**Speaker:** Mario Ullrich

**Time:**

- Wednesday, 8 November at 14-16 (Mathematical Physics seminar)
- Thursday, 9 November at 12-14 (Geometric and Functional Analysis seminar)

**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:**

- Wednesday, 18 October at 14-16 (Mathematical Physics seminar)
- Thursday, 19 October at 12-14 (Geometric and Functional Analysis seminar)
- Tuesday, 24 October at 14-16

**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.'