School of Physical Sciences

Abstract not available

• Speaker: Michael Douglas (Center of Mathematical Sciences and Applications, Harvard University)
• Monday 29 April 2024, 14:00-15:00
• Venue: MR15 Centre for Mathematical Sciences.
• Series: Formalisation of mathematics with interactive theorem provers ; organiser: Anand Rao Tadipatri.

Add to your calendar or Include in your list

A variety X is semismooth if étale locally it is isomorphic to a product of a pinch point (x^2y-z2) with some affine space; equivalently, its normalization is smooth and X is obtained by gluing a smooth divisor to itself via an involution with fixed points in codimension 1. In joint work with Marco Franciosi and Rita Pardini, we calculate the sheaves T1_X and T_X in terms of the normalization and the gluing, and use this to show that all semi-smooth non normal stable Godeaux surfaces are smoothable, and nonsingular points of the moduli space. 

EMG - New equivariant methods in algebraic and differential geometry

• Speaker: Barbara Fantechi (SISSA)
• Friday 26 April 2024, 14:00-15:00
• Venue: Seminar Room 1, Newton Institute.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

We discuss energy and flux budget (EFB) analysis for stably stratified and convective atmospheric turbulence. The EFB analysis shows that that high-Reynolds-number stably stratified turbulence is maintained by shear in any stratification, and the “critical Richardson number,” treated many years as a threshold between the turbulent and laminar regimes, actually separates two turbulent regimes, namely, the strong turbulence typical of atmospheric boundary layers and the weak three-dimensional turbulence typical of the free atmosphere and characterized by a strong decrease in the heat transfer in comparison to the momentum transfer. Large-scale internal gravity waves result in additional vertical turbulent flux of momentum and additional productions of the densities of the turbulent kinetic and potential energies and turbulent flux of potential temperature. We also discuss application of the EFB analysis to convective atmospheric turbulence, which allows to describe the surface layer (unstably stratified and dominated by small-scale turbulence of very complex nature) as well as the convective boundary layer core (dominated by the energy-, momentum-, and mass-transport of semi-organized large-scale structures with a small contribution from small-scale turbulence produced by local structural shears).

ADI - Anti-diffusive dynamics: from sub-cellular to astrophysical scales

• Speaker: Igor Rogachevskii (Ben-Gurion University)
• Thursday 25 April 2024, 16:00-17:00
• Venue: Seminar Room 1, Newton Institute.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

Anderson localization is the absence of diffusion of waves in random media. It is a generic wave phenomenon, which applies to any kind of wave regardless of its nature. Experimentally, Anderson localization has been found for electron gases, acoustic waves, spin waves, matter waves, and more recently also for light waves. The localization occurs because a disordered medium induces multiple scattering paths along which the components of the wave function interfere destructively. Lately, it has been discussed that a weak nonlinearity might destroy the localized state giving rise to unlimited spreading of the wave function along the lattice despite the underlying disorder. The statistics of this spreading process has remained a matter of debate. In this talk, I will review the state of the art, with several toy models predicting asymptotic spreading from the nonlinear Schrödinger dynamics on a lattice. The key words will be continuous time random walks, chaos (strong, weak), percolation, fractional kinetics, Cayley trees. Time permitting, I will touch upon topics concerning nonlinear Schrödinger models with subquadratic power nonlinearity leading to Lévy flights. A summary of the discussion may be found in a recent work [A.V. Milovanov and A. Iomin, Phys. Rev. E. 107, 034203 (2023)].

ADI - Anti-diffusive dynamics: from sub-cellular to astrophysical scales

• Speaker: Alexander Milovanov (ENEA C.R. Frascati)
• Tuesday 23 April 2024, 11:30-12:30
• Venue: Seminar Room 1, Newton Institute.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

