We incorporate the Poisson--Boltzmann (PB) theory of electrostatics into the variational implicit-solvent model (VISM) for the solvation of charged molecules in an aqueous solvent. The principle of VISM is to determine equilibrium solute-solvent interfaces and estimate the molecular solvation free energies by minimizing a mean-field free-energy functional of all possible solute-solvent interfaces. The functional consists mainly of solute-solvent interfacial energy, solute-solvent van der Waals interaction energy, and electrostatic energy. We develop highly accurate numerical methods for solving the dielectric PB equation and for computing the dielectric boundary force. These methods are integrated into a robust level-set method for numerically minimizing the VISM functional. We test and apply our level-set VISM with PB theory to the solvation of some single ions, two charged particles, and two charged plates, and to the solvation of the host-guest system Cucurbit[7]uril and Bicyclo[2.2.2]octane. Our computational results show that VISM with PB theory can capture well the sensitive response of capillary evaporation to the charge in hydrophobic confinement and the polymodal hydration behavior, and can provide accurate estimates of binding affinity of the host-guest system. We also discuss several issues for further improvement of VISM. This is a joint work with Li-Tien Cheng, Joachim Dzubiella, Bo Li, and J. Andrew McCammon.

From Demaratus and Histiaeus to Ceasar, from
Mary Queen of Scots to Hitler, cryptography has played
an important role in world history.
Substitution, transposition and polyalphabetic ciphers,
and the famous Enigma machine will be discussed.
We will finish with personal experiences working outside of academia.

Algebraic K-theory is a deep and subtle invariant of rings and schemes, carrying information about arithmetic and geometry. When applied to the group ring of the loop space of a manifold, it captures information about the diffeomorphism group. Over the past 25 years, the study of algebraic K-theory has been revolutionized by the introduction of trace methods, which use trace (or Chern character) maps to the simpler but related theories of (topological) cyclic and Hochschild homology.

A unifying perspective on the properties of algebraic K-theory and these related theories is afforded by viewing the input as a category of compact modules (i.e., a piece of an enhanced triangulated category). This talk will survey recent work describing the structural properties of these theories using various models of the homotopical category of module categories.

A conjecture of Debarre predicts that the only subvarieties of ppav with minimal cohomology class are Brill-Noether loci in Jacobians and the Fano surface of line in the intermediate Jacobian of a cubic three fold. We present a work in progress with Luigi Lombardi in which we made some progress toward the conjecture. In particular we give a generalization to higher dimension of Matsusaka-Ran criterion.

Natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. This conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from MRI to quantum state tomography. It is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? We explore this question under modern models of low-dimensionality and measurement acquisition.