# Kyoto University Applied Mathematics Seminar (KUAMS)

## Records of past seminars: year 2014

No.7: 2014. 1. 21. (Tue) 16:30-18:00

Dr. Marcio Gameiro, Associate Professor (ICMC-USP Brazil & RIMS)

"Rigorous Numerics for Nonlinear PDEs"

**Abstract: **We present a rigorous numerical method to compute solutions of infinite dimensional
nonlinear problems. The method combines classical predictor corrector algorithms,
analytic estimates and the uniform contraction principle to prove existence of smooth
branches of solutions of nonlinear PDEs. The method is applied to compute equilibria
and time periodic orbits for PDEs defined on two- and three-dimensional spatial domains.

No.8: 2014. 2. 18. (Tue) 16:30-18:00

Dr. Wagatsuma Hiroaki, Associate Professor (Kyushu Institute of Technology)

**Abstract: ** This talk will be given in Japanese.

No.9: 2014. 4. 15. (Tue) 16:30-18:00

Dr. Hiroshi Fujiwara, Assistant Professor (Kyoto University)

"Fast and Reliable Numerical Methods for Near-infrared Light Propagation in Human Bodies"

**Abstract: **We will present a fast and reliable numerical approach for
near-infrared light (NIR) propagation in human bodies. In particular,
an accurate numerical quadrature rule on the unit sphere and degenerate
kernel approximations are shown to realize numerical computations in
the three dimensions.

The use of NIR light is considered as a safe and simple technology for
monitoring our bodies, in particular, brain activities. The radiative
transport equation (RTE) is widely accepted as a mathematical model of
NIR light propagation in our bodies. It is an integro-differential
equations and the boundary value problem in the three dimensions
is essentially a five dimensional large-scale problem. An accurate
numerical quadrature rule and degenerate kernel approximations have been
developed to reduce the size of discretization problems and computational
times. Error analysis and numerical experiments with biomedical data of
a human head are also shown.

This talk is based on a joint work with Prof. Iso (Kyoto Univ.),
Prof. Higashimori (Hitotsubashi Univ.), and Human Brain Research
Center in Kyoto University.

No.10: 2014. 5. 13. (Tue) 16:30-18:00

Dr. Bartosz Protas, Associate Professor (McMaster Univ.)

"Extreme vortex states and the hydrodynamic blow-up problem"

**Abstract: **In the presentation we will discuss our research program
concerning the study of extreme vortex events in viscous
incompressible flows. These vortex states arise as the flows
saturating certain fundamental mathematical estimates, such as the
bounds on the maximum enstrophy growth in 3D. They are therefore
intimately related to the question of spontaneous singularity
formation in the 3D Navier-Stokes system, known as the hydrodynamic
"blow-up" problem. We demonstrate how new insights concerning such
problems can be obtained by formulating them as variational PDE
optimization problems which can be solved computationally using
suitable discrete gradient flows. In offering a systematic approach to
finding flow solutions which may saturate known estimates, the
proposed paradigm provides a bridge between mathematical analysis and
scientific computation. In particular, it allows one to determine
whether or not certain mathematical estimates are "sharp", or if they
may still be improved. In the presentation we will review a number of
new results concerning 2D and 3D vortex flows characterized by the
maximum possible growth of, respectively, palinstrophy and
enstrophy. We will also discuss their relation to the available
theoretical bounds obtained with rigorous methods of mathematical
analysis.

[Joint work with Diego Ayala]

No.11: 2014. 6. 3. (Tue) 16:30-18:00

Kenta Ishimoto (Kyoto Univ., D3)

"Hydrodynamics of cell swimming near boundaries"

**Abstract: **It is known that some microorganisms such as bacteria and spermatozoa accumulate near a boundary, which has been recently considered to be a phenomenon driven by non-linear hydrodynamic interaction between the cel and the boundary, whereby the time-reversal symmetry highlighted by the scallop theorem of the low Reynolds number flow still holds even in presence of boundaries.

In this talk, swimming stability of (i) a squirmer and (ii) a model spermatozoon near a boundary will be discussed after a brief review of the hydrodynamics of swimming microorganism.

The squirmer is a simple mathematical model of a swimming microorganism that propel with surface deformation. The stability behaviour can be characterised by boundary conditions at the surface as well as cell geometry and swimming morphology. The swimming dynamics of the model spermatozoon illustrates the stable planar beat near a no-slip boundary, though the dynamics can be unstable for abnormal cell morphology, implying biological significance of the swimming stability for mammalian spermatozoa in female reproductive tract.

(Joint work with Dr. E. A. Gaffeny at University of Oxford)

No.12: 2014. 10. 14. (Tue) 16:30-18:00

Dr. Michiaki Onodera, Assistant Professor (Kyushu Univ.)

"A flow approach to an inverse problem in potential theory"

**Abstract: **A new geometric flow describing the motion of quadrature surfaces is
introduced, where a quadrature surface is a closed hypersurface
inducing the same electrostatic potential as a given electric charge
density.
This characterization enables us to study quadrature surfaces through
the investigation of the flow.
It is proved that the flow is uniquely solvable under the geometric
condition that the initial surface has positive mean curvature.
As a consequence, a bifurcation criterion for quadrature surfaces is obtained.

This seminar is presented in English. This seminar is also held as an ACCA-JP seminar.

No.13: 2014. 11. 18. (Tue) 16:30-18:00

Dr. Noboru Ito, Assistant Professor (Waseda Univ.)

"Strong and weak (1, 3) homotopy equivalence classes of spherical curves"

**Abstract: **This talk is concerned with two joint works [1] and [2]. Every generic immersed spherical curve can be related to a simple closed curve by a finite sequence of the first, second, and third Reidemeister moves. We decompose the third Reidemeister move into just two kinds of moves: strong and weak third Reidemeister moves. In [1], we show that the strong (1, 3) homotopy equivalence class containing a simple closed curve is represented as any connected sum of finitely many spherical curves, each of which is either a simple closed curve, the curve that appears similar to ｡, or the trefoil projection. In [2], we show that the weak (1, 3) homotopy equivalence class containing a simple closed curve is represented as any connected sum of finitely many spherical curves, each of which is either a simple closed curve or the curve that appears similar to ｡. This talk will obtain the story from our starting point to the present and our motivation of this study.

**Reference:**

[1] N. Ito, Y. Takimura, and K. Taniyama, Strong and weak (1, 3) homotopies on knot projections, to appear in Osaka J. Math.

[2] N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections. J. Knot Theory Ramifications 22 (2013), 1350085 (14 pages).

This seminar will be given in Japanese and the presentation materials are shown in English. This seminar is also held as an CREST seminar.