Date/Time : 10 Dec 2021 10:00
Venue : via Zoom
Speaker : Xu Min, Carnegie Mellon University

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Meeting ID: 977 560 6531
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The cell is the basic structural and functional unit of all living organisms. Understanding how cells function is fundamental to life science. Macromolecules are nano-machines inside cells that govern the cellular processes. To fully understand such processes, it is necessary to know the native structures and spatial organizations of macromolecules inside single cells, and their interactions with other subcellular components. Such information has been extremely difficult to obtain due to a lack of suitable data acquisition techniques. The recent revolution of Cryo-electron tomography (cryo-ET) 3D imaging technology has made collecting such information possible. Cryo-ET captures a 3D image of a single cell’s subcellular structures at sub-molecular resolution and in a near-native state. It provides unprecedented opportunities for systematically studying the native spatial organization of subcellular structures, especially macromolecules. However, cryo-ET has a high degree of structural complexity and imaging limits,such as high structural diversity and crowding, low signal-to-noise ratio, and missing values. These have made the automated systematic analysis of such images extremely difficult. Since 2008, we have been developing image analysis methods to address this challenge. In particular, we focus on systematic recognition and recovery of the structures of large numbers (millions) of macromolecules captured by cryo-ET, without relying on external structural knowledge. To do so, we have developed different cryo-ET image registration, classification, segmentation restoration techniques. Our effort is a key step for systematic analysis of macromolecules’ structures and spatial organizations inside single cells captured by cryo-ET.

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Date/Time : 09 Dec 2021 09:00
Venue : via Zoom
Speaker : Zhong Yiqiao, Joe, Stanford University

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https://nus-sg.zoom.us/j/82977751761?pwd=ZXowNm93bXE3RTE5NVd0bVBPSEhPUT09

Meeting ID: 829 7775 1761
Password: 717960

Abstract:  A mystery of modern neural networks is their surprising generalization power in overparametrized regime: they comprise so many parameters that they can interpolate the training set, even if actual labels are replaced by purely random ones; despite this, they achieve good prediction error on unseen data.

To demystify the above phenomena, we focus on two-layer neural networks in the neural tangent (NT) regime. Under a simple data model where n inputs are d-dimensional isotropic vectors and there are N hidden neurons, we show that as soon as Nd >> n, the minimum eigenvalue of the empirical NT kernel is bounded away from zero, and therefore the network can exactly interpolate arbitrary labels.

Next, we study the generalization error of NT ridge regression (including min-norm interpolation). We show that in the same overparametrization regime Nd >> n, in terms of generalization errors, NT ridge regression is well approximated by kernel ridge regression (infinite-width kernel), which is further approximated by polynomial ridge regression. A surprising phenomenon is the “self-induced” regularization due to the high-degree components of the activation function.

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Date/Time : 08 Dec 2021 09:00
Venue : via Zoom
Speaker : Konstantin Matetski, Columbia University

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https://nus-sg.zoom.us/j/89132660362?pwd=SHB5bkRwVU9UUDRKVUd0UzY0Vk02QT09

Meeting ID: 891 3266 0362
Passcode: 880312

The KPZ universality class includes random growing interfaces, which, after rescaling, are conjectured to converge to the KPZ fixed point. The latter is a Markov process, which has been characterized through the exact solution of TASEP, a particular model in the class. The KPZ equation plays a special role and is conjectured to be the only model connecting the Edwards-Wilkinson (Gaussian) and the KPZ fixed points. In the talk, I will introduce the KPZ fixed point and review recent work on the KPZ universality done by myself and my collaborators.

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Date/Time : 06 Dec 2021 14:30
Venue : via Zoom
Speaker : David Hansen, Max Planck Institute for Mathematics

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https://nus-sg.zoom.us/j/87960829555?pwd=c0ZsRGJXajhDVmI1bnA1WU4vNXBVUT09

Meeting ID: 879 6082 9555
Passcode: 145729

Abstract: Local Shimura varieties are non-archimedean analytic spaces analogous to Shimura varieties, whose cohomology is expected to realize (in a precise sense) both the local Langlands correspondence and the local Jacquet-Langlands correspondence. I’ll give a gentle introduction to the theory of local Shimura varieties, and explain some results that can be proven about their cohomology using current technology. Some of this material is joint work with Tasho Kaletha and Jared Weinstein.

