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Mathematics-Driven Research seminar

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The world is constantly changing, and the use of technologies is rapidly growing. However, lots of challenges still remain unsolved. This is where fundamental mathematics enters the scene.  Mathematics is presented almost everywhere, and the efficiency of the industry highly depends on the mathematical background of the engineers and researchers working towards scientific and technological progress. The more technologically advanced are devices we use in our everyday life, the richer is the mathematics driving them.

Therefore, the Baltic Institute of Mathematics in cooperation with the Taras Shevchenko National University of Kyiv and the Institute of Mathematics of NAS of Ukraine are starting up a research seminar for students from the last two years of high school and first two years of the university, where we will discuss the current technologies and fundamental mathematics behind them. Every participant will be given an opportunity not only to get knowledge but also to give a try on doing research. Each meeting will be followed by a small research project.

To register, please, fill in the registration form

Next meeting is scheduled on MARCH 30th. 
Topics have included but were not limited to mathematical models and simulations, cryptosystems, probability theory, neural networks, computer vision, computational photography, perception, algorithms and self-driving cars, playing music using mathematics and programming, and introduction to data science.
Materials (available only for admitted participants) 
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STEM for Industry

Baltic Institute of Mathematics under the patronage of Prof. Jerzy Buźek in partnership with PZU S.A. and the Institute of Mathematics of NAS of Ukraine as a part of Polish-Ukrainian Science and Culture Forum for the Next Generation announces the STEM for Industry challenge

High school students are eligible to participate in our science challenge!

The process is easy: Industry companies propose a scientific challenge, and students work on a solution. The best solutions will compete for a grant worth 1200 PLN and the opportunity to publish the results in a research publication.

To sign up, please fill in the registration form
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Summer School & Workshop Wisla 23

The Summer School & Workshop Wisla 23 will bring together students, researchers, and practitioners from all over the world to share groundbreaking insights in AI, mathematics, and computer science. The event will take place from  August 21 - September 1, 2023 in a virtual form, and will combine lectures by experts in the field and talks by participants.  This year's theme is Mapping the Interdisciplinary Horizons of AI: Safety, Functional Programming, Information Geometry, and Beyond.

LECTURES

  • Scott Aaronson (The University of Texas at Austin, USA)
    AI Safety: Leaning Into Uninterpretability
    I'll share some thoughts about AI safety, shaped by a year's leave at OpenAI to work on the intersection of AI safety and theoretical computer science.  I'll discuss what I've worked on, including a scheme for watermarking the outputs of Large Language Models such as GPT, as well as proposals such as cryptographic backdoors and a theory of AI acceleration risk.

  • Seth Baum (Global Catastrophic Risk Institute, University of Cambridge, USA)
    Social and Philosophical Dimensions of AI Alignment
    Successful AI alignment requires three steps: (1) concept(s) for what, if anything, to align AI to; (2) design(s) for how to implement the concept(s) in one or more AI systems; and (3) the usage, by human developers of AI systems, of these design(s) instead of other, less suitable design(s). Step (1) is mainly a matter of moral philosophy; step (2) is mainly a matter of computer science; and step (3) is mainly a matter of social science and politics. My lecture(s) will cover steps (1) and (3).

  • Olle Häggström (Chalmers University of Technology, Sweden)
    AI risk and AI alignment
    The planetary dominance over other species that humanity has attained has very little to do with muscular strength and physical endurance: it is all about intelligence. This makes the present moment in history, when we are automating intelligence and handing over this crucial skill to machines, the most important ever. The research area that has become known as AI alignment deals with how to make sure that the first superintelligent machines have goals and values that are sufficiently aligned with ours and that sufficiently prioritize human flourishing. This needs to succeed, because otherwise we face existential catastrophe. In these lectures I will outline key challenges in AI alignment, what is being done to solve them, and how all this relates to the breakneck speed at which AI is presently advancing.

