Date/Time : 29 Nov 2023 15:00
Venue : S17 #06-11 (SR6)
Speaker : Dawei Shen, Sorbonne University

The global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou-Klainerman in 1993. In 2007, Bieri has extended the result of Christodoulou-Klainerman under lower decay and regularity assumptions on the initial data. In this talk, we will introduce a recent work, which extends the result of Bieri to minimal decay assumptions.

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Date/Time : 28 Nov 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Morris Ang, Columbia University

In the first half of the talk, I will introduce the central objects in random conformal geometry. Schramm-Loewner evolution (SLE) is a random planar curve describing the scaling limits of interfaces in statistical physics models (e.g. percolation, Ising model). Liouville quantum gravity (LQG) is a random 2D surface arising as the scaling limit of random planar maps. These fractal geometries have deep connections to bosonic string theory and conformal field theories. There are many powerful theorems which say that, roughly speaking, cutting LQG by independent SLE gives two independent LQG surfaces. In the second half, I present some new developments and their applications.

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Date/Time : 27 Nov 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Lai Lexiao, Columbia University

Lyapunov stability is an important notion for dynamical systems, which has been absent in the optimization literature. When studying optimization methods, the goal has always been proving convergence. However, convergence to critical points could sometimes be unrealistic or undesirable in practice. In this talk, I will introduce the notions of Lyapunov stability for optimization methods. I will then describe my works on characterizing the local and global stability of first-order methods. My results apply to problems arising in data science. In particular, I provide a guarantee of the SGD optimizer for training neural networks.

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Date/Time : 24 Nov 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Fiona Murnaghan, University of Toronto

Let G be a connected reductive group defined over a nonarchimedean local field of odd residue characteristic. Let H be the group of fixed points of an involution of G. A representation of G is said to be H-distinguished if there exists a nonzero H-invariant linear functional on the space of the representation. The H-relatively supercuspidal representations of G are the H-distinguished representations of G whose generalized matrix coefficients are compactly supported modulo HZ, where Z is the centre of G. We will describe results giving a method for constructing H-relatively supercuspidal representations of G.

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Date/Time : 22 Nov 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Anant Raj, University of Illinois Urbana-Champaign

In this talk, we delve into the intricate relationships between heavy-tailed distributions, generalization error, and algorithmic stability in the realm of noisy stochastic gradient descent. Recent research has illustrated the emergence of heavy tails in stochastic optimization and their intriguing links to generalization error. However, these studies often relied on challenging topological and statistical assumptions. Empirical evidence has further challenged existing theory, suggesting that the relationship between heavy tails and generalization is not always monotonic. In response, we introduce novel insights, exploring the relationship between tail behavior and generalization properties through the lens of algorithmic stability. Our analysis reveals that the stability of stochastic gradient descent (SGD) varies based on how we measure it, leading to interesting conclusions about its behavior.Expanding upon these findings, we extend the scope to a broader class of objective functions, including non-convex ones. Leveraging Wasserstein stability bounds for heavy-tailed stochastic processes, our research sheds light on the non-monotonic connection between generalization error and heavy tails, offering a more comprehensive perspective.Additionally, we introduce a unified approach for proving Wasserstein stability bounds in stochastic optimization, emphasizing time-uniform stability and its role in various scenarios, including convex and non-convex losses. Our approach is versatile and applicable to popular optimizers, highlighting the importance of ergodicity.

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Date/Time : 22 Nov 2023 11:00
Venue : S17 #04-05 (SR2)
Speaker : Eva Lütkebohmert, University of Freiburg

Sovereign loan portfolios of Multilateral Development Banks typically consist of only a small number of borrowers and hence are heavily exposed to single name concentration risk. Based on realistic MDB portfolios constructed from publicly available data, this paper quantifies the magnitude of the exposure to name concentration risk using exact Monte Carlo simulations. In comparing the exact adjustment for name concentration risk to its analytic approximation as currently applied by major rating agencies, we further investigate whether current capital adequacy frameworks for MDBs are overly conservative.

(This is joint work with Julian Sester and Hongyi Shen)

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Date/Time : 22 Nov 2023 16:30
Venue : S17 #04-05 (SR2)
Speaker : Wu Hanxin, National University of Singapore

In this talk I purpose a diffuse-interface model for dynamic soft wetting problem with Shuttleworth effect. The model consists of Cahn-Hilliard-Navier-Stokes equations for two-phase fluid on a moving domain and linear elastic equation for soft substrate, and they are coupled on the free fluid-solid interface. An efficient numerical scheme is purposed by combining a monolith fluid-structure-interaction solver and linear convex splitting for Cahn-Hilliard equation on the moving domain. Theorems regarding energy-preserving property of the scheme are proved. We compare numerical results for moving contact line on soft and rigid substrate.

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Date/Time : 20 Nov 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Julian Sester, National University of Singapore

In this talk I introduce a general framework for Markov decision problems (MDPs) under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle, we obtain a local-to-global paradigm, namely solving a local, that is, a one timestep robust optimization problem leads to an optimizer of the global (i.e., infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure.

Based on the theoretical results, we are able to derive a novel Q-learning algorithm to solve numerically the associated robust MDPs where the corresponding ambiguity set of transition probabilities for the underlying  Markov decision process is a Wasserstein ball around a (possibly estimated) reference measure. Moreover, I will discuss two applications of the presented setting to problems in quantitative finance.

First, we apply this framework to portfolio optimization involving data of the S&P 500 where it turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.

Second, we study the optimal market-making problem in order-driven electronic markets while explicitly accounting for model uncertainty.

(The talk is based on joint works with Ying Chen, Ariel Neufeld, Mario Sikic, Hoang Hai Tran and Yijiong Zhang)

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Date/Time : 17 Nov 2023 16:15
Venue : via Zoom
Speaker : Chen Ying, National University of Singapore

Join Zoom Meeting
https://nus-sg.zoom.us/j/83899499647?pwd=U2EvRVBPZXpxeXNmV3lsVi95Z2FTQT09

Meeting ID: 838 9949 9647
Passcode: math

In the digital era, complex data has emerged across various disciplines, characterized by high dimensions, nonlinear temporal and spatial dependencies, and irregular dynamic changes (non-stationarity). Understanding and effectively analyzing such complex temporal structures and achieving high-precision forecasting have become pressing challenges. Additionally, comprehending the stochastic behaviors of forecasting is paramount for executives and managers, aiding them in making resilient and robust decisions in real-life scenarios. In my presentation, I will introduce my research on advancing time series modeling and methodologies. These approaches leverage the power of data science, machine learning, and deep neural networks, all tailored to address the complexities of modern data. Some of the innovative techniques I will discuss include functional neural tangent kernel estimators, mixed-frequency spatio-temporal models, and deep switching state space models. Together, these methods offer insights and tools for tackling the challenges posed by high-dimensional, nonlinear, and nonstationary data in the contemporary landscape.

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