The Scientific Advisory Board of FiRST gathers for the second time in Kumpula, Helsinki on 13.-14.3.2025.
The scientific presentations on both meeting days take place in Chemicum, auditorium A110, and are open to all FiRST researchers and colleagues at the Department of Mathematics and Statistics.
Thursday 13 March
| Time | Place | Event |
| 09:15 – 09:45 | Chemicum, auditorium A110 | Talk: Tuomo Kuusi |
| 09:50 – 10:20 | Chemicum, auditorium A110 | Talk: Osama Abuzaid |
| 10:20 – 10:55 | Coffee break | |
| 10:55 – 11:25 | Chemicum, auditorium A110 | Talk: Julia Sanders |
| 11:30 – 12:00 | Chemicum, auditorium A110 | Talk: Tuomas Hytönen |
Friday 14 March
| Time | Place | Event |
| 09:15 – 09:45 | Chemicum, auditorium A110 | Talk: Pauliina Ilmonen |
| 09:50 – 10:20 | Chemicum, auditorium A110 | Talk: Kalle Kytölä |
| 10:20 – 10:55 | Coffee break | |
| 10:55 – 11:25 | Chemicum, auditorium A110 | Talk: Sarvagya Jain |
| 11:30 – 12:00 | Chemicum, auditorium A110 | Talk: Pekka Koskela |
Please find the abtracts of the talks linked below.
Time: 12 March 2025 at 14:15-17:00
Place: Exactum, room C124
at the Department of Mathematics and Statistics at the University of Helsinki
Speakers: Professor Lillian Pierce (Duke University), Professor Jean-Pierre Eckmann (University of Geneva)
14:15-15:15 Professor Lillian Pierce: "On Superorthogonality"
15:15-16:00 Coffee break at Café Physicum (at own expense)
16:00-17:00 Professor Emeritus Jean-Pierre Eckmann: "Rolling stones reveal new structures in SO(3)"
Professor Lillian Pierce: "On Superorthogonality"
Square functions seem to have arisen, as a tool in harmonic analysis, about one hundred years ago. In the simplest formulation, given a sequence of functions f_1, f_2, f_3,… the associated square function is defined by taking the (discrete) \ell^2 norm of the sequence. Square functions are useful in many ways, and in particular their quadratic structure makes them useful in settings where some type of L^2 behavior, or orthogonality, is at play. More generally, two natural questions arise: first, can we dominate the L^p norm of a sum of functions we are interested in, by the L^p norm of a square function? Second, can we bound the L^p norm of the square function advantageously? Both questions can be quite interesting; in this talk, we will focus on the first question. Ad hoc methods to answer the first question, when p is an even integer, have arisen in a tremendous variety of applications. We will describe how an unlikely sequence of events led us to categorize these methods, so that we can now answer the first question (affirmatively) for any sequence of functions that exhibits an appropriate “type” of “superorthogonality”. Moreover, we will describe new work that answers the first question (affirmatively) for a type of “superorthogonality” that requires less information than any type previously known. The talk will include many motivating examples from the past century, in both harmonic analysis and number theory, as well as new joint work with Phil Gressman, Joris Roos, and Po-Lam Yung.
Professor Jean-Pierre Eckmann: "Rolling stones reveal new structures in SO(3)"
This project started with the question whether one can construct aspecially formed stone so that it rolls along a prescribed curve and its repetitions. We then discovered that for almost all given paths this is indeed possible, but, astonishingly, one needs to traverse two copies of the prescribed line to regain the original orientation of the stone. We finally discovered how this result is related to stochastic and Diophantine properties of rotations in SO(3).
The Finnish Centre of Excellence in Randomness and Structures (FiRST) is organizing a workshop for its young researchers in 11.-12.3. at Kumpula campus of the University of Helsinki, and we invite everyone interested to attend the talks. The speakers represent the different research groups in the CoE and their talks therefore span a wide variety of topics.
See the programme of the workshop below.