  • Anders Sanberg (Future of Humanity Institute, University of Oxford, UK)
    TBA
  • Patrik Jansson (Chalmers University of Technology, Sweden)
    Domain-Specific Languages of Mathematics
    The main idea behind this minicoure is to encourage the students to approach mathematical domains from a functional programming perspective. We will learn about the language Haskell; identify the main functions and types involved; introduce calculational proofs; pay attention to the syntax of mathematical  expressions; and, finally, to organize the resulting functions and types in domain-specific languages.
  • Frank Nielsen (Sony Computer Science Laboratories, Japan)
    Introduction to Information Geometry, Recent Advances, and Applications.
    Information geometry primarily studies the geometric structures, dissimilarities, and statistical invariance of a family of probability distributions called the statistical model. A regular parametric statistical model can be geometrically handled as a Riemannian manifold equipped with the Fisher metric tensor which induces the Fisher-Rao geodesic distance. This Riemannian structure on the Fisher-Rao manifold was later generalized by a dual structure based on pairs of torsion-free affine connections coupled to the Fisher metric: The α-geometry. This dual structure casts light on the close interaction between statistical estimators in inference (maximum likelihood) and parametric statistical models (exponential families obtained from the principle of maximum entropy), and brings into play a generalized Pythagorean theorem useful to prove uniqueness of information projections. We will illustrate applications of information geometry in statistics, information theory, computer vision and pattern recognition, and learning of neural networks. The second part of the minicourse will present recent advances in information geometry and its applications.
  • Dmitri Alekseevsky (University of Hradec Králové, Czech RepublicNeurogeometry of Vision and Information Geometry of Homogeneous Convex Cones
    These lectures will provide a comprehensive overview of information processing. We will start with information processing in early vision in static and geometric models of the primary visual cortex. Then, we will explore information processing in vision in dynamics. Finally, we gonna talk about the information geometry of Chentsov-Amari and homogeneous convex cones.

  • Frédéric Barbaresco (Thales Land and Air Systems, FranceSymplectic Foliation Structures of Information Geometry for Lie Groups Machine Learning
    We present a new symplectic model of Information Geometry  based on Jean-Marie Souriau's Lie Groups Thermodynamics. Souriau model was initially described in chapter IV “Statistical Mechanics” of his book “Structure of dynamical systems” published in 1969. This model gives a  purely geometric characterization of Entropy, which appears as an invariant Casimir function in coadjoint representation, characterized by Poisson cohomology. Souriau has proved that we can associate a symplectic manifold to coadjoint orbits of a Lie group by the KKS 2-form (Kirillov, Kostant, Souriau 2-form) in the affine case (affine model of coadjoint operator equivariance via Souriau's cocycle), that we have identified with Koszul-Fisher metric from Information Geometry. Souriau established the generalized Gibbs density covariant under the action of the Lie group. The dual space of the Lie algebra foliates into coadjoint orbits that are also the Entropy level sets that could be interpreted in the framework of Thermodynamics by the fact that dynamics on these symplectic leaves are non-dissipative, whereas transversal dynamics, given by Poisson transverse structure, are dissipative. We will finally introduce Gaussian distribution on the space of Symmetric Positive Definite (SPD) matrices, through Souriau's covariant Gibbs density by considering this space as the pure imaginary axis of the homogeneous Siegel upper half space where Sp(2n,R)/U(n) acts transitively. We will also consider Gibbs density for Siegel Disk where SU(n,n)/S(U(n)xU(n)) acts transitively. Gauss density of SPD matrices is then computed through Souriau's moment map and coadjoint orbits. Souriau’s Lie Groups Thermodynamics model will be further explored in European COST network CaLISTA and European HORIZON-MSCA project CaLIGOLA.
  • Noémie  Combe (Max Planck Institute for Mathematics in Sciences, Germany)
    Exploring Information Geometry: Recent Advances and Connections to Topological Field Theory
    With the rapid progress of machine learning, artificial intelligence and data sciences, the topic of information geometry is an important domain of research. We aim at introducing the topic of information geometry, as well as presenting some recent progress in this domain. Differential geometry and algebraic aspects shall be developed. The new tight relation between the information geometry and topological field theory will be discussed

MATERIALS (available only for admitted participants)

CALL FOR CONTRIBUTIONS

The school will provide participants with an opportunity to interact with their colleagues and well-known researchers in the field.  Each participant could make a talk about recent research and get independent and constructive feedback on her/his current research and future research directions. Materials from the school and workshop will be published by Springer Nature. All contributions are subject to peer review.

Org. Committee
J. de Lucas, M. Roop, J.Szmit, R.Zawadzki, M.Ulan, M. Wojnowski
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Winter School & Workshop Wisla 22

Online
January 24 - February 6,  2022

The topic of the school:

Differential Geometry and its applications: fluid dynamics, dispersive systems, image processing, and beyond.

