Heron's formula relates the square of the area of a triangle to the 4-dimensional volume of a hyper-rectangle. As such, it should lend itself to a 4-dimensional proof. In this talk, I show how to use a scissors congruence proof of the Pythagorean Theorem to create a scissors congruence proof of Heron's formula. The talk will be an excursion into some interesting aspects of 4-dimensional hyper-solids. 

Recently, imaging techniques in science, engineering and medicine have evolved to expand our ability to visualize internal information of an object such as the human body.  In particular, there has been marked progress in electromagnetic property imaging techniques where cross-sectional image reconstructions of conductivity, permittivity and susceptibility distributions inside the human body are pursued. They will widen applications of imaging methods in medicine, biotechnology, non-destructive testing, monitoring of industrial process and others. 

This lecture focuses on mathematical modeling and analysis on electromagnetic tissue property imaging. The imaging problems can be formulated as inverse problems that are intrinsically nonlinear, and finding solutions with practical significance and value requires deep understanding of underlying physical phenomenon (Maxwell's equations) with data acquisition systems as well as implementation details of image reconstruction algorithms. We will explain strategies dealing with these complicated structures using a simple linear algebra.

The minimal model program (MMP) refers to a series of theorems and
conjectures which arise naturally when one attempts to classify
projective varieties in terms of their pluricanonical line bundles.
The theory of multiplier ideal sheaves has played a central role in
the recent development of MMP.

A multiplier ideal sheaf is determined by a singular hermitian metric
of a line bundle. In fact, a singular hermitian metric contains more
information than its multiplier ideal sheaf. We will give an overview
of these fundamental notions and their applications in the context of
MMP.

On the other hand, in the subclass of algebraic multiplier ideal
sheaves, it is known that not every integrally closed ideal is an
algebraic multiplier ideal. We extend this statement to the full class
of analytic multiplier ideal sheaves, answering a question asked by
Lazarsfeld.

A general goal of noncommutative geometry (in the sense of Alain Connes) is to translate the main tools of differential geometry into the Hilbert space formalism of quantum mechanics by taking advantage of the familiar duality between spaces and algebras. In this setting noncommutative spaces are only represented through noncommutative algebras that play formally the role of algebras of functions on these (ghost) noncommutative spaces. As a result, this allows us to deal with a variety of geometric problems whose noncommutative nature prevent us from using tools of classical differential geometry. In particular, the Atiyah-Singer index theorem untilmately holds in the setting of noncommutative geometry. 

The talk will be an overview of the subject with a special emphasis on quantum space-time and diffeomorphism invariant geometry. In particular, if time is permitted,  it is planned to allude to recent projects in biholomorphism invariant geometry of complex domains and contactomorphism invariant geometry of contact manifolds. 

The Bergman Tau function is a holomorphic function defined over Teichm"uller spaces. This satisfies modular property with repsect to the mapping class group. In this talk, we will explain an infinite product expression of the Bergman Tau function. This can be considered as a generalization of the Dedekind eta function to higher genus case. The complex valued Chern-Simons functional will be introduced for this infinite product expression. We will also explain some corollaries of this result about the eta invariant and a Polyakov type formula.