Date/Time : 11 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Heng Yang, Harvard University

Many fundamental estimation and decision-making problems in robotics are formulated as non-convex mathematical optimization problems. Decades of research have produced numerous algorithms to efficiently solve these problems, yet offering few performance guarantees. In this talk, I aim to advocate using convex semidefinite relaxations to design a new paradigm of algorithms that work out of the box and offer strong performance certificates.

I first briefly introduce the celebrated Lasserre’s hierarchy of moment and sums-of-squares (SOS) relaxations, which relaxes polynomial optimization and decision problems as a hierarchy of convex semidefinite programs (SDP). I then present three applications of this hierarchy in robotics. In the first application, we apply the hierarchy to solve outlier-robust estimation problems in robot perception and show that SDP relaxations provide certificates of optimality. In the second application, we combine this hierarchy with conformal prediction from statistics and set-membership estimation from control theory and demonstrate such relaxations provide certificates of uncertainty. In the third application, we revisit trajectory optimization problems arising in nonlinear optimal control and show that, by exploiting correlative sparsity, SDP relaxations produce near-optimal control trajectories. I end the talk by discussing computational challenges of this framework in our applications and opportunities to tackle them.

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Date/Time : 05 Dec 2023 15:30
Venue : S17 #04-04 (SR3)
Speaker : Lu Qi, Princeton University

In this talk, I will discuss some convexity properties of multiplicities of filtrations on a local ring. In particular, the multiplicity function is convex along geodesics. As a major application, this gives a new proof of a theorem due to Xu and Zhuang on the uniqueness of normalized volume minimizers. In order to characterize strict convexity, we introduce the notion of saturation of a filtration, which turns out to be useful in other settings. For example, it allows us to generalizes a theorem by Rees on characterization of when two filtrations have equal multiplicities. It also allows us to introduce a metric on the space of filtrations. This talk is based on joint work with Harold Blum and Yuchen Liu and some ongoing work.

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Date/Time : 01 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Min Dai, Hong Kong Polytechnic University

We develop a reinforcement learning method to learn an optimal trading strategy in the presence of transaction costs. Using a connection between singular control and a Dynkin game for portfolio choice with transaction costs, we learn the value function and optimal policy of an associated randomized Dynkin game, where a regularization term is incorporated to encourage exploration. We show that the policy efficiently approximates the optimal trading strategy. We design a reinforcement learning algorithm, which is demonstrated by numerical results. This work is jointly with Yuchao Dong.

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Date/Time : 01 Dec 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Chuijia, Technion - Israel Institute of Technology

Kazhdan-Lusztig polynomials play a fundamental role in various Lie theoretical subjects such as Hecke algebras, category O, etc. It is expected that these polynomials only depend on the combinatorial nature of the associated Bruhat intervals. In a recent work of Williamson and his collaborators, they obtain a hypercube decomposition of q derivatives of Kazhdan-Lusztig polynomials via deep geometric machinery, which serves as an important step towards this “combinatorial invariance conjecture”. In this talk, we will try to reinterpret their results from the point of view of parabolic recursions and discuss some applications. This is based on a joint work with Maxim Gurevich.

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Date/Time : 29 Nov 2023 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jaemin Park, University of Basel

In this talk, we study stationary solutions to the 2D incompressible Euler equations in the whole plane. It is well-known that any radial vorticity is stationary. For compactly supported vorticity, it is more difficult to see whether a stationary solution has to be radial. In the case where the vorticity is non-negative, it has been shown that any stationary solution has be radial. By allowing the vorticity to change the sign, we prove that there exist non-radial stationary patch-type solutions. We construct patch-type solutions whose kinetic energy is infinite or finite. For the finite energy case, it turns out that a construction of a stationary solution with compactly supported velocity is possible. This is based on the joint work with Javier Gomez-Serrano and Jia Shi.

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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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