Seminars from until

In the 1960's, W. Leavitt studied a class of universal algebras which do not have a well-defined rank, i.e., algebras $L$ for which $L^m\cong L^n$ as $L$-modules with $m\lt n$, later known as the Leavitt algebra $L(m,n)$. In two simultaneuous but independent studies by G. Abrams and G. Pino, and P. Ara et al., an algebra arising from a directed graph $E$ and a field $K$ has been introduced called the Leavitt path algebra $L_K(E)$. This algebra turned out to be the generalization of $L(1,n)$. In fact, $L(1,n)\cong L_K(R_n)$ where $R_n$ is the graph having one vertex and $n$ loops.

In 2013, R. Hazrat formulated the Graded Classification Conjecture for Leavitt path algebras which claims that the so-called talented monoid is a graded Morita invariant for the class of Leavitt path algebras. In this talk, we will see Leavitt Path algebra modules as a special case of quiver representation with relations and how it will take a role in proving the conjecture.

More concretely, we will provide evidences and a confirmation in the finite-dimensional case of the conjecture.

This is a compilation of joint works with Wolfgang Bock, Roozbeh Hazrat, and Jocelyn Vilela.

When M is a Fano variety and D is an anticanonical divisor in M, mirror symmetry suggests that the quantum cohomology of M should be a deformation of the symplectic cohomology of M \ D. We prove that this holds under even weaker hypotheses on D (although not in general), and explain the consequences for mirror symmetry. We also explain how our methods give rise to interesting symplectic rigidity results for subsets of M. Along the way we hope to give a brief introduction to Varolgunes’ relative symplectic cohomology, which is the key technical tool used to prove our symplectic rigidity results, but which is of far broader significance in symplectic topology and mirror symmetry as it makes the computation of quantum cohomology “local”. This is joint work with Strom Borman, Mohamed El Alami, and Umut Varolgunes.

Duality is an important concept in the study of stochastic interacting particle systems. For arbitrary initial measures duality expresses expectations of a family of functions at time $t$ in terms of the transition probability of a dual process which may be simpler to analyse. Focussing on countable state space we discuss duality from the perspective of the generator. Unlike the more traditional approach of looking at duality in a pathwise manner this allows us to understand straightforwardly how dualities arise from symmetries, or more generally, from invariant subspaces of the generator and leads to constructive methods for finding useful dualities. Also the new concept of reverse duality comes out naturally. It yields the full probability measure of the process at time $t$ for a family of initial measures in terms of transition probabilities of the dual process and thus allows for the computation of arbitrary expectation values.

Shared-parameter joint models link longitudinal and time-to-event data, typically assuming that the conditional logarithm of the hazard function is linearly related over time to baseline covariates. However, this assumption is restrictive, making it crucial to test for linearity in key covariates. A useful approach consists of employing nonparametric smoothing techniques to compare the presumed linear shape with an orthogonal series expansion around it. The number of terms in the expansion is selected using a penalty-modified Akaike information criterion (MAIC). A numerical study validates the nonparametric MAIC-based testing procedure within the shared-parameter joint modeling framework, while the practical utility of the procedure is illustrated with a clinical trial of HIV-infected subjects.

Duality is an important concept in the study of stochastic interacting particle systems. For arbitrary initial measures duality expresses expectations of a family of functions at time $t$ in terms of the transition probability of a dual process which may be simpler to analyse. Focussing on countable state space we discuss duality from the perspective of the generator. Unlike the more traditional approach of looking at duality in a pathwise manner this allows us to understand straightforwardly how dualities arise from symmetries, or more generally, from invariant subspaces of the generator and leads to constructive methods for finding useful dualities. Also the new concept of reverse duality comes out naturally. It yields the full probability measure of the process at time $t$ for a family of initial measures in terms of transition probabilities of the dual process and thus allows for the computation of arbitrary expectation values.

We begin with a brief overview of the rapidly developing research area of active matter (a.k.a. active materials). These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires the development of new mathematical tools. We focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction. We introduce a minimal two-dimensional free-boundary PDE model that captures the evolution of the cell shape and nonlinear diffusion of myosin.

We first consider a linear diffusion model with two sources of nonlinearity: Keller-Segel cross-diffusion term and the free boundary that models moving/deformable cell membranes. Here we establish asymptotic linear stability and derive the explicit formula for the stability-determining eigenvalue.

Next, we consider the effect of nonlinear myosin diffusion, which results in the change of the bifurcation type from super- to subcritical, and we obtain an asymptotic representation of the bifurcation curve (for small velocities). This allows us to derive an explicit formula for the curvature at the bifurcation point that controls the bifurcation type. In the most recent work in progress with the Heidelberg biophysics group, we study the relation between various types of nonlinear diffusion and bistability.

Finally, we discuss novel mathematical features of this free boundary model with a focus on non-self-adjointness, which plays a key role in the spectral stability analysis. Our mathematics reveals the physical origins of the non-self-adjoint of the operators in this free boundary model.

Joint works with A. Safsten & V. Rybalko (Transactions of AMS 2023, and Phys. Rev. E 2022), with O. Krupchytskyi &T. Laux (Preprint 2024), and with A. Safsten & L. Truskinovsky ( Arxiv preprint 2024). This work has been supported by NSF grants DMS-2404546, DMS-2005262, and DMS-2404546.