| Tuesday March 11th | |||
| 13.15-14.00 | Shinji Koshida | Planar algebras for the Young graph and the Khovanov Heisenberg category | B322 |
| 14.15-15.00 | Patrik Nummi | Periodic Solutions to a Stochastic Pressure Equation with Random Permeability | B322 |
| 15.00-15.30 | Coffee break | ||
| 15.30-16.15 | Aapo Laukkarinen | Convex body domination and its applications to commutators | B322 |
| 16.30-17.15 | Julien Roussillon | Fused Specht polynomials and applications | B322 |
| Wednesday March 12th | |||
| 9.15-10.00 | Sauli Lindberg | A priori estimates on nonlinear PDEs via Baire category methods | C124 |
| 10.15-11.00 | Sami Vihko | Reconstruction of log-correlated fields from multiplicative chaos measures | C124 |
| 11.15-12.00 | Sarvagya Jain | Smooth Numbers in Short Intervals: Closing the Gap Between Heuristics and Reality | C124 |
| 12.00-13.15 | Lunch | ||
| 13.15-14.00 | Zofia Grochulska | A look into homeomorphisms in analysis | CK111 |
| 14.15-15.00 | Jaakko Pere | On the Impact of Approximation Errors on Extreme Value Inference: Applications to Multidimensional Extremes | CK111 |
| 15.00-15.30 | Coffee break | ||
| 15.30-16.15 | Aleksis Vuoksenmaa | Chaos via joint cumulants -- the case of the stochastic Kac model | CK111 |
| 16.30-17.15 | Joona Karjalainen | Forgetting properties of particle filter algorithms | CK111 |
at Aalto University on 21.-22.2.2025
This statistics workshop gathers researchers to celebrate research, science and life. The talks cover various topics related (but not limited) to invariant coordinate selection, discrete and continuous stochastic processes, functional data-analysis, extreme value theory, cancer epidemiology, and cancer genetics.
Please visit the web page of the workshop for more information.
Wojciech De Roeck and Francois Huveneers will give a 6 hours set of introductory talks on Many Body Localization.
On the occasion, we also invited Teemu Ojanen to give a talk on entanglement volume-law area-law transitions in 1d qubit systems.
Topics of Wojciech De Roeck and Francois Huveneers mini course:
General Part 1:
More specific Part 2:
Time and Place:
Thursday 13.02 in Y313 Otakaari 1 at Aalto University
Friday 14:02. 10-12 in C124 Exactum building Kumpula Campus.
Speaker: Juan J. Manfredi, University of Pittsburgh
Time: Wednesday 9 October at 12:15—17:00
Place: MaD 302, Mattilanniemi
Department of Mathematics and Statistics at the University of Jyväskylä
Abstract:
These lectures will cover the necessary background in probability theory, discrete stochastic games, and viscosity solutions to study random tug-of-war games with noise. Below are a list of topics and a list of selected references.
(1) Probability tools: Discrete Stochastic Processes, Martingales.
(2) Viscosity Solutions. The Theorem on Sums.
(3) Asymptotic Mean Value Properties, p-Harmonious Functions.
(4) PDEs on Directed Trees.
(5) Regularity for p-harmonic functions via the Ishii-Lions method.
References:
(1) P. Blanc, J. D. Rossi, Game Theory and Partial Differential Equations. De Gruyter, 2019.
(2) P. Lindqvist, Notes on the p-Laplace Equation, BCAM Springer Briefs, 2017.
(3) A. P. Maitra, W. D. Sudderth, Discrete Gambling and Stochastic Games. Applications of Mathematics 32, Springer-Verlag, 1996.
(4) M. Parviainen, Notes on Tug-of-War Games and the p-Laplace Equation, Springer Briefs in PDE and Data Science, 2024.
(5) S.R.S. Varadan; Probability Theory, Courant Lecture Notes in Mathematics 7, New York University.
This four-day mathematical physics conference takes place in Helsinki, September 3-6, 2024. It covers probabilistic and path integral methods in quantum and statistical field theory, highlighting in particular remarkable progress in the mathematical understanding of two-dimensional conformal field theories, exciting developments in higher dimensional field theories, near-critical and integrable theories, and the connections to deep mathematical questions in analysis, geometry, and probability.
For more information, please visit the website of the conference.
This 3-days workshop delves into the multifaceted world of probabilistic field theories, aiming at sharing insights, presenting novel methodologies, and fostering new collaborations.
We plan to gather distinguished experts and young researchers at the intersection of probability theory and field theory, understood in the wide sense encompassing mathematical quantum field theory, stochastic partial differential equations, fluid dynamics and homogenization, also including applications to algebraic field theory, dynamical systems, statistical mechanics, and particle models.
For more information, please visit the website of the workshop.
Speaker: Mario Ullrich
Time:
Lecture room: Exactum, C124
at the Department of Mathematics and Statistics, University of Helsinki
Abstract:
In two talks I'll give an overview of some recent and not so recent developments in the area of high-dimensional integration and approximation of functions based on function evaluations.
The emphasis is on information-based complexity, i.e., we ask for the minimal number of information (aka measurements) needed by any algorithm to achieve a prescribed error for all inputs. Hence, upper error bounds are complemented by lower bounds.