Peter Read + Arrate Antunano, John Barbara, Simon Cabanes, Greg Colyer, Teresa del Río Gaztelurrutia, Agustin Sanchez-Lavega & Roland Young Recent analyses of wind measurements from tracking cloud motions in spacecraft images of Jupiter and Saturn indicate that scale to scale transfers of kinetic energy act from small to large scales over a wide range of length scales, much as anticipated for 2D or geostrophic turbulence paradigms. At the smallest resolvable scales, however, there is evidence of a forward (downscale) transfer, at least at low and middle latitudes on Jupiter. Moreover, the upscale transfers at the largest spatial scales are evidently dominated by direct, spectrally non-local eddy-zonal interactions, in contrast to more classical scenarios, associated with the generation of intense zonal jets, alternating in latitude, via the divergence or convergence of horizontal Reynolds stresses. Most analyses to date have emphasised the global mean interactions for both planets, thereby focusing on the spatially homogeneous components of the turbulence. But more recent observations indicate that the dynamics of these atmospheres varies significantly with latitude from the tropics to the polar regions. Here we present some new analyses of spectral energy transfers on both Jupiter and Saturn that resolve variations in latitude. The (preliminary) results indicate significant variability between different locations, with a clear distinction between the tropics, the extratropical middle latitudes and the polar regions. We discuss these in light of other observations and models of gas giant circulation. 

ADIW03 - Climate Applications of Layering

• Speaker: Peter Read (University of Oxford)
• Tuesday 21 May 2024, 10:00-11:00
• Venue: Seminar Room 1, Newton Institute.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

Combinatorial optimisation is a central area in computer science, applied mathematics, and operational research. Ideas and notions developed within the area of combinatorial optimisation include linear programming, flows, and matchings. In this talk I will describe the formalisation, in Isabelle/HOL, of some results from the theory of combinatorial optimisation, mainly focusing on the theory of matching. I will briefly discuss mathematically interesting findings and also formalisation/methodological findings.

—-

WATCH ONLINE HERE : https://www.microsoft.com/en-gb/microsoft-teams/join-a-meeting?rtc=1 Meeting ID: 370 771 279 261 Passcode: iCo7a5

Online

• Speaker: Mohammad Abdulaziz (King's College London)
• Thursday 25 April 2024, 17:00-18:00
• Venue: Live-streamed at MR14 Centre for Mathematical Sciences.
• Series: Formalisation of mathematics with interactive theorem provers ; organiser: Anand Rao Tadipatri.

Add to your calendar or Include in your list

In this talk, I will give an overview of recent developments linking wave theory and multidimensional complex analysis. I will explain how a procedure of complex deformation of the integration surface of Fourier-like highly oscillatory double integrals can lead to closed-form far-field asymptotics results in wave diffraction theory. Each far-field component will be shown to be connected to a special point on the singularity set of the integrand. The procedure will be illustrated through two examples that can be reformulated as two-complex-variables scalar Wiener-Hopf problems: the three-dimensional problem of plane wave diffraction by a quarter-plane and the two-dimensional problem of plane wave diffraction by a penetrable wedge. I will also show how it can be used to shed some light on wave propagation in periodic structures. The talk will cover aspects of several articles written jointly with great collaborators who should be acknowledged: Andrey V. Shanin, Andrey K. Korolkov, Valentin D. Kunz and I. David Abrahams.

WHTW02 - WHT Follow on: the applications, generalisation and implementation of the Wiener-Hopf Method

• Speaker: Raphael Assier (University of Manchester)
• Monday 01 July 2024, 14:00-14:30
• Venue: Seminar Room 1, Newton Institute.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

Abstract not available

• Speaker: Michael Douglas (Center of Mathematical Sciences and Applications, Harvard University)
• Monday 29 April 2024, 14:00-15:00
• Venue: Live-streamed at MR14 Centre for Mathematical Sciences.
• Series: Formalisation of mathematics with interactive theorem provers ; organiser: Anand Rao Tadipatri.

Add to your calendar or Include in your list

—-

WATCH ONLINE HERE : https://www.microsoft.com/en-gb/microsoft-teams/join-a-meeting?rtc=1 Meeting ID: 370 771 279 261 Passcode: iCo7a5

Online

• Speaker: Mohammad Abdulaziz (King's College London)
• Thursday 25 April 2024, 17:00-18:00
• Venue: Live-streamed at MR14 Centre for Mathematical Sciences.
• Series: Formalisation of mathematics with interactive theorem provers ; organiser: Anand Rao Tadipatri.

