Date/Time : 05 Apr 2022 16:00
Venue : S17 #04-04 (SR3)
Speaker : John Loftin, Rutgers University

Abstract: Consider a Riemann surface S of genus g at least 2 equipped with a holomorphic cubic differential U. For a given g, such pairs (S,U) can be used to parametrize the Hitchin component of the space of representations of the fundamental group into PSL(3,R), and also the space of convex projective structures on S. This generalizes and extends the space Fuchsian representations into PSL(2,R) (the Teichmuller space of hyperbolic structures), which corresponds to the locus U=0. This parametrization of the Hitchin component is not explicit but involves the Hitchin system of PDEs. For nonzero U, we study the case sU as s approaches infinity. In particular, we show the geometry in this limit can be read off explicitly from U, in terms of an embedding of the universal cover of S into the real building given by the asymptotic cone of the symmetric space SL(3,R)/SO(3). We are able to provide explicit pictures for most triangle groups.

This is joint work with Andrea Tamburelli and Mike Wolf.

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Date/Time : 05 Apr 2022 14:00
Venue : S17 #06-11 (SR6)
Speaker : Huanchen Bao, National university of Singapore

Abstract:  The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) J-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.

We show that the J-totally nonnegative flag variety has a cellular decomposition into totally positive J-Richardson varieties. Moreover, each totally positive J-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive J-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam.

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Date/Time : 01 Apr 2022 10:00
Venue : via Zoom
Speaker : Rafael Peres Da Silva, National University of Singapore

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https://nus-sg.zoom.us/j/9775606531?pwd=cVZ6eTRYS3ZzVUlFNXRuSlhMVjJSUT09

Meeting ID: 977 560 6531
Passcode: 123321

Large-scale cancer omics studies have highlighted the diversity of patient molecular profiles and the importance of leveraging this information to deliver the right drug to the right patient at the right time. Key challenges in learning predictive models for this include the high-dimensionality of omics data and heterogeneity in biological and clinical factors affecting patient response. The use of multi-task learning techniques has been widely explored to address dataset limitations for in vitro drug response models, while domain adaptation (DA) has been employed to extend them to predict in vivo responses. In both of these transfer learning settings, noisy data for some tasks (or domains) can substantially reduce the performance for others compared to single-task (domain) learners, i.e. lead to negative transfer (NT).

This talk introduces TUGDA (ISMB 2021), a novel multi-task unsupervised DA method, that addresses these limitations in a unified framework by quantifying uncertainty in predictors and weighting their influence on shared feature representations. TUGDA’s ability to rely more on predictors with low-uncertainty allowed it to notably reduce cases of NT for in vitro models (94% overall) compared to state-of-the-art methods. For DA to in vivo settings, TUGDA improved over previous methods for patient-derived xenografts (9 out of 14 drugs) as well as patient datasets (significant associations in 9 out of 22 drugs). TUGDA’s ability to avoid NT thus provides a key capability as we try to integrate diverse drug-response datasets to build consistent predictive models with in-vivo utility.

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Date/Time : 31 Mar 2022 09:00
Venue : via Zoom
Speaker : Stefan Dawydiak, University of Toronto

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https://nus-sg.zoom.us/j/89752886605?pwd=Z1V3N2lPbEtodjViZXVmMXRCNGFpZz09

Meeting ID: 897 5288 6605
Passcode: 343140

Abstract:  In the early 1980s Lusztig defined a ring J that serves as a limit of an affine Hecke algebra as the parameter v tends to 0, and showed that J can be realized as a subalgebra of a completion of the affine Hecke algebra, with elements of J being infinite linear combinations of elements of the affine Hecke algebra with Laurent series coefficients. Using work of Braverman-Kazhdan and the Plancherel formula for p-adic groups, we prove that the above coefficients are in fact rational functions whose denominators all divide a fixed polynomial that depends only on the affine Weyl group. We conjecture, and prove in the case that the p-adic group is GL_n, that this polynomial is precisely the Poincare polynomial of the finite Weyl group of G. We will explain a possible application to the representation theory of affine Hecke algebras at root of unity.

