Date/Time : 16 Dec 2021 09:00
Venue : via Zoom
Speaker : Guillaume Remy, Columbia University

Join Zoom Meeting
https://nus-sg.zoom.us/j/82279401256?pwd=ZTg0M3dueXVNaUUrMzlIcTlsUUlLQT09

Meeting ID: 822 7940 1256
Passcode: 044547

Abstract: Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced in physics by A. Polyakov to describe a canonical random 2d surface. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will take a look at the integrable structure of Liouville CFT on a domain with boundary by proving exact formulas for its correlation functions. We will also discuss the connection with the conformal blocks of CFT which are fundamental functions determined by conformal invariance that underlie the exact solvability of CFT. Based on joint works with Morris Ang, Promit Ghosal, Xin Sun, Yi Sun and Tunan Zhu.

Add to calendar :

Date/Time : 15 Dec 2021 09:00
Venue : via Zoom
Speaker : Souvik Dhara, MIT

Join Zoom Meeting
https://nus-sg.zoom.us/j/85163689192?pwd=cXdBOE9oa2NCU2tPZUFlMWZTV2s4Zz09

Meeting ID: 851 6368 9192
Passcode: 322049

Abstract: In this talk, I will discuss some recent challenges arising in the study of large and complex networks. Over the last decades, improved data collection methods have significantly increased the availability of empirical data on networks, and the study of networks has become relevant across multiple disciplines such as biology, computer science, physics, and statistics. This has stimulated enormous activity in the mathematics research community to understand (1) local and global structure of networks, (2) dynamic processes on them, and (3) algorithms to discover hidden patterns in networks. In this talk, I will discuss my research on the above three topics in the context of percolation process, graph limits, and community detection on networks.

Add to calendar :

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

Join Zoom Meeting
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.

Add to calendar :

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

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

Meeting ID: 977 560 6531
Passcode: 123321

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.

Add to calendar :

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

Join Zoom Meeting
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.

Add to calendar :

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

Join Zoom Meeting
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.

Add to calendar :

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

Join Zoom Meeting
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.

Add to calendar :

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

Join Zoom Meeting
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.

Add to calendar :

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

Join Zoom Meeting
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.

Add to calendar :