Add to your calendar or Include in your list

Alpha-beta pruning is an efficient search strategy for two-player game trees. It was invented in the late 1950s and is at the heart of most implementations of combinatorial game playing programs. In this talk I will survey my recent formalizations and verifications of a number of standard variations of alpha-beta pruning. Findings include:

- Basic variants already having a property ascribed to an improved version

- Authors being confused about which algebraic structure they actually work in

- Generalizations to new algebraic structures

- The implementation in a famous paper is flawed

• Speaker: Tobias Nipkow (Technische Universität München)
• Thursday 13 June 2024, 17:00-18:00
• Venue: Live-streamed at MR14 Centre for Mathematical Sciences.
• Series: Formalisation of mathematics with interactive theorem provers ; organiser: Anand Rao Tadipatri.

Add to your calendar or Include in your list

Abstract not available

• Speaker: Patrick Massot (Université Paris-Saclay and Carnegie Mellon University)
• Thursday 16 May 2024, 17:00-18:00
• Venue: Live-streamed at MR14 Centre for Mathematical Sciences.
• Series: Formalisation of mathematics with interactive theorem provers ; organiser: Anand Rao Tadipatri.

Add to your calendar or Include in your list

Condensed sets form a topos, and hence admit an internal type theory. In this talk I will describe a list of axioms satisfied by this particular type theory. In particular, we will see two predicates on types, that single out a class CHaus of “compact Hausdorff” types and a class ODisc of “overt and discrete” types, respectively. A handful of axioms describe how these classes interact. The resulting type theory is spiritually related Taylor’s “Abstract Stone Duality”.

As an application I will explain that ODisc is naturally a category, and furthermore, every function ODisc → ODisc is automatically functorial. This axiomatic approach to condensed sets, including the functoriality result, are formalized in Lean 4. If time permits, I will comment on some of the techniques that go into the proof.

Joint work with Reid Barton.

• Speaker: Johan Commelin (Utrecht University)
• Thursday 02 May 2024, 17:00-18:00
• Venue: Live-streamed at MR14 Centre for Mathematical Sciences.
• Series: Formalisation of mathematics with interactive theorem provers ; organiser: Jonas Bayer.

Add to your calendar or Include in your list

This is the joint talk with N.V. Movchan   The lecture addresses Wiener-Hopf formulations for mathematical models which describe wave propagation in discrete semi-infinite elastic systems, containing resonators. The dynamic response of resonators, attached to the main waveguide, appears in the governing equations as the inertial input, which may be positive or negative  depending on the phase shift of a resonator relative to the supporting structure.   When the structure is semi-infinite, the Wiener-Hopf approach provides an elegant model for a time-harmonic wave propagation, and the corresponding functional equation includes the kernel function, which is linked to the lattice Green’s function of a periodic lattice. This connection is exploited here to discuss different dynamic regimes for semi-infinite structured waveguides.   Examples and applications are discussed for flexural elastic waves in structured plates and beams, including modelling of waves induced by a moving load. 

WHTW02 - WHT Follow on: the applications, generalisation and implementation of the Wiener-Hopf Method

• Speaker: Alexander Movchan (University of Liverpool), Natasha Movchan (University of Liverpool)
• Friday 05 July 2024, 09:30-10:15
• Venue: Seminar Room 1, Newton Institute.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

We will discuss the modeling and simulations of active suspensions in complex confinement when they interact with each-other through chemotaxis and fluid-mediated forces. Then we will outline the possibilities of separating by motility active suspensions utilizing confinement or chemotaxis. Theoretical results will be compared to recent experimental results of systems involving bacterial suspensions.    Collaborators: Yangrui Chen and Xiang Cheng at the University of Minnesota

ADIW04 - Anti-Diffusion in Multiphase and Active Flows

• Speaker: Enkeleida Lushi (New Jersey Institute of Technology)
• Thursday 13 June 2024, 14:50-15:20
• Venue: Seminar Room 1, Newton Institute.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

OFBW65 - Connecting Heavy Tails and Differential Privacy in Machine Learning

• Speaker: Bikramjit Das (Singapore University of Technology and Design)
• Friday 26 April 2024, 15:10-15:45
• Venue: External.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

We analyze a continuous diffusion approximation of SGD , called homogenized stochastic gradient descent, show that it behaves asymptotically heavy-tailed, and give explicit upper and lower bounds on its tail-index. We validate these bounds in numerical experiments and show that they are typically close approximations to the empirical tail-index of SGD iterates. In addition, their explicit form enables us to quantify the interplay between optimization hyperparameters and the tail-index. Our results show that also continuous diffusions, not only Lévy-driven SDEs, can accurately represent the emergence of heavy tails in SGD . In addition, our results suggest skew Student-t-distributions, not alpha-stable distributions, as surrogates of parameter distributions under SGD .    

