**Topics
in Differential Geometry and its Discretizations**

**Poster session**

**Title:** Discrete-time quantum walk on the square lattice

**Author:** Takashi Komatsu* (Tohoku University)

**Abstract:**

We study discrete-time quantum walks on graphs. The
notion of discrete- time quantum walks was introduced as a quantum version of
random walks. Recently, quantum walks have been intensively studied in
connection with quantum computing and quantum physics.

In this poster, we will propose a model of discrete-time quantum walks on
the square lattice without localization and give the limiting distribution of
our quantum walk. Next, we see that the Konno function appears as the density
function with respect to radial direction in our quantum walk.

**Title:** Translating soliton in arbitrary codimension PDF

**Author:** Keita Kunikawa* (Tohoku University)

**Abstract:**

We study the translating solitons of mean curvature flow. Although many authors
study translating solitons in codimension
one, there are few references and examples for higher codimensional
cases except for Lagrangian translating solitons. Hence we observe non-trivial examples of
translating solitons in arbitrary codimension.
We will see that they have the property called parallel principal normal (PPN).
Conversely, we classify the complete translating solitons
with PPN.

**Title:** Hodge-Kodaira Decomposition of Evolving
Neural Networks PDF

**Authors:** Keiji Miura* (Tohoku University), Takaaki Aoki (Kagawa University)

**Abstract:**

Here we applied the Hodge-Kodaira decomposition, a topological method, to an evolving
neural network model in order to characterize its loop structure. By
controlling a learning rule parametrically, we found that a model with an
STDP-rule, which tends to form paths coincident with causal firing orders, had
the most loops. Furthermore, by counting the number of global loops in the
network, we detected the inhomogeneity inside the chaotic region, which is
usually considered intractable.

**Title: **Neural Implementation of Shape-Invariant Touch Counter Based on Euler Calculus PDF

**Authors:** Keiji Miura* (Tohoku University), Kazuki Nakada (University of
Electro-Communications )

**Abstract:**

Here we propose a fully parallelized
algorithm for a shape-invariant touch counter for 2-D pixels. The number of touches is counted by the Euler integral, a
generalized integral, in which a connected component counter (Betti number) for the binary image was used as elemental
module. The proposed circuit architecture embodies the Euler integral in
the form of recurrent neural networks for iterative vector operations. Our
parallelization can lead the way to Field-Programmable Gate Array or Digital
Signal Processor implementations of topological algorithms with scalability to
high resolutions of pixels.

Title:The DPW method for discrete constant mean curvature surfaces in Riemannian spaceforms

**Author:** Yuta Ogata* (Kobe University)

**Abstract:**

Bobenko and
Pinkall discretized constant mean curvature (CMC) surfaces
in Euclidean 3-space in terms of Lax representations. Applying matrix factorizing theorems, Hoffmann gave a generalized Weierstrass-type representation for discrete CMC surfaces
in the sense of Bobenko and Pinkall.
In this poster, we will explain that such construction can be extended to
discrete CMC surfaces in other Riemannian spaceforms.
This poster is based on the joint work with M. Yasumoto.

**Title:** Phase transition property of $l_{p}$-product spaces

**Author:** Ryunosuke Ozawa* (Tohoku University)

**Abstract:**

We consider metric measure spaces $X_{n}$
close to a one-point metric measure space if for any $1$-Lipschitz
function on $X$ is close to a constant function. This phenomenon is called the
measure concentration. The $\infty$-dissipation property
is opposite from the measure concentration and means that the metric measure
spaces disperse into many small pieces far apart each other. A sequence $\{ X_{n} \}_{n=1}^{\infty}$ of
metric measure spaces has the phase transition property if there exists a
sequence $\{ c_{n} \}_{n=1}^{\infty}$ satisfying the
following (1) and (2). (1) For any sequence $\{ t_{n}
\}_{n=1}^{\infty}$ with $t_{n}/c_{n} \to 0$ as $n \to
\infty$, the scaled metric measure space $t_{n}X_{n}$
close to the one-point metric measure space as $n \to \infty$.
(2) For any sequence $\{ t_{n} \}_{n=1}^{\infty}$ with $t_{n}/c_{n} \to +\infty$
as $n \to \infty$, the sequence $\{ t_{n}X_{n}
\}_{n=1}^{\infty}$ $\infty$-dissipates.
We call such a sequence $\{ c_{n} \}_{n=1}^{\infty}$ a sequence of critical scale order. In this poster,
we give a sequence of critical scale order of a sequence of the $l_{p}$-product
spaces. This poster is based on a joint work with Takashi Shioya
(Tohoku University).

**Title:** Wave splitting solution for the FitzHugh-Nagumo
equations

**Author:** Tomoyuki Terada* (Tohoku University)

**Abstract:**

Using numerical simulations, we discover
a wave splitting solution for the FitzHugh-Nagumo
equations under temporal switching between two states of mono- and bi-stable nonlinearities.
Mono(bi)-stable
nonlinearity has several unstable steady states and only one(two) stable
steady state, respectively. To
analyze the pattern formation dynamics containing the wave splitting solution,
we define of a wave splitting solution describing the discovered solution.

**Title:** Parallel surfaces of cuspidal edges PDF

**Author:** Keisuke Teramoto* (Kobe University)

**Abstract:**

We investigate
parallel surfaces of cuspidal edges. We give a
criterion for the parallel surfaces of cuspidal edges
to have swallowtail singularities. Moreover, we also clarify relations between
singularities of parallel surfaces and differential geometric properties of
initial cuspidal edges.

**Title:** Final state problem for a system of nonlinear Schrödinger equations with three wave interaction

**Author:** Kota Uriya* (Tohoku University)

**Abstract:**

In this poster, we consider a system
of nonlinear Schrödinger equations with three wave interaction. We derive the
asymptotic behavior of a solution to the system by using a particular
solution to a system of ordinary differential equations.