Past talks

Janne Siipola

Neural networks are functions that have gained growing popularity in recent years. In this talk I will discuss about the basics of AI. We will begin with thinking what does one mean with the black box of AI and take a look at some of the recent applications. After that I show some of the efforts people have made to understand neural networks solutions within mathematical problems, especially within partial differential equations.

Expected Length: 30-45 minutes

Goal: Advertise neural networks, familiarize the audience to neural networks, give a glimpse into the mathematical theory on neural networks.

Target audience: PhD students

Matilda Häggblom

In team semantics formulas are evaluated in a set of assignments, called a team, as opposed to a single assignment under the usual semantics. Team semantics allow us to examine various relationships between variables, such as dependence, independence, exclusion, and inclusion.
We consider the implication problems for exclusion and inclusion: Does a finite set of exclusion (inclusion) atoms entail a given exclusion (inclusion) atom? We show that the implication problems are axiomatizable and prove them by building counterexample teams.

Expected Length: 25-30 minutes.

Goal: Explain in a simple way how to build teams to prove some implication problems for dependence atoms.

Target audience: PhD and MSc students.

Petri Laarne

You can make some divergent sums converge just by randomly flipping the signs of the summands. The price to pay is that the convergence is only almost sure, but in exchange one gets almost unbelievably good probabilistic bounds. In this talk we’ll explore the proof and its ingredients, and also peek at some applications.

Expected Length: 30 to 45 minutes.

Goal: Showcase a nice result, and try to unravel the ideas behind a proof to a general audience.

Target audience: Some knowledge of probability (e.g. Todennäköisyyslaskenta II) and analysis (L^p spaces) might make the talk more relevant, but is not strictly necessary.

Susanna Heikkilä

We define the quasiregular ellipticity of a closed Riemannian manifold; the focus is on the intuitive idea and not on technical details. We also discuss examples of both quasiregularly elliptic and not quasiregularly elliptic closed Riemannian manifolds.

Presentation slides

Expected Length: 30 minutes.
Goal: Practice planning and giving a talk on my research topic, which is accessible to a general mathematical audience.
Target audience: PhD and MSc students.

Sami Vihko

What lies beyond Laurent Schwartz’s theory of distributions? In this talk, I aim to provide one possible answer to this question: Mikio Sato’s theory of hyperfunctions.

Distributions and hyperfunctions are different classes of generalized functions, mathematical objects that have “function-like” qualities but do not strictly fit into the classical definition of a function of real or complex variables. While distributions are linear functionals from the space of test functions of real variables to the real numbers, hyperfunctions can be viewed as boundary values of complex analytic functions in some sense. Distributions can also be viewed as boundary values of analytic functions, but then there will be some growth conditions. Formally, if we remove these growth conditions, we obtain the hyperfunctions.

In this talk, after the formal introduction, we will first take a brief look at the theory of distributions in an arbitrary dimension d. Towards the end of this part, we will restrict our discussion to the case of d = 1 and present a representation result for distributions to pave the way for hyperfunctions. Subsequently, we will define the space of hyperfunctions and present some properties of them. Finally, we will demonstrate that the theory of hyperfunctions encompasses the theory of distributions.

We aim to keep the prerequisites to a minimum, but we do assume some background in complex/real analysis. At the very least, you should have a good understanding of what a complex/real analytic function is. While no knowledge of distribution theory is required, some familiarity with it is probably useful. Our objective is to provide a brief overview of this subject from an elementary point of view, so we will not present detailed proofs.

Presentation slides

Expected Length: 45 minutes.
Goal: The goal of this talk is to practice presenting a subject that I have recently learned in a manner that is easily understandable.
Target audience: PhD students.

Patrik Nummi

In this talk, I will discuss concentration inequalities in probability theory. Broadly speaking, these are mathematical tools that seek to quantify how likely it is for a random variable to deviate from some number (e.g. its mean) by a certain amount. In addition to being interesting on their own right, these bounds are useful in many applied fields of stochastics, such as in Extreme value theory (quantifying probabilities of extremely rare events); Algorithmic analysis (proving bounds on the error probability of randomized algorithms); Statistical inference (providing bounds on the probability that an estimator deviates significantly from its true value); and Machine learning (providing bounds on the difference between empirical and expected risk). I will start with the well-known workhorses of real analysis, the Markov and the Chebyshev inequalities, and discuss how one can obtain further tail probability bounds from these elementary inequalities; in particular, we will consider sums of independent random variables, for which the Hoeffding and the Bernstein inequalities offer exponentially decaying bounds on the probability of deviations from expected values. I will also briefly discuss how these ideas can be generalised to higher dimensions (e.g. to Banach-valued random elements) and heavier-tailed distributions by introducing suitable Orlicz spaces, and their alternatives.