TMLW02 - SGD: stability, momentum acceleration and heavy tails

• Speaker: Martin Keller-ressel (Technische Universität Dresden)
• Thursday 25 April 2024, 15:15-16:00
• Venue: External.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

We establish inequalities for assessing the distance between the distribution of errors of partially observed high-frequency statistics of multidimensional L\’evy processes and that of a mixed Gaussian random variable. Furthermore, we provide a general result guaranteeing stable  convergence. Our arguments rely on a suitable adaptation of the Stein’s method perspective to the context of mixed Gaussian distributions, specifically tailored to the framework of high-frequency statistics. 

TMLW02 - SGD: stability, momentum acceleration and heavy tails

• Speaker: Chiara Amorino (Université du Luxembourg)
• Thursday 25 April 2024, 14:30-15:15
• Venue: External.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

This talk will first review the so-called `ODE’ (for `Ordinary Differential Equations’) approach for the analysis of stochastic approximation algorithms of which stochastic gradient descent is a special case. Using that as a backdrop, certain analogous results will be presented for a class of algorithms with heavy tailed noise. This is joint work with V. Anantharam of Uni. of California at Berkeley.    

TMLW02 - SGD: stability, momentum acceleration and heavy tails

• Speaker: Vivek Borkar (Indian Institute of Technology Bombay)
• Thursday 25 April 2024, 11:45-12:30
• Venue: External.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

Stochastic gradient descent (SGD) is a powerful optimization technique, particularly useful in online learning scenarios. Its convergence analysis/effectiveness is relatively well understood under the assumption that the data samples are independent and identically distributed (iid). However, applying online learning to policy optimization problems in operations research involves a distinct challenge: the policy changes the environment and thereby affects the data used to update the policy. The adaptively generated data stream involves samples that are non-stationary, no longer independent from each other, and are affected by previous decisions. The influence of previous decisions on the environment introduces estimation bias in the gradients, which presents a potential source of instability for online learning. In this paper, we introduce simple criteria for the adaptively generated data stream to guarantee the convergence of SGD . We show that the convergence speed of SGD with adaptive data is largely similar to the classical iid setting, as long as the mixing time of the policy-induced dynamics is factored in. Our Lyapunov-function analysis allows one to translate existing stability analysis of systems studied in operations research into convergence rates for SGD , and we demonstrate this for queuing and inventory management problems. We also showcase how our result can be applied to study an actor-critic policy gradient algorithm. This is joint work with Ethan Che and Xin Tong.  

TMLW02 - SGD: stability, momentum acceleration and heavy tails

• Speaker: Jing Dong (Columbia University)
• Thursday 25 April 2024, 11:00-11:45
• Venue: External.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list

The empirical success of deep learning is often attributed to the mysterious ability of stochastic gradient descents (SGDs) to avoid sharp local minima in the loss landscape, as sharp minima are believed to lead to poor generalization. To unravel this mystery and potentially further enhance such capability of SGDs, it is imperative to go beyond the traditional local convergence analysis and obtain a comprehensive understanding of SGDs’ global dynamics within complex non-convex loss landscapes. In this talk, we characterize the global dynamics of SGDs through the heavy-tailed large deviations and local stability framework. This framework systematically characterizes the rare events in heavy-tailed dynamical systems; building on this, we characterize intricate phase transitions in the first exit times, which leads to the heavy-tailed counterparts of the classical Freidlin-Wentzell and Eyring-Kramers theories. Moreover, applying this framework to SGD , we reveal a fascinating phenomenon in deep learning: by injecting and then truncating heavy-tailed noises during the training phase, SGD can almost completely avoid sharp minima and hence achieve better generalization performance for the test data.  

TMLW02 - SGD: stability, momentum acceleration and heavy tails

• Speaker: Xingyu Wang (Northwestern University)
• Thursday 25 April 2024, 09:45-10:30
• Venue: External.
• Series: Isaac Newton Institute Seminar Series; organiser: nobody.

Add to your calendar or Include in your list