Upcoming talks
Wednesday 21 June
Speaker: Erika Hausenblas (Montanuniversität Leoben)
Title: A stochastic version of a Schauder-Tychanoff Theorem and systems of nonlinear SPDEs arising in biochemistry.
Abstract: Nonlinear partial differential equations appear naturally in many biological or chemical systems. E.g., activator-inhibitor systems play a role in morphogenesis and may generate different patterns. Noisy random fluctuations are ubiquitous in the real world. The randomness leads to various new phenomena and may have a non–trivial impact on the behaviour of the solution. The presence of the stochastic term (or noise) in the model often leads to qualitatively new types of behaviour, which helps to understand the real processes and is also often more realistic. Due to the interplay of noise and nonlinearity, noise-induced transitions, stochastic resonance, metastability, or noise-induced chaos may appear. Noise in stochastic Turing patterns expands the range of parameters in which Turing patterns appears.
The topic of the talk is a nonlinear partial differential equation disturbed by stochastic noise. Here, we will present recent results about the existence of martingale solutions using a stochastic version of a Tychanoff-Schauder type Theorem. In particular, we will introduce the stochastic Klausmeier system, a system which is not monotone, nor do they satisfy a maximum principle. So, the existence of a solution can only be shown using compactness arguments.
In the talk, we first introduce the System. Secondly, we will introduce the notion of martingale solutions and present our main result. Finally, we will outline the proof of our main result, i.e., the proof of the existence of martingale solutions.
Past talks
Wednesday 10 May
Title: B-splines, volumes of slices through simplices, and log-concavity.
Abstract: I'll go through some of the basic properties of B-splines outlined in the first few pages of the ground-breaking paper
H. B. Curry and I. J. Schoenberg, On Polya frequency functions IV: the fundamental spline functions and their limits, J. Analyse Math. 17 (1966), 71–107,
including the fact that a B-spline of degree d is the volume function of d-dimensional slices through a simplex in R^{d+1}. Using this and Brunn's theorem (which follows from the Brunn-Minkowski inequality), one can thus show that B-splines are log-concave.
Wednesday 26 April (two talks!)
Patterns & Non-Patterns in Permutations
Speaker: Richard A. Brualdi
Abstract: Permutations are fundamental objects in all of mathematics; permutation matrices are their matrix analogues. A pattern in a permutation is a subsequence of a specified type, for instance, a permutation with the pattern 4321 has a decreasing subsequence of length 4, such as 536214 (viz. 5321). The permutation 236142 avoids the pattern 4321. A square (0,1)-matrix A contains in general many permutation matrices ($P\le A$); they may all contain a specified pattern or they may all avoid a specified pattern. In this talk we will focus on the pattern 12...k, i.e., increasing subsequences of length k.
11.30: Inertias of matrices and sign patterns related to a system of second order ODEs
Speaker: Adam Berliner
Abstract: An inverse eigenvalue problem is any problem concerning construction of a matrix given desired information about eigenvalues. Much of the research in this area involves matrices whose zero-nonzero patterns correspond to graphs or directed graphs. Real matrices of the form $C = \begin{bmatrix} A & D \\ I & O\end{bmatrix}$, where $A$ is irreducible and $D$ is a positive diagonal matrix, occur in applications that give rise to systems of linear second order differential equations. In this talk, we discuss the inverse eigenvalue problem, in particular results relating to refined inertia, for such a matrix $C$ and its corresponding sign pattern matrix $\mathcal{C}$. The set of possible refined inertias of general sign pattern matrices $\mathcal{C}$ of this form is determined, and is completely specified in certain cases.
About the speakers:
Richard Brualdi is prof. em. at the University of Wisconsin, Madison. He is editor-in-chief of the journal Linear Algebra and Its Appl., he has written several books in combinatorics and in combinatorial matrix theory, and he has published more than 300 papers in these and related areas.
Adam Berliner is professor at St. Olaf College, Minnesota. His research is in matrix theory and graph theory, including the study of eigenvalues of structured classes of (square) matrices associated with graphs or certain differential equations.
Wednesday 8 March
Speaker: Kunlun Qi
Title: Stability and Convergence Analysis of the Fourier-Galerkin Spectral Method for the Boltzmann Equation
Abstract: Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by utilizing the "spreading" property of the collision operator. In this talk, we introduce a new proof based on a careful $L^2$ estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions. This talk is based on some joint works with Tong Yang (PolyU Hong Kong) and Jingwei Hu (University of Washington).
Wednesday 1 February
Speaker: Giuseppe Coclite (University of Bari)
Title: Adhesion and decohesion in a thermoelastic continuum
Abstract: Motivated by a cryogenic process for the recycling of photovoltaic crystalline modules, we present a model that describes the evolution of the temperature and displacement fields in a one dimensional string attached to a rigid substrate through an adhesive layer. This adhesive interaction is characterized by a nonlinear term describing the adhesion force exhibiting discontinuities when a critical value of the displacement is reached. We study the well-posedness of the problem under Neumann boundary conditions in the Fourier regime of heat propagations and investigate the long time dynamics.
This talk is based on joint works with N. De Nitti, G. Devillanova, G. Florio, M. Ligabo', F.Maddalena, G. Orlando, and E. Zuazua.