# Applied math seminar

Title: How the Tulips get their Stripes

Speaker: Thomas Hillen, University of Alberta

Date and time:
01 Nov 2023,
2:30pm -
3:30pm

Location: DTB A203

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Zoom link: https://uvic.zoom.us/j/5826187847?pwd=RlVrb0RoU0xDWTlLUDVkZW54ZThyQT09

Abstract: Tulips have captivated human interest for centuries, with their vibrant colors and unique shapes. Particularly striped tulips have been highly popular, leading to the “tulipomania” in the Dutch Golden Age. But how do the tulips get their stripes? Maybe Turing can help?

Title: Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries

Speaker: Vakhtang Putkaradze, University of Alberta

Date and time:
30 Oct 2023,
2:30pm -
3:30pm

Location: DTB A-203

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Abstract: Physics-Informed Neural Networks (PINNs) have received much attention recently due to their potential for high-performance computations for complex physical systems, including data-based computing, systems with unknown parameters, and others. The idea of PINNs is to approximate the equations and boundary and initial conditions through a loss function for a neural network. PINNs combine the efficiency of data-based prediction with the accuracy and insights provided by the physical models. However, applications of these methods to predict the long-term evolution of systems with little friction, such as many systems encountered in space exploration, oceanography/climate, and many other fields, need extra care as the errors tend to accumulate, and the results may quickly become unreliable.

We provide a solution to the problem of data-based computation of Hamiltonian systems utilizing symmetry methods. Many Hamiltonian systems with symmetry can be written as a Lie-Poisson system, where the underlying symmetry defines the Poisson bracket. For data-based computing of such systems, we design the Lie-Poisson neural networks (LPNets). We consider the Poisson bracket structure primary and require it to be satisfied exactly, whereas the Hamiltonian, only known from physics, can be satisfied approximately. By design, the method preserves all special functions of the bracket (Casimirs) to machine precision. LPNets yield an efficient and promising computational method for many particular cases, such as rigid body or satellite motion (the case of SO(3) group), Kirchhoff's equations for an underwater vehicle (SE(3) group), and others.

Joint work with Chris Eldred (Sandia National Lab), Francois Gay-Balmaz (CNRS and ENS, France), and Sophia Huraka (U Alberta). The work was partially supported by an NSERC Discovery grant.

Title: Stability analysis of a single-species logistic model with time delay and constant inflow

Speaker: Yasuhiro Takeuchia, Aoyama Gakuin University, Kanagawa, Japan

Date and time:
19 Oct 2023,
11:30am -
12:30pm

Location: DSB C114

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We consider a single-species system with distributed delay and constant inflow. First, we consider the case with time delay in which the constant inflow does not exist. The condition of Hopf bifurcation for the delay order of k = 2 is known, but is not given clearly for the delay order of k ≥ 3. We have obtained the relationship among systems parameters for Hopf bifurcation. Next, we consider the case in which the constant inflow exists. In this case, the positive equilibrium changes to be unstable from being stable first, and return to be stable again by increasing with average time delay T for small intrinsic growth rate r. It is found that there exists important difference between the delay orders k = 2 and k ≥ 3. For k = 2, the equilibrium can be stable for large T and any r(> 0), but for k ≥ 3, the equilibrium is unstable for large T and r .

Title: Convex optimization for statistical mechanics

Speaker: Minjae Cho, Princeton

Date and time:
11 Oct 2023,
2:30pm -
3:30pm

Location: DTB A203

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Abstract: In this talk, we will discuss how convex optimization can be used to study statistical mechanics, with a focus on the example of the lattice Ising model. After introducing the Gibbs measure that describes the system, we will explain how convex optimization produces results that asymptotically converge to the Gibbs measure. If time permits, we will also explore the relationship between Markov chain Monte-Carlo and convex optimization for the Ising model.

Title: Fokas Diagonalization

Speaker: Dave Smith, Yale-NUS

Date and time:
27 Sep 2023,
2:30pm -
3:20pm

Location: DTB A203

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Abstract: We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included.

Title: A walk with relative equilibria from the plane to the sphere S2

Speaker: Ernesto Prerez Chavela, ITAM Mexico

Date and time:
13 Sep 2023,
2:30pm -
3:30pm

Location: ECS 130

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