In Part 1, I'll present that in many cases, certain (unregularized) least squares methods based on "random" information, like function evaluations, can catch up with arbitrary algorithms based on arbitrary linear information, i.e., the best we can do theoretically.
After a detailed introduction to the field, we will discuss the following:
(1) random data for L_2-approximation in Hilbert spaces,
(2) approximation in other norms for general classes of functions, and
(3) "Does random data contain optimal data?" (Spoiler: The answer is often: Yes!)
In Part 2, the focus is on high-dimensional integration and approximation, and the dependence of the error on the dimension. Here, we mainly discuss the "curse of dimension" for classical (isotropic) spaces C^k on domains, and that the (expectedly ineffective) product rules are indeed optimal in high-dimensions.
I'll mention several open problems in the field.
In both parts, I'll try to introduce all the necessary concepts in detail and therefore think that no expertise is required to follow the talk.
Speaker: Kari Vilonen
Time:
Lecture room: Exactum, C124
at the Department of Mathematics and Statistics, University of Helsinki
Abstract: I will give a leisurely introduction to a result which is basic to geometric approaches to the Langlands program. In particular, I will explain how the dual group of a reductive group arises naturally from the geometry of the affine Grassmannian.
Speaker: Ewain Gwynne (University of Chicago)
Time: Monday, August 28 to Friday, September 1. Afternoons 14-16.
Lecture room: Mon, Tue, Thu, Fri 14-16: Otakaari 1, R001/M234 M3; Wed 14-16: Otakaari 1, R001/M232 M1
at the Department of Mathematics and Systems Analysis, Aalto University
Abstract: Liouville quantum gravity (LQG) is a universal one-parameter family of random fractal surfaces. These surfaces have connections to string theory, conformal field theory, and statistical mechanics, and are expected to describe the scaling limits of various types of random planar maps.
Recent works have shown that one can endow an LQG surface with a metric (distance function). This metric has many interesting geometric properties. For example, it induces the same topology as the Euclidean metric, but its Hausdorff dimension is strictly greater than two and its geodesics merge into each other to form a tree-like structure.
I will discuss the definition of and motivation for LQG, the construction and properties of the metric, and some of the techniques for proving things about it.
Course lecture notes Course exercises Recording of the Lectures
Speaker: Richard Kenyon (Yale University)
Time: June 13-15, 2023. Tue, Wed, Thu at 10-12
Lecture room: Exactum, D122
at the Department of Mathematics and Statistics, University of Helsinki
Abstract: The dimer model is the study of random dimer covers (perfect matchings) of a bipartite graph on a surface. The dimer model has remarkable connections with other parts of mathematics, from conformal field theory to integrable systems to representation theory.
Webs are representation-theoretic objects, defined by Greg Kuperberg to study invariants in tensor products of SL_3 representations. We’ll discuss recent results on large-scale structures (webs) in random multiple-domino tilings, and their conformal invariance properties. These talks are based on joint work with David Wilson, Daniel Douglas, Haolin Shi.
Speaker: Adam Harper (University of Warwick)
Time: May 22-26, Mon-Fri 10:15-12:00
Seminar room: QM3 (Quantum building, third floor)
at the University of Turku
Abstract: Random multiplicative functions provide a probabilistic model for number theoretic functions like Dirichlet characters. They were introduced by Wintner in the 1940s, but in the last twenty years or so there has been an explosion of interest in them and of new, sometimes unexpected, results. I will try to give a gentle introduction to this area, and in particular to explain some of the ideas involved in my work on low moments of random multiplicative functions. Then I will explain how we can (sometimes) transfer results about random multiplicative functions to results about the deterministic multiplicative functions that we most want to understand.
Lecturer: Henna Koivusalo (University of Bristol)
Mon.-Thu. 16.-19.1.2023
at the Department of Mathematics and Statistics, University of Helsinki
on Monday 16.1. and on Wednesday 18.1. at 10-12: Exactum, C123
on Tuesday 17.1. and on Thursday 19.1. at 10-12: Exactum, B222
Course abstract: Cut and project sets are obtained by taking an irrational slice through a lattice and projecting it to a lower dimensional subspace. This usually results in a set which has no translational period, even though it retains a lot of the regularity of the lattice. As such, cut and project sets are one of the archetypical examples of point sets featuring aperiodic order. The definition of cut and project sets allows for many interpretations and generalisations, and they can naturally be studied in the context of dynamical systems, discrete geometry, harmonic analysis, or Diophantine approximation, for example, depending on one's own tastes and interests.
We will discuss the definition and basic properties of cut and project sets, and give an overview of their uses in the various contexts they arise.
