Mathematics

To Be Announced

Masahiro Nozaki (University of Chicago)

Sensing with Whiskers: From Geometry and Mechanics to the Statistics of the Array

Hayley Belli (Northwestern University)

Abstract: The rodent whisker system is a widely used model to study the sense of touch. Its neuroanatomy parallels that of the human, but its mechanics are vastly simplified compared to that of the human tactile system. Just like other modalities with complex sensor accessory structures, the geometry, mechanics, and material properties of the whiskers will highly affect the data a rat can acquire through these sensors. In the present work, we provide a detailed characterization of the geometry and mechanics of the whisker at both the level of individual whiskers, as well as the morphology of the entire whisker array. We perform a meta-analysis of seven studies, with over 500 rat whiskers, to show key relationships between geometric and mechanical parameters of individual whiskers. We also improve our existing model of the rat whisker array by developing normalized parameters that are applicable to whiskered species regardless of the number and arrangement of whiskers. Using these normalized parameters, we quantify the whisker arrays of the rat, mouse, and harbor seal, and draw comparisons between the three species.

Ramanujan's life and earlier notebooks

Bruce Berndt   [email] (UIUC)

Abstract: Generally regarded as India's greatest mathematician, Srinivasa Ramanujan was born in the southern Indian town of Kumbakonam on December 22, 1887 and died in Madras at the age of 32 in 1920. Before going to England in 1914 at the invitation of G.~H.~Hardy, Ramanujan recorded most of his mathematical discoveries without proofs in notebooks. The speaker devoted over 20 years to the editing of these notebooks; his goal was to provide proofs for all those claims of Ramanujan for which proofs had not been given in the literature. In this lecture, we give a brief history of Ramanujan's life, a history of the notebooks, a general description of the subjects found in the notebooks, and examples of some of the more interesting formulas found in the notebooks.

From Physics to Mathematics and back again: an exploration of generalized Kähler geometry

Marco Gualtieri (University of Toronto)

Abstract: In 1984, physicists Gates, Hull and Rocek, experts in the then-burgeoning field of supersymmetry, realized that their physical model required the existence of a unknown geometric structure, involving a Riemannian metric with a pair of compatible complex structures. Until relatively recently, we lacked a basic understanding of the features of this generalization of Kähler geometry. It was only after Hitchin's introduction of the concept of a generalized complex structure that we were able to prove the conjectures made by the physicists and provide various examples of what is now known as a generalized Kähler structure. I will explain the basic features of this fascinating geometric structure and outline the many relations we have discovered to other parts of geometry, including twistor theory, Poisson geometry, and Dirac geometry.