Expected Length: 45 minutes.
Goal: To introduce some basic techniques in estimating tail probabilities of random variables to a non-specialist audience.
Target audience: Anyone who knows basic measure theory and also some undergraduate-level stochastics, i.e. has taken an ‘Intro to probability’ type of course at some point.

Lauri Tarpila

Phrases like ”All models are wrong”, “This could all be just happening be in your head” or “We need assumptions for science to work” are common for someone who delves deep into science. In my presentation I am trying to clarify the origin of these phrases and explain it in a scientific way. My conclusion is that science could be understood as a tool for “unscientific” matters rather than as an objective and justified point of view on its own. This interpretation is, in my opinion, the most consistent with all the issues in the previously shown phrases and with scientific thinking. I will try to validate this by showing how, in my opinion, it is impossible to verify assumptions with statistical tests and unambiguous model selection is impossible.

Presentation slides

Expected Length: 30 minutes.
Goal: Sharing new information and testing my hypothesis about the issues of statistical testing.
Target audience: Anyone who knows the basics of statistical testing and preferably linear models. A high understanding of mathematics or statistics is beneficial.

Sara Parikka

Mathematical flexibility refers to the combination of mathematical skills and understanding that allows for an optimal approach to solving equations. Flexibility is an essential aspect of a deeper understanding of strategies, their application, and the development of mathematical competence and thinking. The role of flexibility in mathematical proficiency has gained increasing attention in both research and teaching. Previous studies on flexibility have primarily focused on mental arithmetic and elementary algebra among middle school students. At the university level, mathematical flexibility has been examined in the context of differential calculus and basic computation. Knowledge about flexibility in different areas of mathematics remains limited, and research on flexibility in the context of university students is scarce, despite its crucial role in mathematical competence and learning.

Expected Length: 30 minutes.
Goal: The aim of this study is to investigate the mathematical flexibility of students in the first-year mathematics course, particularly in the context of solving equations.
Target audience: Anyone who is interested in the type of educational research conducted in mathematics at the university.

Evading Epsilons with Ultrafilters
Ultrafilters and ultralimits provide a generalized and convenient (though non-constructive) theory of convergence and other related topological notions such as compactness that avoids many of the technicalities of managing epsilons and taking subsequences of subsequences. I will discuss some of the basic definitions and results related to ultrafilters such as the fact that every bounded sequence of real numbers has an ultralimit.

Expected time: 20 or maybe 30 minutes.
Goal: Practise giving a talk and explain the basics of ultrafilters.
Target audience: Regular DOMAST seminar attendees.

Presentation slides

Szemeredi’s Regularity via Counting Measures
In this talk I plan to sketch a proof of Szemerédi’s graph regularity lemma which uses tools from pseudofinite model theory. I will review the basic ultraproduct construction and I explain how one can extend measures over a class of finite structures to their ultraproduct. Finally, if time permits, I will briefly describe the strengthenings of Szemeredi’s regularity that follow once we consider graphs with nice model theoretic properties.

Expected length: 45 minutes.
Goal: Show an application of ultrafilters and model theory to graph theory.
Target audience: Regular DOMAST seminar attendees.

Presentation slides

High school students and vectors: What kind of misconceptions arise?
The study is based on the Spring 2020 Advanced mathematics matriculation examination, question 2, with over 13,000 students taking the exam. Most of the students do very well. However, there are some widespread errors. We will briefly present the format of the Finnish electronic matriculation examination and then go further in to the found misconceptions the students had.

Expected length: 20 min.
Goal: Introduce the Finnish matriculation examination and to present the findings from our research on the found misconceptions.
Target audience: Fellow PhD students, but also master students, especially in the teacher program.

Presentation slides

Summamutikka: Mathematics science class
Do you know what all the mathematics science class, Summamutikka, a part of the department and of LUMA-center Finland does? How could you take part in it?

Expected length: 20 min.
Goal: Introduce Summamutikka, show some of our recent material and tell about possible collaborations.
Target audience: PhD students, but also anyone in the department or a student of the programs.

Bilevel optimization with single-step inner method
We propose a new approach to solving bilevel optimization problems. Main motivation for developing new approach is that some problems are computationally demanding with the previous approaches. The overall idea is to solve the full-system optimality conditions, but to precondition them to alternate between taking steps of simple conventional methods for the inner problem, the adjoint equation, and the outer problem. While the inner objective has to be smooth, the outer objective may be nonsmooth. We demonstrate good performance on learning the regularization parameter and the convolution kernel for image deconvolution with anisotropic total variation regularization. This talk also introduces ideas providing linear convergence proof for the approach.