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Date/Time : 06 Dec 2021 09:00
Venue : via Zoom
Speaker : Yuhua Zhu, Stanford University

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https://nus-sg.zoom.us/j/87958362587?pwd=OThycThnUXNUZXcxQzBndlUxaFNVdz09

Meeting ID: 879 5836 2587
Passcode: 628767

Abstract: As the continuous limit of many discretized algorithms, PDEs can provide a qualitative description of algorithm’s behavior and give principled theoretical insight into many mysteries in machine learning. In this talk, I will give a theoretical interpretation of three machine learning algorithms using Fokker-Planck (FP) equations. In the first one, we provide a mathematically rigorous explanation of why resampling outperforms reweighting in correcting biased data when stochastic gradient-type algorithms are used in training. In the second one, we propose a new method to alleviate the double sampling problem in model-free reinforcement learning, where the FP equation is used to do error analysis for the algorithm. In the last one, inspired by an interactive particle system whose mean-field limit is a non-linear FP equation, we develop an efficient gradient-free method that finds the global minimum exponentially fast.

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Date/Time : 02 Dec 2021 13:00
Venue : via Zoom
Speaker : Yang Lei, Sichuan University

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https://nus-sg.zoom.us/j/89266647804?pwd=K2p0K2V3R1VMcEQxdjlKMFFFYlJEQT09

Meeting ID: 892 6664 7804
Passcode: 445251

Abstract: In this talk, we will prove the convergence part of Khintchine’s theorem on non-degenerate manifolds. This confirms a conjecture of Kleinbock and Margulis in 1998. Our approach uses geometric and dynamical ideas together with a new technique of `major and minor arcs’. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near `major arcs’ and give explicit exponentially small bounds for the measure of `minor arcs’. This is joint work with Victor Beresnevich.

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Date/Time : 01 Dec 2021 16:00
Venue : via Zoom
Speaker : Pierre-Francois Rodriguez, Imperial College, London

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https://nus-sg.zoom.us/j/85694125031?pwd=eGJuMWk5SXRCS2hWc1JITXpRWmdRUT09

Meeting ID: 856 9412 5031
Passcode: 918464

Abstract: I will report on recent progress regarding the near-critical behavior of certain statistical physics models in dimension 3. Our results deal with the critical scaling exponents associated to the second-order phase transition for two percolation problems involving the Gaussian free field in 3D. In one case, they determine a unique “fixed point” corresponding to the transition, which is proved to obey one of several scaling relations. Such laws are classically conjectured to hold by physicists on the grounds of a corresponding scaling ansatz.

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Date/Time : 30 Nov 2021 09:00
Venue : via Zoom
Speaker : Goh Jun Le, University of Wisconsin-Madison

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https://nus-sg.zoom.us/j/83885192152?pwd=a2RDZTFvalVMM3VFT3lGazhyV2doQT09

Meeting ID: 838 8519 2152
Passcode: 769950

Abstract: Effective mathematics seeks to measure the algorithmic content of mathematical objects. One framework for doing so is the Turing degrees (Turing 1939), which allows us to measure the complexity of a variety of objects such as real numbers, sets of words in groups, and countable graphs.

However, the Turing degrees are inadequate for measuring the complexity of continuous real-valued functions (Miller 2004). Such objects (and more) can be measured in the broader structure of the enumeration degrees.

We shall discuss two lines of research in the enumeration degrees: (1) relationships between its subclasses, of which there are many, arising from various areas of mathematics; and (2) the enumeration degrees as a partial order, with the goal of algorithmically deciding all order-theoretic questions with a particular syntactic form.

The new results in this talk are joint work with Kalimullin, Miller, Soskova and Lempp, Ng, Soskova.

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Date/Time : 26 Nov 2021 10:00
Venue : via Zoom
Speaker : Wenyu Pan, University of Chicago

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https://nus-sg.zoom.us/j/82434099670?pwd=VlhicUs1SlZPd0IrTzZ5d0NrMm1oQT09

Meeting ID: 824 3409 9670
Passcode: 772471

Abstract: Let  Hⁿ be the hyperbolic n-space and Γ be a geometrically finite discrete subgroup in Isom + (Hⁿ) with parabolic elements. We investigate whether the geodesic flow (resp. the frame flow) over the unit tangent bundle T¹ (Γ\Hⁿ) (resp. the frame bundle F(Γ\Hⁿ)) mixes exponentially. This result has many applications, including spectral theory, orbit counting, equidistribution, prime geodesic theorems, etc. 

I will start with a survey of the past results and related problems on this topic. Then I will present the joint work with Jialun Li in which we show that the geodesic flow mixes exponentially. I will describe some ingredients in the proof. If there is time, I will also discuss the exponential mixing of the frame flow.

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