# 세미나 및 콜로퀴엄

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Backward stochastic differential equation (BSDE) is a generalization of martingale representation theorem and it has been widely used for financial derivative pricing and stochastic optimization. Traditionally, most of well-posedness result of BSDE were based on contraction mapping theorem on the space of stochastic processes.

In our work, we were able to transform BSDEs into fixed point problems in the space of Lp random variables. The simplicity of our framework enables us to apply various kind of fixed point theorems which have not been tried in previous literature. In particular, this enables to remove infinite dimensionality arise from time and we were able to use white noise analysis to use topological fixed point theorems. As a result, we were able to generalize previous well-posedness results: e.g. time-delayed type, mean-field type, multidimensional super-linear type.

The talk is aimed for those who are not familiar with BSDE and it is based on BSE's, BSDE's and fixed point theorem (joint work with Patrick Cheridito)

I will explain the construction of coordinates on the moduli spaces of maximal globally hyperbolic 3 dimensional spacetimes generalising the shear coordinates on the Teichmüller space. I will then discuss the Poisson structure and the mapping class group action on these moduli spaces. These constructions are motivated by the problem of quantum gravity, so I’ll also give a brief description of some of the ideas behind quantisation and how they apply in the context of 3d gravity. This talk is based on joint work with Catherine Meusburger and ongoing work with Hyun Kyu Kim.

What is the probability that in a telecommunication system, an atypically large proportion of users experiences a bad quality of service induced by lack of capacity? Although today's wireless networks exhibit nodes embedded in the Euclidean plane, large deviations have so far predominantly been analyzed in mean-field approximations. We explain how to prove a large deviation principle for a random spatial telecommunication network featuring capacity-constrained relays. This talk is based on joint work with Benedikt Jahnel and Robert Patterson (WIAS Berlin)

In this talk, I will talk on transverse stability line solitary waves for KP-II and the Benney Luke equation. Both equations are long wave long wave models for 3D water waves with weak surface tension. I will explain that the resonant continuous eigenmodes which we can find in an exponentially weighted space have to do with modulation of line solitary waves.

We study the dynamical properties of the topological generalized beta transformations, which generalizes the concept of generalized beta transformations defined by Gora. In particular, we generalize the result on admissible sequence for unimodular maps to the case of generalized beta maps, and also study the properties of the topological entropy and its Galois conjugates, generalizing some results by Tiozzo. This talk represents an ongoing collaboration with Diana Davis, Kathryn Lindsey and Harry Bray.

A novel high-order numerical scheme is proposed to solve the shallow water equations (SWEs) on arbitrary rotating curved surfaces. Based on the method of moving frames (MMF), the proposed scheme not only has the smallest dimensionality of two in space, but also does not require either of (i) metric tensors, (ii) composite meshes, or (iii) the surrounding space. The MMF-SWE formulation is numerically discretized using the discontinuous Galerkin method of arbitrary polynomial order p in space and an explicit Runge-Kutta scheme in time. In this talk, we start with the fundamental concepts of the innovational moving frames for Riemannian geometry developed by the famous French mathematician Elie Cartan in the early 20th century. Then, we discuss its adaptation and validity in the discrete space for scientific computing by overviewing the past works on conservational laws and diffusion equations. Applications to SWEs will be explained in details in views of algorithmic novelty to overcome the classical issues of PDEs on the closed surface such as geometric singularities and rotational effects. Results of six standard tests on the sphere will be displayed with the optimal order of convergence of p+1. Also, its general applicability and stability on arbitrary rotating surfaces such as ellipsoid, irregular, and non-convex surfaces will be demonstrated.

We proved a Kazhdan type theorem for the canonical metrics of finite graphs. Namely, we show that the canonical metric of finite normal coverings of the graph converges when the covering converges, and the limit depends only on the limit of the coverings. We also generalize the argument to higher dimensional simplicial complexes. The proof is mostly based on an analogous argument in the case of Riemann surfaces and Lück's approximation theorem for L^2 cohomology. This is joint work with Farbod Shokrieh.