Expected length: 20 minutes (or little more)
Goal: Develop my conference presentation of optimization topic better suited for audience whose main field is not optimization. Audience might be focused on some field that use optimization methods as a tool in their research.
Target audience: PhD students or more experienced scientists.

Presentation slides

Exceptionally in room Exactum B120.

There is way to associate a topological space to each abstract logic, in such a manner that important properties of the logic reflect as topological properties of the space. For instance, a logic is compact if and only if its corresponding space is compact.

In this talk, I will discuss the notion of a logic, define its associated topological space, and sketch ways to obtain information of the logic by studying its corresponding space. The ultimate motivation comes from the fact that in first-order model theory, the most important tool is a topological one: the so-called space of types. I will try to start from the beginning and motivate everything through a lot of examples.

Expected length: about 30 minutes
Goal: Practice giving a presentation at short notice
Target audience: PhD and MSc students

A brief introduction to non-standard analysis

In the 1690s, Leibniz and Newton developed calculus based on the notion of infinitesimally small quantities; e.g. the integral of a function f on an interval [a,b] is the (infinite) sum of the areas f(x)dx of infinitely thin rectangles, where x ranges over [a,b], f(x) is the height of the rectangle and dx, an infinitesimally small number, is the base of the rectangle. However, the notion of infinitesimals was ultimately deemed non-rigorous and was replaced as the basis of calculus by the epsilon-delta definition of a limit, due to Weierstrass in the 1870s. This mindset of epsilons and inequalities, as opposed to infinitely small distances, is still the basis of today’s analysis. However, Robinson in 1960 discovered a way to treat infinitesimals with the rigor of modern mathematics which lead to the birth of the field of non-standard analysis. In this talk, I will briefly describe the framework of hyperreal numbers and how infinitesimals can be used to give the basic definitions of calculus, following the intuition of Leibniz and Newton but with the rigor that is acceptable to a modern mathematician.

Expected length: 45 minutes
Goal: To practice giving a talk to a general mathematical audience
Target audience: PhD and MSc students, although advanced bachelor students with some knowledge of model theory (and first-year analysis courses) may also be able to follow

Presentation slides

Exceptional date; room Exactum CK111.

The categoricity of theories in second-order logic

Mathematicians are often interested in characterising various objects by describing what axioms they satisfy. A theory (think: a list of axioms) is categorical if it has only one model up to isomorphism: thus the theory completely characterises the model. Any finite structure can be finitely axiomatized with first-order logic, but if a first-order theory has an infinite model it will be non-categorical. Conversely, many natural structures (such as the real numbers as a complete ordered field) have categorical second-order axiomatizations. Therefore one may wonder how often this happens: are most second-order theories categorical? The answer to this question, in most cases, turns out to be independent of the axioms of set theory.

I will define second-order logic, motivate the question and (hand-wavingly) define and discuss various set-theoretic assumptions that have implications on the matter of second-order categoricity.

Expected length: 30 minutes
Goal: Modify my conference presentation to be somewhat understandable to a larger audience
Target audience: Regular DOMAST seminar audience

Numerical Solution of a Schrodinger Bridge Problem using Forward-Backward Iteration

The Schrodinger bridge problem was originally posed in 1931 and since then has found many applications. Here, we fix initial and final probability distributions and find a stochastic process that satisfies these while minimising a cost function (e.g. the Kullback-Leibler divergence) between a reference process. The problem can be rephrased as a system of partial differential equations (PDEs), which, in a particular case, simplify to two linear PDEs with coupled boundary conditions. I will describe an algorithm for computing a numerical solution for this type of problem that uses a forward-backward iterative procedure.

Expected length: 30 minutes
Goal: Explaining a numerical algorithm in an understandable way
Target audience: Regular DOMAST seminar audience

Sparse domination and weighted norm inequalities

Sparse domination is a recent technique in harmonic analysis that allows one to dominate various complicated operators by sums of averages taken over cubes that have “large” disjoint subsets. This type of domination is primarily motivated by the sharp quantitative weighted norm inequalities that arise from it.

In this talk I will introduce the fundamental definitions related to sparse domination and discuss some sparse domination results and weighted norm inequalities that follow.