The Scientific Advisory Board of the Finnish Centre of Excellence in Randomness and Structures (FiRST) gathers in Helsinki on 14-15th November 2022. As part of the meeting programme, the researchers of FiRST give scientific talks that are open to everyone interested.
| Time | Place | Event |
|---|---|---|
| 09:15 – 09:45 | Chemicum, hall A110 | Talk: Matomäki, Chair: Saksman |
| 09:50 – 10:20 | Chemicum, hall A110 | Talk: Vuorinen, Chair: Hytönen |
| 10:20 – 10:55 | Coffee break | |
| 10:55 – 11:25 | Chemicum, hall A110 | Talk: Laarne, Chair: Lukkarinen |
| 11:30 – 12:00 | Chemicum, hall A110 | Talk: Koski, Chair: Zhong |
| Time | Place | Event |
|---|---|---|
| 09:15 – 09:45 | Chemicum, hall A110 | Talk: Vihola, Chair: Kuusi |
| 09:50 – 10:20 | Chemicum, hall A110 | Talk: Pere, Chair: Ilmonen |
| 10:20 – 10:55 | Coffee break | |
| 10:55 – 11:25 | Chemicum, hall A110 | Talk: Adame-Carrillo, Chair: Kytölä |
| 11:30 – 12:00 | Chemicum, hall A110 | Talk: Peltola, Chair: Koskela |
Abstract: conformal field theory (CFT) is a powerful tool with astonishing predictions about statistical models. A key feature of CFT is the Virasoro algebra – it encodes the conformal symmetries of the theory. This information is encapsulated in the algebraic structure of the field content, which, in CFTs, is a representation of the Virasoro algebra. In this short presentation, I will discuss our methods to (rigorously) construct a Virasoro representation on the space of local fields of the double-dimer model at lattice level with central charge c = −2.
Abstract: the Burkholder functional Bp is perhaps the most important rank-one convex functional in the plane due to its various connections to singular integrals, martingales, and the vector valued calculus of variations.Bp is therefore a prime candidate to consider the validity of the notorious Morrey conjecture for. The Iwaniec conjecture concerning the exact norms of the Beurling transform is also equivalent to the quasiconvexity of the Burkholder functional. In this talk we discuss recent joint work together with K. Astala, D. Faraco, A. Guerra, and J. Kristensen. We study the Burkholder functional from the context of nonlinear elasticity, where the axiom of non-interpenetration of matter gives natural motivation to consider a restricted class of mappings which satisfy Bp(Df) ≤ 0 pointwise. In particular, we show the quasiconvexity of Bp in this class.
Abstract: I will introduce Jean Bourgain’s (1994) probabilistic approach to solving certain PDEs where deterministic methods cannot give enough regularity. Particular examples are the nonlinear Schrödinger and wave equations on d-dimensional tori, which are connected e.g. to the ϕ 4 quantum field theory. I will then outline our ongoing research (with Nikolay Barashkov) on extending these results to infinite volume.
Abstract: I will first introduce some fundamental concepts in analytic number theory such as primes and the Liouville function as well as briefly discuss classical results concerning them and their local distribution. Then I will move on to discussing recent progress concering the local distribution of almost primes and the values of the Liouville function.
Abstract: when studying large deviations (LDP) of Schramm-Loewner evolution (SLE) curves, a ”Loewner energy", and "Loewner potential”, that describe the rate function for the LDP were recently introduced. This object turned out to have several intrinsic, and perhaps surprising, connections to various fields. For instance, it has a simple expression in terms of zeta-regularized determinants of Laplace-Beltrami operators. On the other hand, minima of the Loewner potential solve a nonlinear first order PDE that arises in a semiclassical limit of certain correlation functions in conformal field theory, arguably also related to isomonodromic systems. Finally, and perhaps most interestingly, the Loewner potential minimizers classify rational functions with real critical points, thereby providing a novel proof for a version of the now well-known Shapiro-Shapiro conjecture in real enumerative geometry. Joint work with Yilin Wang (MIT / IHES).
Abstract: consider extreme value theory in functional settings. Extreme value index estimators of L p -norms of functional observations can be used to infer about extreme behaviour of infinite dimensional observations. However, in practice we do not have complete access to the L p -norm, as the underlying integral needs to be approximated or we have only observed, for example, the discretised process. In this work, we consider extreme value index estimators calculated with discretized L p norms. Assuming that L p norm of the sampled process is in the domain of attraction, we show that extreme value index estimators calculated with discretized L p norms are consistent and, under suitable additional regularity assumptions, asymptotically normal provided that the quality of the discretization is good enough (Joint work with Benny Avelin, Valentin Garino, Pauliina Ilmonen and Lauri Viitasaari).
