Schedule
Monday 19 July 2021, from 15:00 AEST
Boris Springborn (TU Berlin)
15:00 : What is a discrete conformal map?
16:00 : Discrete uniformization and ideal hyperbolic polyhedra, see https://www.youtube.com/watch?v=dzZYDUexzIk
Abstract: This talk will be about two seemingly unrelated problems:
a discrete version of the uniformization problem for piecewise flat surfaces and
constructing ideal hyperbolic polyhedra with prescribed intrinsic metric.
I will explain these problems, how they are equivalent, and how they can both be solved with the same variational principles.
Tuesday 20 July 2021, from 15:00 AEST
Herbert Edelsbrunner (IST Austria)
15:00 : What is Crofton's formula?
16:00 : Random triangles and random inscribed polygons
Abstract: Given three random points on a circle, the triangle they form is acute with probability 1/4. In contrast, the triangle formed by three random points in the 2-sphere is acute with probability 1/2. Both of these claims have short geometric proofs. We use the latter fact to prove that a triangle in the boundary of a random inscribed 3-polytope is acute with probability 1/2.
Picking n points uniformly at random on the 2-sphere, we take the convex hull, which is an inscribed 3-polytope. The expected mean width, surface area, and volume of this polytope are 2 (n-1)/(n+1), 4\pi [(n-1)(n-2)]/[(n+1)(n+2)], and (4\pi/3) [(n-1)(n-2)(n-3)]/[(n+1)(n+2)(n+3)]. These formulas are not new but our combinatorial proofs are.
Work with Arseniy Akopyan and Anton Nikitenko.
Wednesday 21 July 2021, from 10:00 AEST
Sabetta Matsumoto (GaTech)
10:00 : What is knot theory doing in biology?
11:00 : Twisted topological tangles or: the knot theory of knitting
Abstract: Imagine a 1D curve, then use it to fill a 2D manifold that covers an arbitrary 3D object – this computationally intensive materials challenge has been realized in the ancient technology known as knitting. This process for making functional materials 2D materials from 1D portable cloth dates back to prehistory, with the oldest known examples dating from the 11th century CE. Knitted textiles are ubiquitous as they are easy and cheap to create, lightweight, portable, flexible and stretchy. As with many functional materials, the key to knitting’s extraordinary properties lies in its microstructure.
At the 1D level, knits are composed of an interlocking series of slip knots. At the most basic level there is only one manipulation that creates a knitted stitch – pulling a loop of yarn through another loop. However, there exist hundreds of books with thousands of patterns of stitches with seemingly unbounded complexity.
The topology of knitted stitches has a profound impact on the geometry and elasticity of the resulting fabric. This puts a new spin on additive manufacturing – not only can stitch pattern control the local and global geometry of a textile, but the creation process encodes mechanical properties within the material itself. Unlike standard additive manufacturing techniques, the innate properties of the yarn and the stitch microstructure has a direct effect on the global geometric and mechanical outcome of knitted fabrics.
Thursday 22 July 2021, from 10:00 AEST
Feng Luo (Rutgers)
10:00 : What is discrete conformal geometry of surfaces?
Abstract: Discrete conformal geometry of surfaces attempts to establish computable discretizations of classical Riemann surface theory. This talk will focus on answering questions like, what are the discrete conformal equivalences and discrete Riemann surfaces, are their discrete counterparts of the uniformization theorem, Riemann-Roch theorem, and Abel-Jacobi theorem? We will also discuss the relationship among discrete conformal geometry, convexity and hyperbolic geometry.
Title: Recent developments in discrete conformal geometry of surfaces
Abstract: Classical theory of Riemann surfaces is a pillar in mathematics and has many applications within and outside of mathematics. There have been many approaches to establish discrete versions of conformal geometry since the pioneer work of W. Thurston in 1978. In this talk, we will report some recent developments in this area. These include the vertex scaling conformal equivalence, a discrete uniformization theorem for polyhedral surfaces and its relationship to the classical Weyl problem and the Koebe circle domain conjecture. This is a joint work with D. Gu, J. Sun and T. Wu.