Expected length: 30-45 minutes perhaps
Goal: Give an accessible exposition to sparse domination and its applications to weighted norm inequalities
Target audience: Regular DOMAST seminar audience

Presentation slides

Forcing and the continuum hypothesis – how to prove that we cannot prove

What Does Time Have to Do with Logic?: a Brief History of Logic, Time and Computation

Multiple Stochastic Integration and Homogenous Chaos

Presentation slides: patrik_nummi_DOMAST_presentation

Renormalization Group approach to Stochastic Quantization

Inverse problem meets computational topology

Exponential functionals of stochastic processes with applications

Geometry of quasiregular mappings and curves

Ritz method meets neural nets

In this domast seminar talk we turn from the highway of variational calculus to the path which starts in the Ritz method and ends, at the moment, to Deep Ritz method. In the talk I shortly present the original Ritz method and how it influenced the finite elements method and the Deep Ritz method, which is the latest computational method in this line of research.

Goal of this talk is presented above; present those connections to the audience. I try to skip the heavy details and proofs and focus on the main ideas as clearly as possible.

Fractional Gaussian fields and the multiplicative chaos measure

The talk is intended for anyone interested in obtaining some rudimentary knowledge about Gaussian random generalised functions from a real analysis perspective, and minimal prior knowledge of stochastics is required.

Presentation slides: fractional_gaussians_and_GMC_slides

Using games for measuring similarity between mathematical structures

We give a brief introduction to the so called Ehrenfeucth–Fraïssé games and how they can be used to determine whether two mathematical structures look similar in the eyes of logic. We also discuss some recent developments in metric variants of the games.

Presentation slides: Domast_Student_Seminar_2022

On the regularity theory of p-harmonic functions

Floquet transform and its applications to PDEs

We present the basics of Floquet transform in periodic domains. We then show some nice results for elliptic and or parabolic PDEs in periodic domains.

Invariant measures: what and why

Invariant measures arise in statistical mechanics. We illustrate their definition and properties on finite-dimensional spaces, and then move on to PDEs. We take a nonlinear Schrödinger equation on 1D torus as our example, and see how a random Fourier series yields an invariant measure. As the result, we get improved regularity theory of “almost all” solutions.

I assume that the audience has heard of differential equations, L^p spaces, and Fourier transform — but willingness to accept what I claim about them is enough. I will skip technicalities liberally, and each tool is independent of the others. The talk roughly follows my MSc thesis.

Presentation slides: Laarne_Domast_invariant_handout

Basic structure and methods of proof for random probability measures

Random probability measures serve as an interesting example of metric space-valued random variables. In this presentation, we will give a high-level overview of some basic methods of proof concerning uniform tightness, weak convergence, and the distributions of random probability measures.

Our hope is that by the end of the presentation, the audience will appreciate the notions of expectations of expectations and distributions of distributions in the context of random probability measures.

Presentation slides: random_measures

Insights into the regularity theory of elliptic partial differential equations: from Hilbert’s 19th problem to the 21st century

4D Dual-tree complex wavelet transform for time dependent data

Regularization is the key to solving ill-posed inverse problems, such as computing tomographic reconstructions from noisy and limited data, and sparsifying representation systems such as wavelets are a common and reliable choice. If we consider repeated cone-beam measurements of a dynamic object of interest, the regularized object is 4-dimensional (3D + time)! Motivated by this challenge we extend the dual-tree complex wavelet transform to 4D and apply it to sparse dynamic tomography regularization.

The aim is to give a rough idea on the construction of the 4D dual-tree complex wavelets, how the implementation can be made relatively simple and briefly illustrate what are the advantages of using complex valued wavelets over traditional real valued ones. Finally we compate both in 4D sparse dynamic tomography setting.

Evaluating Decision Makers over Selectively Labelled Data: A Causal Modelling Approach

(moved to take place one week earlier due to AGENT)

Commutator bounds using paraproducts and duality estimates

(postponed by one week due to DOMAST day, the new date is visible)

Wick polynomials and their application in the kinetic theory of the Discrete Nonlinear Schrödinger equation (DNLS)

Wick polynomials (also known as Wick products or Wick monomials) are linear combinations of random variables that satisfy nice additional properties that make them useful in the study of the time evolution cumulants of random variables. In this talk, I will give a basic overview of Wick polynomials and cumulants, some of their properties, as well as different ways of defining them. Time permitting, I will discuss how Wick polynomials can be used in the study of the kinetic theory of the Discrete Nonlinear Schrödinger equation.

Free Boundary Problems: An introduction

In this colloquium-style talk I will tell you about state-of-the-art in free boundary regularity. We will see what problems there are in the field, what methods are used to deal with them and what remains unsolved. No prerequisites, although standard PDE regularity knowledge wouldn’t hurt.