia, which provides the maximum propulsion efficiency for all ciliates. The propulsion efficiency in this case decreases inversely proportionally to body length. Our estimated optimal density of cilia corresponds to those of actual microorganisms, including species of ciliates and microalgae, which suggests that now-existing motile ciliates and microalgae may have survived by acquiring the optimal propulsion efficiency.