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.
An important geometric invariant of a hypersurface singularity is its Fukaya–Seidel category. In this talk, I will motivate and describe the study of two special families of isolated singularities. Time permitting, I will introduce “type A symplectic Auslander correspondence”, a purely geometrical construction which realises a notable result in representation theory.
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.
I will discuss geometric, algebraic, and combinatorial constructions related to loop spaces in topology. The talk will revolve around a functorial construction that models the passage from a topological space X to its free loop space LX. The input is a coalgebra equipped with additional structure and the output is a chain complex with a compatible “rotation” operator. The construction is dual in an appropriate sense to the Hochschild complex of a dg algebra/category. When applied to the coalgebra of chains on X, suitably interpreted, it produces a chain complex that is naturally quasi-isomorphic to the chains on LX with rotation operator corresponding to the circle action. This statement does not require any hypotheses on X (such as simple connectivity, nilpotence, finite type, etc…) or on the underlying ring of coefficients. The model turns out to be useful when studying and computing explicitly the structure of the free loop space of a manifold.
Mirror Symmetry predicts a correspondence between the complex geometry (the B-side) and the symplectic geometry (the A-side) of suitable pairs of objects. In this talk I will consider certain orbifold del Pezzo surfaces falling outside of the standard mirror symmetry constructions. I will describe the derived category of coherent sheaves of the surfaces (their B-side), and discuss early results on the A-side. This is joint work with Franco Rota.
The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm that iteratively samples from the conditional distributions of the probability measure of interest. Under the assumption of log-concavity, for its random scan version, we provide a sharp bound on the speed of convergence in relative entropy. Assuming that evaluating conditionals is cheap compared to evaluating the joint density, our results imply that the number of full evaluations required for the Gibbs sampler to mix grows linearly with the condition number and is independent of the dimension. This contrasts with gradient-based methods such as overdamped Langevin or Hamiltonian Monte Carlo (HMC), whose mixing time typically increases with the dimension. Our techniques also allow us to analyze Metropolis-within-Gibbs schemes, as well as the Hit-and-Run algorithm. This is joint work with Filippo Ascolani and Giacomo Zanella.
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.
In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.
In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.
In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.
