Minnesota

Persistent homology is a tool for identifying topological features of (often high-dimensional) data. Typically, the data is indexed by a one-dimensional parameter space, and persistent homology is easily visualized via a persistence diagram or "barcode." Multi-dimensional persistent homology identifies topological features for data that is indexed by a multi-dimensional index space, and visualization is challenging for both practical and algebraic reasons. In this talk, I will give an introduction to persistent homology in both the single- and multi-dimensional settings. I will then describe an approach to visualizing multi-dimensional persistence, and the algebraic and computational challenges involved. Lastly, I will demonstrate an interactive visualization tool, the result of recent work to efficiently compute and visualize multi-dimensional persistent homology. This work is in collaboration with Michael Lesnick of the Institute for Mathematics and its Applications.

Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well.
The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.

Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well.
The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.

On parabolic and elliptic equations with VMO coefficients.

Abstract: An L_p-theory of divergence and non-divergence form elliptic and parabolic equations is presented.

The main coefficients are supposed to belong to the class VMO_x, which, in particular, contains all functions independent of x.

Weak uniqueness of the martingale problem associated with such equations is obtained

Bacteriophage T4 is a virus that attacks bacteria by a unique mechanism. It

lands on the surface of the bacterium and attaches its baseplate to the cell

wall. Aided by Brownian motion and chemical bonding, its tail fibres stick to

the cell wall, producing a large moment on the baseplate. This triggers an

amazing phase transformation in the tail sheath, of martensitic type, that

causes it to shorten and fatten. The transformation strain is about 50%. With a

thrusting and twisting motion, this transformation drives the stiff inner tail

core through the cell wall of the bacterium. The DNA of the virus then enters

the cell through the hollow tail core, leading to the invasion of the host.

This is a natural machine. As we ponder the possibility of making man-made

machines that can have intimate interactions with natural ones, on the scale of

biochemical processes, it is an interesting prototype. We present a mathematical

theory of the martensitic transformation that occurs in T4 tail sheath.

Following a suggestion of Pauling, we propose a theory of an active protein

sheet with certain local interactions between molecules. The free energy is

found to have a double-well structure. Using the explicit geometry of T4 tail

sheath we introduce constraints to simplify the theory. Configurations

corresponding to the two phases are found and an approximate formula for the

force generated by contraction is given. The predicted behaviour of the sheet is

completely unlike macroscopic sheets. To understand the position of this

bioactuator relative to nonbiological actuators, the forces and energies are

compared with those generated by inorganic actuators, including nonbiological

martensitic transformations. Joint work with Wayne Falk, @email

Wayne Falk and R. D. James, An elasticity theory for self-assembled protein

lattices with application to the martensitic transformation in Bacteriophage T4

tail sheath, preprint.

K. Bhattacharya and R. D. James, The material is the machine, Science 307

(2005), pp. 53-54.