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

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.