Date/Time : 14 Dec 2023 16:20
Venue : S17 #04-06 (SR1)
Speaker : Xiao Manyu, Northwestern Polytechnical University

The common issue of surrogate models is to make good use of sampling data. In theory, the higher the fidelity of sampling data obtained, the more accurate the approximation model built. However, in practical engineering problems, high-fidelity data may be less available, and such data may also be computationally expensive. On
the contrary, we often obtain low-fidelity data under certain simplifications. Although low-fidelity data is less accurate, such data still contains much information about the real system. So, combining both high- and low-fidelity data in the construction of a surrogate model may lead to better representation of the physical phenomena.

In this work, the original principle of Multi-Fidelity Proper Orthogonal Decomposition will consist in projecting the high-dimensional system of equations into a low-dimensional space, with reduced basis vectors. These vectors are updated with incremental “on-the-fly” merges between costly high-fidelity simulation snapshots and more rapid lower-fidelity vectors. Successful achievements of this work will lead to potential benefits in the fields of large-scale mechanical engineering.

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Date/Time : 13 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Xin Sun, Peking University and University of Pennsylvania

Smirnov’s proof of Cardy’s formula for percolation on the triangular lattice leads to a discrete approximation of conformal maps, which we call the Cardy-Smirnov embedding. Under this embedding, Holden and I proved that the uniform triangulation converge to a continuum random geometry called pure Liouville quantum gravity. There is a variant of the Gaussian free field governing the random geometry, which is an important example of conformal field theory called Liouville CFT. A key motivation for understanding Liouville quantum gravity rigorously is its application to the evaluation of scaling exponents and dimensions for 2D critical systems such as percolation. Recently, with Nolin, Qian and Zhuang, we used this idea and the integrable structure of Liouville CFT to derive a scaling exponent for planar percolation called the backbone exponent, which was unknown for several decades.

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Date/Time : 12 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Maxim Raginsky, University of Illinois at Urbana-Champaign

The majorizing measure theorem of Fernique and Talagrand is a fundamental result in the theory of random processes. It relates the boundedness of random processes indexed by elements of a metric space to complexity measures arising from certain multiscale combinatorial structures, such as packing and covering trees. In this talk, based on joint work with Yifeng Chu, I will build on the ideas first outlined in a little-noticed preprint of Andreas Maurer to present an information-theoretic perspective on the majorizing measure theorem, according to which the boundedness of random processes is phrased in terms of the existence of efficient variable-length codes for the elements of the indexing metric space. I will close by discussing the relevance of these results to the analysis of statistical performance of machine learning algorithms through the lens of data compression and the Minimum Description Length principle.

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Date/Time : 12 Dec 2023 16:30
Venue : S17 #04-04 (SR3)
Speaker : Lucas Mann, University of Muenster

Based on classical constructions in algebraic topology and algebraic geometry, we introduce abstract 6-functor formalisms, which are a form of “abstract cohomology theories”. It turns out that the abstract theory of 6-functor formalisms is surprisingly rich and has applications in non-geometric fields as well. As an example we provide new tools for studying smooth representations of locally profinite groups — in particular in natural characteristic, where many of the classical representation theoretic methods fail. An application of the theory is a conceptual construction of an anti-involution on the (derived) Iwahori-Hecke algebra, which previously could only be constructed on the associated Ext-algebra using explicit computations.

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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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