First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities.  Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles.

We consider one-dimensional nonlinear Schrodinger equations around a traveling wave. We prove its asymptotic stability for general nonlinearities, under the hypotheses that the orbital stability condition of Grillakis-Shatah-Strauss is satisfied and that the linearized operator does not have a resonance and only has 0 as an eigenvalue. As a by-product of our approach, we show long-range scattering for the radiation remainder. Our proof combines for the first time modulation techniques and the study of space-time resonances. We rely on the use of the distorted Fourier transform, akin to the work of Buslaev and Perelman and, and of Krieger and Schlag, and on precise renormalizations, computations, and estimates of space-time resonances to handle its interaction with the soliton. This is joint work with Pierre Germain.

Results from a few years ago of Kennedy and Schafhauser attempt to characterize the simplicity of reduced crossed products, under an assumption which they call vanishing obstruction. 

However, this is a strong condition that often fails, even in cases of finite groups acting on finite dimensional C*-algebras. In this work, we give complete C*-dynamical characterization, of when the crossed product is simple, in the setting of FC-hypercentral groups. 

This is a large class of amenable groups that, in the finitely-generated setting, is known to coincide with the set of groups with polynomial growth.

We discuss Connes’ quantized calculus on quantum tori and Euclidean spaces, as applications of the recent development of noncommutative analysis.
This talk is based on a joint work in progress with Xiao Xiong and Kai Zeng.
 

I will give an introduction to quasidiagonality of group actions wherein an action on a C^*-algebra is approximated by actions on matrix algebras.  This has implications for crossed product C^*-algebras, especially as pertains to finite dimensional approximation.  I'll sketch the proof that all isometric actions are quasidiagonal, which we can view as a dynamical Petr-Weyl theorem.  Then I will discuss an interplay between quasidiagonal actions and semiprojectivity of C^*-algebras, a property that allows "almost representations" to be perturbed to honest ones.  

To every action of a discrete group on a compact Hausdorff space one can canonically associate a C*-algebra, called the crossed product. The crossed product construction is an extremely popular one, and there are numerous results in the literature that describe the structure of this C* algebra in terms of the dynamical system. In this talk, we will focus on one of the central notions in the realm of the classification of simple, nuclear C*-algebras, namely Jiang-Su stability. We will review the existing results and report on the most recent progress in this direction, going beyond the case of free actions both for amenable and nonamenable groups. 

Parts of this talk are joint works with Geffen, Kranz, and Naryshkin, and with Geffen, Gesing, Kopsacheilis, and Naryshkin. 

The (finite) Hilbert matrix is arguably one of the single most well-known matrices in mathematics. The infinite Hilbert matrix H was introduced by David Hilbert around 120 years ago in connection with his double series theorem. It can be interpreted as a linear operator on spaces of analytic functions by its action on their Taylor coefficients. The boundedness of H on the Hardy spaces H^p for 1 < p < ∞ and Bergman spaces A^p for 2 < p < ∞ was established by Diamantopoulos and Siskakis. The exact value of the operator norm of H acting on the Bergman spaces A^p for 4 ≤ p < ∞ was shown to be π /sin(2π/p) by Dostanic, Jevtic and Vukotic in 2008. The case 2 < p < 4 was an open problem until in 2018 it was shown by Bozin and Karapetrovic that the norm has the same value also on the scale2 < p < 4. In this talk, we introduce some background, review some of the old results, and consider the still partly open problem regarding the value of the norm on weighted Bergman spaces. We also consider a generalised Hilbert matrix operator and its (essential) norm. The talk is partly based on a joint work with Mikael Lindström, David Norrbo, and Niklas Wikman (Åbo Akademi University).
 

In ring theory, Morita equivalence is an invariant for many properties, generalising the isomorphism of commutative rings. A strong Morita equivalence for selfadjoint operator algebras was introduced by Rieffel in the 60s, and works as a correspondence between their representations. In the past 30 years, there has been an interest to develop a similar theory for nonselfadjoint operator algebras and operator spaces with much success. Taking motivation from recent work of Connes and van Suijlekom, we will present a Morita theory for operator systems. We will give equivalent characterizations of Morita equivalence via Morita contexts, bihomomoprhisms and stable isomorphisms, while we will highlight properties that are preserved in this context. Time permitted we will provide applications to rigid systems, function systems and non-commutative graphs. This is joint work with George Eleftherakis and Ivan Todorov.

One of the oldest open problems in representation theory is to classify the irreducible unitary representations of a semisimple Lie group G_R. Such representations play a fundamental role in harmonic analysis and the Langlands program and arise in physics as the state space of quantum mechanical systems in the presence of G_R-symmetry. Most unitary representations of G_R are realized, via some kind of induction, from unitary representations of proper Levi subgroups. Thus, the major obstacle to understanding the unitary dual of G_R is identifying the "non-induced" unitary representations of G_R. In previous joint work with Losev and Matvieievskyi, we have proposed a general construction of these non-induced representations, which we call "unipotent" representations of G_R. Unfortunately, the methods we employ do not provide a proof that these representations are unitary. In this talk, I will explain how one can apply Saito's theory of mixed Hodge modules to overcome this difficulty, giving a uniform proof of the unitarity of all unipotent representations. This is joint work in progress with Dougal Davis