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Date/Time : 30 Mar 2022 16:00
Venue : via Zoom
Speaker : Zexing Li, University of Cambridge

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https://nus-sg.zoom.us/j/86113239460?pwd=anRLYzEvaGFQVGVLNVF3dGVRM1pOUT09

Meeting ID: 861 1323 9460
Passcode: 220330

Abstract: The Chern-Simons-Schr\”odinger equation is a nonrelativistic quantum model describing the dynamics of a large number of charged particles in the plane interacting both directly and via a self-generated electromagnetic field. Mathematically, it is a gauged $L^2$-critical Schr\”odinger equation has soliton behavior. Considering the equivariant and self-dual case, Liu-Smith has proved below threshold charge wellposedness via concentration compactness method. In this talk, I will present our result on characterizing dynamics at the threshold charge. This is a joint work with Baoping Liu.

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Date/Time : 30 Mar 2022 09:00
Venue : via Zoom
Speaker : Wei Kuo Chen, University of Minnesota

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https://nus-sg.zoom.us/j/81402485618?pwd=WlBSN2NhM2xXaGlTc1NhTFBqRjRIUT09

Meeting ID: 814 0248 5618
Passcode: 266695

Abstract: The Grothendieck L_p problem is defined as an optimization problem that maximizes the quadratic form of a Gaussian matrix over the unit L_p ball. The p=2 case corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble, while p=\infty this problem is known as the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In this talk, I will describe the limit of this optimization problem for general p and discuss some results on the behavior of the near optimizers along with some open problems. This is based on a joint work with Arnab Sen.

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Date/Time : 30 Mar 2022 16:00
Venue : via Zoom
Speaker : Wu Liuzhen,

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https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09

Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

Abstract: The continuum function is a key and long-studied object inside set theory. We will survey the study on the behavior of continuum function in presence of strongly compact cardinals. We will also introduce some major research problems in this area. Finally, We discuss our recent work on forcing continuum function of some special pattern.

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Date/Time : 28 Mar 2022 10:00
Venue : via Zoom
Speaker : Guanglian Li, University of Hong Kong

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https://nus-sg.zoom.us/j/87674182782?pwd=dEtkOWxldGl4RFVFNkJpV2syalJhUT09

Meeting ID: 876 7418 2782
Passcode: 684943

Abstract: I will present in this talk the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm recently proposed in [Li and Hu, J. Comput. Phys., 2021] to efficiently solve subdiffusion equations with heterogeneous coefficients in long time. This algorithm combines the advantages of multiscale methods that can effectively deal with heterogeneity in the spatial domain, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. Compared with the previous work for parabolic problem, the main challenge in both the analysis and simulation arises from the nonlocality of the fractional derivative. To conquer this obstacle, an auxiliary problem is constructed on each coarse temporal subdomain to uncouple the temporal variable completely. In this manner, the approximation properties of the correction operator is proved. In addition, a new summation of exponential sums is derived to generate single-step time stepping scheme, with the number of terms of O(|log τ_f|) independent of final time. Here, τ_f is the fine-scale time step size. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step size and the fine-scale time step size. Several numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly.

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Date/Time : 25 Mar 2022 09:00
Venue : via Zoom
Speaker : Chao Li, Columbia University

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https://nus-sg.zoom.us/j/87647484260?pwd=Zys0M211ekFkeE9OZlRCOHcvS0NLZz09

Meeting ID: 876 4748 4260
Passcode: 511096

Abstract: Can an integer n be represented as a sum of two squares n=x^2+y^2? If so, how many different representations are there? We begin with the answers to these classical questions due to Fermat and Jacobi and put it in the modern perspective of the Siegel-Weil formula. After illustrating Kudla’s influential program on geometric and arithmetic generalizations using the example of modular curves, we report on recent theorems establishing arithmetic Siegel-Weil formulas and discuss applications including BSD-type conjectures for higher dimensional varieties.

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