High-dimensionality is one of the major challenges in stochastic simulation of realistic physical systems. The most appropriate numerical scheme needs to balance accuracy and computational complexity, and it also needs to address issues such as multiple scales, lack of regularity, and long-term integration.

In this talk, I will review state-of-the-art numerical techniques for high-dimensional systems including low-rank tensor approximation, sparse grid collocation, and ANOVA decomposition. The presented numerical methods are tested and compared in the joint response-excitation PDF equation that generalizes the existing PDF equations and enables us to do kinetic simulations with non-Gaussian colored noise. The alternative to the numerical approach, I will discuss dimension reduction techniques such as Mori-Zwanzig approach and moment closures that can obtain reduced order equations in lower dimensions. I will also present numerical results including stochastic Burgers equation and Lorenz-96 system.

For a reliable simulation of systems subject to noise, it is necessary to characterize the noise properly and develop efficient algorithms.

In the first part of this talk, I will present a numerical technique to model and simulate multiple correlated random processes. The method finds the appropriate expansion for each correlated random process by generalizing the Karhunen-Loeve (K-L) expansion, in particular, by releasing the bi-orthogonal condition of the K-L expansion. I will address the convergence and computational efficiency, in addition to some explicit formulae and analytical results.

In the second part, I will present an adaptive reduced basis method that enables an efficient simulation of parameterized stochastic PDEs. The method is employed by using an adaptive ANOVA and probabilistic collocation method to automatically identify the important dimensions and appropriate resolution in each dimension. The effectiveness of the method is demonstrated in anisotropic high-dimensional stochastic PDEs.

We provide a combinatorial characterization of all testable properties of k-graphs (i.e. k-uniform hypergraphs).

Here, a k-graph property 𝒫 is testable if there is a randomized algorithm which quickly distinguishes with high probability between k-graphs that satisfy 𝒫 and those that are far from satisfying 𝒫. For the 2-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. This is joint work with Felix Joos, Deryk Osthus and Daniela Kühn.

Based on the quantum white noise theory, we introduce the new concept of quantum white noise derivatives of white noise operators. As applications we solve implementation problems for the canonical commutation relation and for a quantum extension of Girsanov transformation.

‣ Date & Time : 7/21, 10:00~11:00, 11:10~12:10

In the second lecture we continue the discussion of orthogonal polynomials, now dealing with multi-variable functions. By introducing creation, annihilation, and preservation operators for the multi-variables, we construct again an interacting Fock space (IFS). Thereby we extend the theory of orthogonal polynomials in the 1-dimensional space to that in the multi-dimensional space. As a byproduct we show the relationship between the support of the measure and the deficiency rank of the form generator, which appears in the construction of the IFS. We finish with some open problems. This lecture is based on the joint work with A. Dhahri (Chungbuk) and N. Obata (Tohoku).

- Date & Time : 7/21, 13:30~14:30, 15:00~16:00

We start with the standard construction of generalized white noise functionals as infinite dimensional distributions and we study the analytic characterization theorem for S-transform of generalized white noise functionals. Then we study basic concepts and results on white noise operators which is necessary for the study of quantum white noise calculus. The analytic characterization of operator symbols and the Fock expansion theorem are of particular importance.

In the first part of the lectures we will discuss 1-dimensional orthogonal polynomials. Main topics that will be discussed are the followings.

- Three-term recurrence relation and the Jacobi coefficients

- Examples

- Graph spectrum

- Interacting Fock spaces

- Accardi-Bozejko formula

The main reference for this lecture is <Quantum probability and spectral analysis of graphs>, Springer, 2007, by A. Hora and N. Obata.

‣ Date & Time : 7/20, 13:30~14:30, 15:00~16:00

I will discuss some aspects of the algebraic structure of finitely generated groups of diffeomorphisms of compact one-manifolds. In particular, we show that if G is not virtually metabelian then (G x Z)*Z cannot act faithfully by C^2 diffeomorphisms on a compact one-manifold. Among the consequences of this result is a completion of the classification of right-angled Artin groups which admit faithful C^{\infty} actions on the circle, a program initiated together with H. Baik and S. Kim. This represents joint work with S. Kim.