Type of activity Time  in CET
Virtual coffee break 16:00 - 16:30
Lecture 1 16:30 - 17:30
Virtual coffee break 17:30 - 18:00
Lecture 2 18:00 -19:00
Virtual coffee break 19:00 - 19:30
Talks by participants 19:30 - 20:00
Networking 20:00 - 21:00

January 24 - January 27: Lecture 1 by Valentin Lychagin, Lecture 2 by Peter J. Olver
January 28: Lecture 1 & Lecture 2 by Klas Modin
January 29: Lecture 1 & Lecture 2 by Boris Khesin
January 31: Lecture 1 by Klas Modin, Lecture 2 by Boris Khesin
February 1- February 3: Lecture 1 by Ian Roulstone, Lecture 2 by Volodya Roubtsov
Full program and details (available  only for admitted participants) 

ABSTRACTS

Boris Khesin (University of Toronto, Canada)
Hamiltonian Fluid Dynamics
The course outlines group-theoretic and Hamiltonian approaches to hydrodynamics. We start by describing the Eulerian dynamics of an ideal fluid and the Korteweg—de Vries equation of shallow water from the group-theoretic and geometric points of view. The Hamiltonian framework will allow us to visualize the geometry of Casimirs for the Euler equation and helicity of vector fields, as well as to recover the motion of point vortices, vortex filaments and membranes.


Valentin Lychagin  (University of Tromsø, Norway)
Continuum Mechanics of Media with Inner Structure
Short description:
Continuum mechanics on Riemannian manifolds
-Thermodynamics of Newtonian media.
-Conservation laws, Navier-Stokes and Euler equations.
Continuum mechanics of media with inner structures
Continuum mechanics of molecular media


Klas Modin (Chalmers University of Technology & University of Gothenburg, Sweden)
Introduction to Geometric Hydrodynamics
In three lectures I trace the work of three legends of mathematics and mechanics: Euler, Poincaré, and Arnold. This leads up to the aim of the lectures: to explain Arnold’s discovery from 1966 that solutions to Euler’s equations for the motion of an incompressible fluid correspond to geodesics on the infinite-dimensional Riemannian manifold of volume preserving diffeomorphisms. In many ways, this discovery is the foundation for the field of geometric hydrodynamics, which today encompasses much more than just Euler’s equations, with deep connections to many other fields such as optimal transport, shape analysis, and information theory.


Peter J. Olver  (University of Minnesota, USA)
Fractalization and Quantization in Dispersive Systems
These talks will survey recent results on linear and nonlinear dispersive wave equations on periodic domains.  The Talbot effect, also known as dispersive quantization. The evolution, through spatially periodic linear dispersion, of rough initial data produces fractal, non-differentiable profiles at irrational times and, for asymptotically polynomial dispersion relations, quantized structures at rational times.  Such phenomena have been observed in dispersive wave models, optics, and quantum mechanics, and lead to intriguing connections with exponential sums arising in number theory.   Ramifications and recent progress on the analysis, numerics, and extensions to nonlinear wave models, both integrable and non-integrable, will be presented.  Related results for the Fermi-Pasta-Ulam-Tsingou problem of the dynamics of nonlinear mass-spring chains and the Lamb model for radiation damping of a vibrating body in an energy conducting medium will also be discussed.

Symmetry, invariance, and equivalence in image processing
Symmetry recognition is fundamental in human vision and thus also plays a key role in image processing algorithms.  These talks will survey old and new mathematical perspectives on symmetry, invariance, and equivalence based on transformation groups and groupoids.  Cartan's solution to the equivalence and symmetry problem for submanifolds relies on the associated geometric invariants, through what is now known as the differential invariant signature.  Applications arise in art, computer vision, medicine, geometry, and beyond, including automated assembly of broken objects: jigsaw puzzles, egg shells, broken bones, and lithics (stone age tools).


Volodya Roubtsov  (University of Angers, France)
Introduction to Hamiltonian Mechanics
I shall give a short account of Hamiltonian methods of classical mechanics. After a minimal reminder from symplectic geometry, I shall concentrate on explicit examples of mechanical systems and shall demonstrate main features of the Hamiltonian approach and Liouville integrability: a construction of action-angle variables for the chosen examples. My lectures do not contain new results and have a fully methodological flavor.


