Domast PhD seminar

The Domast student seminar is an informal seminar for doctoral students of mathematics and statistics, as well as (advanced) Master’s degree level students interested in research. The aim is to give students an opportunity to develop science communication and presentation skills and to get a peek into other fields than their own. Students of Domast may obtain study credits for presenting in or organising the seminar. After every speaker there will be a relaxed discussion as well as a feedback session, so be prepared to take part in these if you attend. The seminar is organised for the first time in Autumn 2020 and will be developed along the way.

Everyone is welcome and we hope to have speakers from all Domast fields!

If you wish to give a talk in the Autumn 2022 edition of the seminar or have other questions, contact the organisers Jaakko Sinko or Aleksis Vuoksenmaa (firstname.lastname@helsinki.fi). Patrik Nummi (firstname.lastname@helsinki.fi) was the organiser of the seminar in the Spring of 2022.

The seminar is held bi-weekly on Fridays at 14-16, in room D123 in Exactum.

The seminar schedule for the current semester is found below.

Autumn 2022 Speaker
September 2nd 2022

Susanna Heikkilä

Differential forms in Euclidean spaces

In this talk, we will discuss the basics of differential forms. In particular, we will only work with differential forms in Euclidean spaces and manifolds will not appear in the talk. Differential forms in Euclidean spaces provide an alternative, more general, approach to multivariable calculus.

We will begin by presenting necessary preliminaries in multilinear algebra. After introducing the definition of differential forms, we will study the exterior derivative. For the remainder of the talk, we will consider closed and exact differential forms. We will also cover multiple examples throughout the talk.

The talk is intended for anyone interested in expanding their general knowledge in mathematics. Thus, the talk is aimed at a broad audience and elementary knowledge of linear algebra and vector analysis is sufficient for attending the talk.

Presentation slides: differential_forms_domast

September 16th 2022 Otto Rajala

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

I will present the main ideas of set theoretical forcing and sketch a proof showing that the continuum hypothesis cannot be proven true or false from the axioms of set theory.

The most famous result in the history of set theory is the proof that the continuum hypothesis (CH)  cannot be proven true or false from the usual axioms of set theory. CH claims that there is no set that is larger than the set of natural numbers but still strictly smaller than the set of real numbers. Forcing is the main technique used to construct models that show that the axioms do not prove or disprove a given statement.  It was first developed specifically to prove the independence of CH.

I will start by introducing the basics of set theory and define ordinals and cardinals and then continue to present the basic ideas of forcing. I will then sketch how we can apply forcing to show the independence of CH by building a model where CH does not hold and another model where CH is true. If time permits, I will briefly present my own work.

The talk does not presuppose any background knowledge in set theory or logic. I try to present the ideas in an intuitive and untechnical manner, so that the talk can be interesting to any mathematician who would like to know a bit more about set theory.

September 30th 2022 Max Sandström

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

In this talk I aim to give a conceptual and historical background to my research, in order to give perspective on the field of temporal logic and its applications in computer science. I will take you on a sightseeing tour across millennia, showing how considerations of time have played a part in the development of logic, how the emergence of digital computers gave these considerations a renewed focus, and how formal approaches to the study of dependence can plot a course for the field into the future.

The talk will not be technical at all and is suitable for a general audience regardless of background or experience with logic.

October 14th 2022 Patrik Nummi

Multiple Stochastic Integration and Homogenous Chaos

In this talk, I will give a brief introduction to the theory of stochastic integration with respect the Brownian motion process, with a particular focus on multiple integration. I will then move to a slightly more abstract Hilbert space setting to introduce the concept of homogenous chaos, or Wiener polynomial chaos, dealing with representing L^2-random variables in terms of polynomial functions of Gaussian random variables. Finally, the aim of the talk will be to introduce the Wiener-Itô theorem, which states a fundamental connection between multiple stochastic integrals and the orthogonal decomposition of L^2​ given by homogenous chaos.

The talk is given from a real / functional analysis point of view, and is intended for anyone interested in obtaining some rudimentary knowledge about stochastic integration and its connection to Hilbert space theory. Thus minimal prior knowledge of stochastics is required.

Presentation slides: patrik_nummi_DOMAST_presentation

October 28th 2022 Ilaria Pia

An introduction on statistical model comparison and selection: objectives, methods and applications

Statistics is a science based on data. When implementing a statistical model using reliable and unbiased data as well as defining the objective of our model is crucial.Furthermore, to select a good model, we need to be able to properly evaluate its performance.

In this talk I will introduce statistical model selection methods, while addressing the following questions: what are the main steps to follow when implementing a statistical model and how do we select a good model?

After a brief introduction about statistical models and their principal objectives, I will present an overview of the most popular techniques in model selection, taking a decision theory point of view, with a focus on Bayesian statistics. I will conclude with some applied example in ecological statistics.

The talk is targeted to anyone interested in gaining some knowledge about statistical model selection, from a general perspective. Not much technical terminology is involved, and no previous statistical knowledge is required.

November 11th 2022 Antti Aro

Renormalization Group approach to Stochastic Quantization

In this talk I explain the ideas behind using Renormalization Group to study physical systems at different scales and the use of Stochastic Quantization to study quantum mechanics through the use of stochastic partial differential equations. I will also give the broader historical context and justification for these concepts. Finally these ideas are combined into a simple calculation aimed at understanding quantum mechanics on arbitrarily small scales. The talk is aimed at a broad audience with general knowledge of mathematics and physics.

The goal of the talk is to explain what Renormalization Group and Stochastic Quantization are, their broader historical context and how they look in an actual calculation.

Presentation slides: RGSQpresentation

November 25th 2022 Elli Karvonen

Inverse problem meets computational topology

In this talk, I will present the main ideas of my current research, which combines inverse problems and computational topology, namely limited angle tomography and persistent homology.

Limited angle tomography is an X-ray imaging technique that allows one to study the interior structures of an object. It results in a much harder reconstruction problem than a full-angle case. Despite an algorithm, certain parts of the object’s boundaries cannot be detected stably in limited angle cases. This means parts of the boundaries are missing. We aim to estimate the missing parts with the help of persistent homology.

Persistent homology is a computational tool to measure topological features of the data, i.e., different dimensional holes. Since the boundaries of the object form loops, we can implement homology to detect does estimated boundaries’ parts occur together with known boundaries.

The talk does not presuppose any background knowledge of inverse problems or homology. The content of the talk is not highly technical and should be accessible to a broad audience.

December 9th 2022 Jonas Lindblad

[Title TBA]

[Abstract TBA]