phd seminar

Minna Hirvonen

A dependence or independence atom is a statement that some variables are in some sense (in)dependent. Examples of these include functional dependence from database theory and conditional independence from probability theory. A set of atoms S is said to logically imply another atom s if every suitable object (e.g. a database or a distribution) that satisfies all of the atoms in S also satisfies the atom s. An implication problem is the task of deciding whether a given set of atoms S logically implies another given atom s.

In this talk, I will define some well-known classes of atoms and explain how their implication problem can be applied in practice. I will also present some of my axiomatization results for implication problems for classes that combine qualitative and quantitative atoms.

Expected Length: 30-45 minutes.

Goal: To introduce the concept of implication problems to a general mathematical audience, to explain how they can be studied by constructing axiomatizations and algorithms, and to give examples of their applications.

Target audience: PhD and MSc students.

Julia Sanders

Diffusion models are used for image generation: if we add noise to an image we get noise, but, when reversed, we can use this process to generate new images. In the talk, I will give an overview of how these types of models work, and in particular take a more detailed look at the theory behind score based generative models.

Expected Length: Approx 30 mins.

Goal: Summarise a new concept in detail and help to improve my own understanding of it.

Target audience: Students in Domast.

Tapio Saarinen

Set theorists are known for independence results: showing that various statements cannot be proven or disproven from ZFC, the standard axioms of set theory. However, a classical result of Shoenfield’s in the field of descriptive set theory places limits on the possibility of independence. Namely, his absoluteness theorem shows that the truth of any \Sigma^1_2 statement is absolute between any two models of set theory that have the same ordinals, i.e. it is either true in both or false in both models. (A \Sigma^1_2 statement is an arithmetical statement about real and natural numbers of a certain form: for example, the Riemann hypothesis can be cast in such a form.)

I will define each unfamiliar word in the abstract, and then give a sketch of the proof.

Expected Length: 30-45 minutes.

Goal: Give a taste of descriptive set theory by sketching the proof of a nice result.

Target audience: Regular DOMAST attendees; of course any of the courses in the logic program will help.

Janne Nurmela

Greenhouse gas emissions and their contribution to the global warming are a major concern in today’s world. A huge part of these emissions originate from atropogenic sources (human activity) such as farming, avrious factories and power plants. However, natural sources have also clear contribution and they are not fully understood yet. Methane is the second most prominent contributor to the global warming, right after carbon dioxide, which is naturally emitted from decaying plants, cattle, wetlands etc.

In this talk, I will focus on wetland emissions and briefly show how (partial) differential equations can be numerically simulated and/or approximated with a steady state solution. I will also introduce how measurements of methane over wetland can be collected utilizing a drone and what kind of results we are hoping to see using the data.

Expected Length: ~40 min

Goal: Introduction on how gas dispersion can be numerically simulated and how these models can be used together with measurements in order to perform mathematical inversion for emission estimation.

Target audience: MSc students

Ensio Suonperä

Bilevel problems have been traditionally solved through either treating the inner problem as a constraint, and solving the resulting Karush-Kuhn-Tucker conditions using a Newton-type solver; or by trivialising the inner problem to its solution mapping. The latter approach in principle requires near-exact solution of the inner problem for each outer iterate. Moreover, an adjoint equation typically needs to be solved to calculate the differential of the inner problem solution mapping. Recently, intermediate approaches have surfaced that solve the inner problem and occasionally the adjoint as well to a low precision, and still obtain some form of convergence. In this talk, we discuss the methods based on taking interleaved steps of proximal-type methods on both the inner and outer problem, and computationally cheap steps for the adjoint. We demonstrate numerical performance on imaging applications.

Expected Length: Approx 30 minutes

Goal: Give a talk about my current research with less required background knowledge than my typical conference talks.

Target audience: PhD and MSc students. (Domast or related field)

Sami Vihko

One can always choose to study the mathematical questions or objects arising from physics almost completely independent of their origin. In this talk, I will take a different approach and aim to provide the background for the mathematical axioms of Quantum field theory. At least half of the talk will be on a formal level and trying to explain the road of physics formalism from Newton through Lagrange, Hamilton and Einstein to quantum mechanics and finally to quantum field theory. On the latter part of the talk, we are going to introduce two sets of axioms for quantum field theory, The Euclidean axioms originally due to Osterwalder and Schrader, and the axioms in the Minkowski space due to Wightman and Gårding. The mathematical framework for the first set is probability theory and for the latter Operator theory in Hilbert spaces. Both of them require some amount of distribution theory. The axioms themselves are quite technical and unfortunately due to the view point we are not going to give the detailed definitions of the objects involved. Thus, to get the most out of the latter part it is useful to have a relatively strong background in probability, operator and distribution theory. However, we hope that most of the talk can be followed with a wide general background in mathematics.

Expected Length: 45min

Goal: To introduce two sets of equivalent axioms of QFT to an audience with no prior experience on QFT or physics in general.

Target audience: Phd-students of mathematics