Ian Roulstone (University of Surrey, UK)
Monge–Ampère Geometry and the Navier–Stokes Equations
In this series of three lectures, we shall apply ideas from Monge–Ampère  geometry to the partial differential equations governing incompressible fluid flow in two and three spatial dimensions. Despite the apparent randomness of turbulent flows, vortices play a key role in determining the structure and evolution of the flow. We shall show how almost-complex structures in four and six dimensions are naturally associated with regimes in which vorticity dominates over strain and, furthermore, how such flows are associated with Riemannian metrics of positive scalar curvature. In turn, these insights suggest a route to obtaining topological information about vortex structures.

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REGISTRATION

To register, please, fill in the registration form.

The school fee is 125 EUR
The fee includes online participation,  lectures, and covers technical issues.

Financial Support
We expect that some support will be available to fund students and young researchers.
If you are requesting financial support, please complete the registration form and send your CV as soon as possible.

Org. Committee
R. Kycia, J. de Lucas, M. Roop, J.Szmit, R.Zawadzki, M.Ulan, M. Wojnowski
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Proceedings of Winter School & Workshop Wisla 22

The school will provide young researchers with an opportunity to interact with their colleagues and well-known researchers in the field:
- each participant could make a talk about recent research or present a poster
- each participant will get independent and constructive feedback on her/his current research and future research directions
- a decision about including participantwork to the book will be made based on experts feedback
- each participant will also be given an opportunity to improve her/his work during the school

Materials of the school and workshop will be published by Springer Nature. All contributions are subject to peer review.

Lecture notes from previous editions:

Nonlinear PDEs, Their Geometry, and Applications. Proceedings of the Wisła 18 Summer School

Differential Geometry, Differential Equations, and Mathematical Physics. The Wisła 19 Summer School

Proceedings of Winter School & Workshop Wisla 21 and Summer School & Workshop Wisla 20 are in progress
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An invitation to symplectic geometry and classical mechanics.

Online
May 18, 2021 11:00 - 12:00 CET

Speaker:
Javier de Lucas Araujo (Assistant professor at the Department of Mathematical Methods in Physics of the Faculty of Physics of the University of Warsaw)

Abstract:
In this talk, I will introduce some notions and results on symplectic geometry and their applications to classical mechanics. Although omitting technical proofs, I will explain the main ideas and results about this theory and its applications. In detail, I will comment on the definition of symplectic manifolds, Hamiltonian vector fields and functions, Poisson brackets, momentum maps, and the Marsden-Weinstein theorem. More physically, I will explain how these structures explain the Hamilton equations of a classical mechanical system and how they are used to study systems with symmetries.

Registration:
To register, please, fill in the registration form.

Organizers:
- Baltic Institute of Mathematics (Warsaw, Poland, http://www.baltinmat.com)
- Institute of Mathematics of NAS of Ukraine (Kyiv, Ukraine, http://www.imath.kiev.ua)
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From Goals to Actions - How to EdTech

May 17 - 28, 2021
Online

Baltic Institute of Mathematics in cooperation with the Dnipro Academy of Continuous Education will organize a seminar of 6 mini-courses:

- Digital transformation in education
- AR, MR, and VR technologies for learning
- Gamification of learning
- Blended and flipped learning
- Team-based and collaborative learning
- Design thinking: creativity, innovation, and empathy

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As a part of each mini-course, we will discuss the growing role of technologies, challenges, and opportunities for their implementation in the educational process. We will touch on both theoretical and practical aspects.

Registration is closed
More Info & Slides (Password Available Upon Request)

Organizers:
Baltic Institute of Mathematics
Dnipro Academy of Continuous Education

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Symplectic Methods and Dynamical Systems

Online
January 15, 2021 11:00 - 12:00 CET

Speaker:
Javier de Lucas Araujo (Assistant professor at the Department of Mathematical Methods in Physics of the Faculty of Physics of the University of Warsaw)

Abstract:
This talk provides a brief introduction to symplectic geometry and its applications to dynamical systems. In particular, I will explain the most fundamental notions on symplectic geometry such as the Darboux theorem, the canonical symplectic structure on the cotangent bundle, and the main properties of the so-called Hamiltonian vector fields.  Concerning the applications of symplectic geometry to dynamical systems, the Liouville theorem, the Gromov’s non-squeezing theorem, and the main properties of the referred to as integrable Hamiltonian systems will be analysed.

Slides & Recording (Password Available Upon Request)

Organizers:
- Baltic Institute of Mathematics (Warsaw, Poland, http://www.baltinmat.com)
- Institute of Mathematics of NAS of Ukraine (Kyiv, Ukraine, http://www.imath.kiev.ua)
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