Pizza Seminar Baylor Mathematics pizza margherita:

from subwavelength metaoptics to intracellular signaling pathways

@academiabarilla.com

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 1 / 14 a background search for new energy motivations:

Pipelines and fossil energy exhaustion... ⇒ A new search

web pictures

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 2 / 14 subwavelength metamaterials energy transform lights through subwavelength holes:

Theory: Wave propagation control at the deep subwavelength scale in metamaterials, by F. Lemoult et. al., Nature (physics), 2013 Applications: Areas in the energy engineering, also electromagnetic radiation, seismology, and acoustics Designs of metamaterials that mold electromagnetics

web pictures Researcher taps high-performance computers to design metamaterials that mold electromagnetics.

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 3 / 14 subwavelength metamaterials math models mathematical models

@JMD

Professor John Davis and his PDE book ⇒ pg. 170:

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 4 / 14 subwavelength metamaterials partial differential equations Maxwell’s equations:

James Clerk Maxwell (1831–1879)

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 5 / 14 subwavelength metamaterials partial differential equations paraxial Helmholtz equation:

For a monochromatic beam, consider the complex envelope E of the complex wavefunction in the paraxial sense:

2iκEz = Exx + Eyy + Ezz ,

(x, y) ∈ D2, z > z0,

where D2 is the transverse domain, z is the beam propagation direction, and κ = 2πν/c > 102 is the wave number of the light. Hermann Ludwig Ferdinand von Helmholtz (1821–1894)

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 6 / 14 subwavelength metamaterials partial differential equations Ritz and his criticism:

Walther Ritz (1878 –1909)

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 7 / 14 subwavelength metamaterials Helmholtz in polar system our concern:

 1  E = β E + E + E , 0 < r ≤ R, z ≥ 0. z rr r r zz

i) Paraxial Models; ii) Kerr Effects

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 8 / 14 subwavelength metamaterials multiscale configurations a “possible” resolution:

Micro and Macro Regions

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 9 / 14 subwavelength metamaterials multiscale configurations multiscale scheme:

Due to the singularity as r → 0+, we replace the polar PDE by coupled partial differential equations in connected transverse domains. Definition (a decomposed system)

Ez = β (2Err + Ezz + g(E)) , 0 ≤ r < ˜r,  1  E = β E + E + E , ˜r ≤ r ≤ R. z rr r r zz

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 10 / 14 subwavelength metamaterials multiscale configurations a multiscale infrastructure:

Micro and Macro Grids

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 11 / 14 cancer treatments a cell biology model and its solution hes1 and p53-mdm2 intracellular signaling pathways:

∂q α n = κ ∇2 + q µ , ∈D , qn qn h qqn x n ∂t 1 + (pn/p) ∂q c = κ ∇2q − µ q , x ∈D , ∂t qc c q c c ∂p c = κ ∇2p + H (x)α q − µ p , x ∈D , ∂t pc c 1 p c p c c ∂p n = κ ∇2p − µ p , x ∈D . ∂t pn n p n n

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 12 / 14 cancer treatments multiscale configurations high resolution multiscale computations:

Can we do the computing and numerical analysis?

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 13 / 14 cancer treatments cell model computing and analysis what else to expect?

Challenges are waiting!

qin sheng (tim) pizza seminar @ mathematics Feb. 28, 2018 14 / 14