Date/Time : 10 Jan 2022 13:00
Venue : via Zoom
Speaker : Hadi Daneshmand, Princeton University

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https://nus-sg.zoom.us/j/83955852990?pwd=VUpkT1o2M2pzN01YWWxVTTJ1QmpLQT09

Meeting ID: : 839 5585 2990
Passcode: 158210

Abstract:  Deep neural networks have achieved incredible performance in various domains by automatizing feature extraction with less need for conventional feature engineering. When neurons are well-connected and optimized, their performance goes beyond the human level for image processing. Yet, wiring and optimizing many neurons is a cumbersome engineering task. Over 70 years of research and optimization breakthroughs have advanced the training of deep networks. In this talk, we will cover the main outcomes of these breakthroughs. The talk presents recent studies on batch normalization, the fundamental building of deep neural networks. Leveraging tools from Markov chain theory, we will shed light on the inner workings of batch normalization for deep learning. Then, we will review escaping challenging attractors for neural learning with first-order optimization methods.

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Date/Time : 05 Jan 2022 09:00
Venue : via Zoom
Speaker : Pan Wenyu, University of Chicago

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https://nus-sg.zoom.us/j/84440678768?pwd=RUZ3Y2RCTTB3SStuQWtuOWE0YlNDQT09

Meeting ID: 844 4067 8768
Passcode: 224158

Abstract:  Let $\Gamma$ be a Zariski dense Kleinian Schottky subgroup of $\operatorname{PSL}_2(\mathbb{C})$. Let $\Lambda_{\Gamma}\subset \mathbb{C}$ be its limit set, endowed with a Patterson-Sullivan measure $\mu$. I will present the joint work with Jialun Li and Frédéric Naud in which we showed that the Fourier transform of $\mu$ enjoys polynomial decay.  

For $\operatorname{PSL}_2(\mathbb{R})$, this was proved by Bourgain-Dyatlov.  As a corollary, all limit sets of Zariski dense Kleinian groups have a positive Fourier dimension. Another application is uniform “essential” spectral gap for any finite cover of $\Gamma \backslash \mathbb{H}^3$.  I will outline the main ingredients of the proof and talk about some future problems.

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Date/Time : 13 Dec 2021 09:00
Venue : via Zoom
Speaker : Reza Gheissari, University of California, Berkeley

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https://nus-sg.zoom.us/j/85754837628?pwd=ZytjTUI2TVBtU3l2Y01PM0lpazBNUT09

Meeting ID: 857 5483 7628
Passcode: 359207

Abstract: Since its introduction in 1920, the Ising model has been one of the most studied models of phase transitions in statistical physics. In its low-temperature regime, the model has two thermodynamically stable phases, which, when in contact with each other, form an interface: a random curve in 2D and a random surface in 3D. In this talk, I will survey the rich phenomenology of this interface in 2D and 3D, and describe recent progress in understanding its geometry in various parameter regimes where different surface phenomena and universality classes emerge.

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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 : 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 : 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 : 25 Nov 2021 09:00
Venue : via Zoom
Speaker : Minh Chieu Tran, University of Notre Dame du lac

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https://nus-sg.zoom.us/j/84777353294?pwd=cklXS3NTN2lIbUJxOC9ob245d0JHQT09

Meeting ID: 847 7735 3294
Passcode: 821366

Abstract: Let $G$ be a connected locally compact group with a left Haar measure $\mu$, and let $A,B \subseteq G$ be nonempty and compact. Assume further that $G$ is unimodular, i.e., $\mu$ is also the right Haar measure; this holds, e.g., when $G$ is compact, a nilpotent Lie group, or a semisimple Lie group. In 1964, Kemperman showed that

$$ \mu(AB) \geq \min \{\mu(A)+\mu(B), \mu(G)\} .$$

The inverse Kemperman problem (proposed by Griesmer, Kemperman, and Tao) asks when the equality happens or nearly happens. I will discuss the recent solution of this problem, highlighting the roles played by model theory and descriptive set theory. (Joint with Jinpeng An, Yifan Jing, and Ruixiang Zhang)

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