phd seminar

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.

Siiri Kivimäki
The notion of a logic, topologically

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

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