Abstract: particle filters are important stochastic algorithms for statistical inference of a general class of dynamic models (the general state-space/hidden Markov models). The interaction of the particles in the algorithm is caused by so-called resampling step, which can be implemented in a number of different ways. We discuss the behaviour of the particle filter with different resamplings with a time-discretised path-integral model. Our theoretical results suggest that (certain variants of) existing resampling algorithms can be preferable in such a case. Empirical evidence support the theoretical findings.
Abstract: singular integral operators adapted to so-called Zygmund dilations are certain kind of multi-parameter operators. We discuss weighted norm inequalities, T(1) theory and dyadic representation of such operators.
Lecturers: Zaher Hani (University of Michigan), Yu Deng (University of Southern California)
Tue-Fr. 8.11.-11.11.2022
at the Department of Mathematics and Statistics, University of Helsinki
On Tuesday at 12-14 Exactum, C123: Lecture 1 (Slides)
On Wednesday at 14-16: Exactum, C124 Lecture 2 (Slides)
On Thursday at 14-16 Exactum, C123: Lecture 3
On Friday at 12-14 Exactum, C123: Lecture 4
ABSTRACT: Wave turbulence is the theory of nonequilibrium statistical mechanics for wave systems. Initially formulated in pioneering works of Peierls, Hasselman, and Zakharov early in the past century, wave turbulence is widely used across several areas of physics to describe the statistical behavior of various interacting wave systems. We shall be interested in the mathematical foundation of this theory, which for the longest time had not been established.
The central objects in this theory are: the "wave kinetic equation" (WKE), which stands as the wave analog of Boltzmann’s kinetic equation for interacting particle systems, and the "propagation of chaos” hypothesis, which is a fundamental postulate in the field that lacks mathematical justification. Mathematically, the aim is to provide a rigorous justification and derivation of those two central objects; This is Hilbert’s Sixth Problem for waves. In this minicourse, we shall describe our recent results in which we give a full resolution of this problem, namely a rigorous derivation of the wave kinetic equation, under a full range of scaling laws, and a justification of the propagation of chaos, in the context of the nonlinear Schrodinger equation.
Lecturer: Ellen Powell (Durham University)
Mo-Fr. 17.10.-21.10.2022
at the Department of Mathematics and Statistics, University of Helsinki
Monday at 14-16: Exactum, B222
From Tuesday to Friday at 14-16: Exactum, D122
ABSTRACT: One simple way to think of the Gaussian Free Field (GFF) is that it is the most natural and tractable model for a random function defined on either a discrete graph (each vertex of the graph is assigned a random real-valued height, and the distribution favours configurations where neighbouring vertices have similar heights) or on a subdomain of Euclidean space. The goal of these lectures is to give an elementary, self-contained introduction to both of these models, and highlight some of their main properties.
We will start with a gentle introduction to the discrete GFF, and discuss its various resampling properties and decompositions. This will assume knowledge only of Gaussian random variables and elementary properties of random walks (that will be reviewed along the course). We will then move on to the continuum GFF, which can be obtained as an appropriate limit of the discrete GFF when it is defined on a sequence of increasingly fine graphs. We will explaining what sort of random object (i.e, generalised function) it actually is, and how to make sense of various properties that generalise those of the discrete GFF.
Wednesday 24.8.-Thursday 25.8.2022
Department of Mathematics and Statistics
University of Helsinki, Exactum, Room D123
WEDNESDAY:
11:00 - 11:40 Matti Vihola: 'Conditional particle filters with diffuse initial distributions.'
Lunch
13:00 - 13:40 Tomas Soto: 'Resampling schemes for particle filters with weakly informative observations.'
14:00 - 14:40 Johanna Tamminen: "Uncertainty quantification in satellite remote sensing – From adaptive MCMC to climate change."
Coffee
15:20 - 16:00 Eric Moulines: 'Federated Learning meets Langevin Dynamics.'
THURSDAY:
11:00 - 11:40 Heikki Haario: 'Gaussian likelihoods for ’intractable’ situations.'
Lunch
13:00 - 13:40 Miika Kailas: '"Mass matrix adaptation for HMC and NUTS.'
Coffee
14:20 - 15:00 Alain Durmus: 'Boost your favorite MCMC: the Kick-Kac Teleportation algorithm.'
Thematic program "The Analysis and Geometry of Random Spaces" at the The Simons Laufer Mathematical Sciences Institute was co-organized by the Centre of Excellence in Randomness and Structures. Please visit the website of the program for more information.