Mathematicians, statisticians, students, researchers - Everyone is welcome.

**In the spring we will award the best speaker with a surprise prize. The awarded person is chosen by a public vote.**

If you are willing to give a talk or have any questions you can contact

Antti Kykkänen (antti.k.kykkanen[at]jyu.fi) or

Janne Nurminen (janne.s.nurminen[at]jyu.fi).

**Abstract:**
In this seminar, we exhibit (and proof) some fundamental properties of Helmholtz equation (e.g. time-harmonic wave equation). By constructing a suitable **real-valued** fundamental solution, we able to obtain some consequences such as the mean value property and the maximum principle. For simplicity, here we only focus on 3-dimensional case. See the appendix in the arXiv version (arXiv:2204.13934) of my recent work (doi:10.1007/s11118-022-10054-5) for general results. These facts are quite elementary, but surprisingly not so well-known.

**Abstract:** I will talk about solutions to an elliptic PDE and how the boundary condition to this PDE can be used to control critical points (points, where the gradient of the solution vanishes) in the interior of the domain. In this context I will mention the weak maximum principle and Sobolev spaces (the natural space for the solutions). Furthermore, I will elaborate what happens, when the boundary condition is a discontinuous function so that the solution is not in the natural Sobolev space anymore. I will highlight how these insights can be used for an inverse problem of recovering the electrical condutivity from interior measurements of the domain.

**Abstract:**

**Abstract:** The real Hardy space is uniquely determined by the property that its dual (H)* = BMO is the space of bounded mean oscillation. I give a short introduction to both spaces, state and record some of their basic properties and give an elementary proof of the classical atomic factorization of the real Hardy space.

**Abstract:** I will introduce a method to show that a boundary value problem for a nonlinear PDE is well-posed, in some sense that will be specified in the talk. This method uses the implicit function theorem which in turn boils down to Banach fixed point theorem. In this talk the method will also be applied to an explicit PDE.

**Abstract:** In this talk, we discuss some qualitative properties of the Laplace equation such as maximum principal, mean value theorem, regularity and density property.

**Abstract:** In this seminar, I will again talk about something not related to my research and take a look at this simple sounding but suprisingly tricky geometric problem. We will discuss where the problem started, prove some basics results and try to understand what makes it so tricky to solve.

**Abstract:** I will give a short introduction to forward-backward stochastic differential equations, focusing on existence results and applications.

**Abstract:** In this talk, I will discuss the Harnack type inequalities for some elliptic and parabolic PDEs. I will also discuss an intrinsic Harnack inequalities for some nonhomogeneous parabolic equations in non-divergence form.

**Abstract:** I will give a short introduction on Shannon entropy and discuss the Asymptotic Equipartition Property in the simple case where the underlying stochastic process is a time series of an i.i.d. source.

**Abstract:** This talk is about second-order linear hyperbolic PDEs. The most famous, and simplest example of such a PDE is the wave equation $u_{tt} - \Delta u = 0$, where $\Delta$ is the Laplace operator. General second-order hyperbolic PDEs model the propagation of waves in a non-uniform or porous medium, with a possible source/sink and some underlying flow of the liquid, due to for example gravity. I will present results for the 3 main questions that arise whenever one has a PDE in their hands, 1) does a weak solution always exists, 2) is the weak solution unique, and 3) how smooth is the weak solution given the initial data. Depending on time I will also briefly discuss the behavior of the solutions, namely the finite propagation speed of the initial data.

**Abstract:** This is the second out of a three part talk on algebraic geometry. In this talk, we will look at projective geometry and projective varieties (geometric objects that are cut out by homogeneous polynomial equations). We will see why projective space is the natural space to work in within the realm of algebraic geometry. In the previous talk, saw that there is an algebro-geometric dictionary between different types of affine varieties and different types of ideals. Similarly, we will see that there is an algebro-geometric dictionary between projective varieties and homogeneous ideals.

**Abstract:** In this talk I aim to give a non-technical high-level description of the Standard Model (SM) of particle physics for the scientifically curious, with a focus on Quantum Chromodynamics, the theory of the strong nuclear force. First, I will introduce SM's elementary features and give a sample of interesting and exciting open research questions of contemporary research, both theoretical and experimental. In the second part, I describe the intricate non-Abelian behavior of the strong nuclear force as viewed through the internal structure of the proton. Finally, I will give a theorist's explanation of experimental particle physics such as is done at the LHC at CERN, before finishing with the key results of my PhD thesis.

**Abstract:** In this talk we briefly recall the classical result of Hilbert and Haar involving the bounded slope condition and Lipschitz minimizers. Then we discuss its generalizations to evolutionary settings.

**We will be awarding the prestigious young researcher's seminar award for the best speaker of the seminar series. The recipient is chosen by a public vote by the participants of the seminars. Also, we will hear a though or two from the winner.**

Fall 2022

Spring 2020

Autumn 2019

Autumn 2018

Autumn 2017

Spring 2014