3-d printing gives us unprecedented ability to tailor microstructures to achieve desired goals. From the mechanics perspective one would like, for example, to know how to design structures that guide stress, in the same way that conducting fibers are good for guiding current. In that context the natural question is: what are the possible pairs of (average stress, average strain) that can exist in the material. A more grand question is: what are the possible effective elasticity tensors that can be achieved by structuring a material with known moduli. This is a highly non-trivial problem: in 3-dimensions elasticity tensors have 18 invariants and even an object as simple as a distorted hypercube in 18 dimensions requires about 4.7 million numbers to specify it. Here we review some of the progress that has been made on this question. This is joint work with Marc Briane and Davit Harutyunyan

The behavior of shape memory materials, the response of composites, and inverse problems (where one seeks to determine what is inside a body from boundary measurements) would at first sight seem to have little in common. However there are unifying mathematical themes that underlie them all. Finding composites that have the best properties, for design applications, often reduces to a type of energy minimization problem with a non-linear energy, even though the underlying physical equations may be linear. The same sort of energy minimization problems govern the response of shape memory materials. Similarly, the response of inhomogenous bodies, as governed by the appropriately defined Dirichlet to Neumann map, is similar in many respects to an effective tensor in the theory of composites, and this connection can be made more mathematically explicit. As a result of these connections, mathematical tools developed in one area can be applied to problems in the other areas.

In this talk I will introduce tensor networks for breaking the curse-of-dimensionality in large scale optimizaition problems. I will mainly focus on the tensor train (TT) format, which is one of the simplest tensor networks. I will show how large-scale optimization problems,

which are intractable by standard numerical methods, can be solved by using the concept of tensorization and TT format. In addition, I will discuss several state-of-the art numerical schemes for tensor networks including truncated iteration scheme, alternating linear scheme, and Riemannian optimization approach.

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E6-1 Room 1409
Discrete Math
권창현 (Industrial and Management Systems Engineering, Uni)
On the Price of Satisficing in Network User Equilibria

When network users are satisficing decision-makers, the resulting traffic pattern attains a satisficing user equilibrium, which may deviate from the (perfectly rational) user equilibrium. In a satisficing user equilibrium traffic pattern, the total system travel time can be worse than in the case of the PRUE. We show how bad the worst-case satisficing user equilibrium traffic pattern can be, compared to the perfectly rational user equilibrium. We call the ratio between the total system travel times of the two traffic patterns the price of satisficing, for which we provide an analytical bound. Using the sensitivity analysis for variational inequalities, we propose a numerical method to quantify the price of satisficing for any given network instance.

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E6-1 Room 1409
Discrete Math
Edgardo Roldán-Pensado (Instituto de Matemáticas, UNAM, Mexico)
On the colourful Helly theorem

Let F be a family of convex sets in R^d coloured using d+1 colours. Lovasz’s Colourful Helly Theorem states that if any colourful subfamily of convex sets is intersecting, then one of the monochromatic families is intersecting. We study what happens with the rest of the families.

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E6-1 Room 1409
Discrete Math
이의웅 (Computer Science Department, Carnegie Mellon Unive)
FPT Approximation Algorithms for Graph Problems

Approximation algorithms and fixed-parameter tractable (FPT) algorithms have been two major ways to deal with NP-hardness of combinatorial optimization problems. The notion of FPT approximation can be naturally defined, and it is getting significant attention recently. Starting from gentle introductions to approximation algorithms and FPT algorithms, I will talk about my three recent results on FPT approximability.

– Given a graph G = (V, E) and an integer k, we study k-Vertex Separator, where the goal is to remove the minimum number of vertices such that each connected component in the resulting graph has at most k vertices. We give an O(log k)-FPT approximation algorithm for k-Vertex Separator. Our result improves the best previous graph partitioning algorithms.

– We also study k-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. We present an O(log k)-FPT approximation algorithm for k-Path Transversal. There was no nontrivial approximation algorithm for k > 4 before this work.

– Finally, k-cut is the problem where we want to remove the minimum number of edges such that the graph has at least k connected components. We give a (2 – ε)-FPT approximation algorithm for some epsilon > 0, improving upon a (non-FPT) 2-approximation.

The third result is joint work with Anupam Gupta and Jason Li.