department of mathematics dcamm 11. symposium silkeborg march 2007 integrals of heat kernels...

Department of Mathematics DCAMM 11. SymposiumSilkeborg March 2007

Integrals of Heat KernelsObserved from a distance

Steen Markvorsen


Integrals of heat kernels

What integrals?


Heat integrals



They solve PDEs


PDEs


Sources of heat kernels

Brownian Motion


Brownian motion


Brownian motion (2D)

Animation courtesy of Prof. T. Spencer, Institute of Advanced Study


Discrete sources of heat kernels

Random Walk in 2D

Animation courtesy of Prof. T. Spencer, Institute of Advanced Study


Random walk with trace (2D)


Escape time

Mean Exit Time


Discrete calculation of escape time

Random walk in 1D


Mean Exit Time E(x)


Mean Exit TimeTransition probability matrix


Mean Exit TimeChord version


Mean Exit Time is:


Mean Exit TimeMixed chord version


Mean Exit Time is STILL:


Building minimal websA 2D joint for a web


Building minimal websA 3D joint for a web


Building minimal websA web tower block


Building minimal webs and surfacesA minimal surface block


Building minimal webs and surfacesScherk’s minimal surface


Building minimal webs and surfacesWeb tower blocks approximate Scherk’s surface


Building and cuttingA Scherk surface spherical R-block



The quest for transience


Transience


TransienceScherk’s minimal surface is transient


Topology


Heat integrals

Minimal webs - of low vertex degree


Building webs againA radial minimal web


Minimal webs


Minimal websThe germ of a regular Hex


Minimal websThe building of a regular Hex


Minimal webs


Minimal websThe building of a nonregular HexUsing the parametric freedom


Minimal websA nonregular Hex


Minimal webs


Minimal websA nonregular polyhex


Minimal webs


Minimal websA minimal wild polyhex


Spider web


Model of a spider web


Web approximation again


Web approximationDiscrete isothermic minimal surfaces


Enneper’s Minimal Surface


Enneper (discrete with a random walk)


Helicoid


Helicoid (discrete with a random walk)


Catenoid


Catenoid (discrete with a random walk)


Catenoid (discrete with a circle/sphere packing)


Cate-Heli-coid (discrete with a circle packing)


Heat integralsEnd

department of mathematics dcamm 11. symposium silkeborg march 2007 integrals of heat kernels...

Documents

dbrownian motion

random walkcatenoid

surfacesweb tower

transiencebrownian motion

heat kernelsthe quest

d joint

danimation courtesy

circlesphere packingcateheli