|
4:00pm to 5:00pm - RH 306 - Applied and Computational Mathematics Leonid Berlyand - (Penn State) Asymptotic stability in a free boundary PDE model of active matter. 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 development of new mathematical tools. We next focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two-dimensional free-boundary PDE model that generalizes a previous one-dimensional model by combining a Keller-Segel model, Hele-Shaw kinematic boundary condition, and the Young-Laplace law with a novel nonlocal regularizing term. This nonlocal term precludes blowup or collapse of the cell by ensuring that membrane-cortex interaction is sufficiently strong. We found a family of asymmetric traveling wave solutions bifurcating from stationary solutions. Our main result is the nonlinear asymptotic stability of traveling wave solutions that model observable steady cell motion. We derived and rigorously justified an explicit asymptotic formula for the stability determining eigenvalue via asymptotic expansions in a small speed of cell. Our spectral analysis reveals the physical mechanisms of stability/instability. It also leads to a novel spectral properties due to the non-self-adjointness of the linearized problem which is a signature of active matter out-of-equilibrium systems. This results in striking math features such as collapse of eigenspaces and presence of generalized eigenvalues and we determine their physical origins. This is joint work with V. Rybalko and C. Safsten published in Transactions of AMS (2023) and Phys. Rev.B, 2022. If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading. |
|
4:00pm - RH 306 - Mathematical Physics Leonid Berlyand - (Penn State) Asymptotic stability in a free boundary PDE model of active matter.
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 development of new mathematical tools. We next focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two-dimensional free-boundary PDE model that generalizes a previous one-dimensional model by combining a Keller-Segel model, Hele-Shaw kinematic boundary condition, and the Young-Laplace law with a novel nonlocal regularizing term. This nonlocal term precludes blowup or collapse of the cell by ensuring that membrane-cortex interaction is sufficiently strong. We found a family of asymmetric traveling wave solutions bifurcating from stationary solutions. Our main result is the nonlinear asymptotic stability of traveling wave solutions that model observable steady cell motion. We derived and rigorously justified an explicit asymptotic formula for the stability determining eigenvalue via asymptotic expansions in a small speed of cell. Our spectral analysis reveals the physical mechanisms of stability/instability. It also leads to a novel spectral properties due to the non-self-adjointness of the linearized problem which is a signature of active matter out-of-equilibrium systems. This results in striking math features such as collapse of eigenspaces and presence of generalized eigenvalues and we determine their physical origins. This is joint work with V. Rybalko and C. Safsten published in Transactions of AMS (2023) and Phys. Rev.B, 2022. If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading. |
|
1:00pm to 2:00pm - 440R Rowland Hall - Combinatorics and Probability Sam Mattheus - (UCSD) The geometry of bilinear forms in extremal graph theory Problems in extremal graph theory typically aim to maximize some graph parameters under local restrictions. In order to prove lower bounds for these kinds of problems, several techniques have been developed. The most popular one, initiated by Paul Erdős, is the probabilistic method. While this technique has enjoyed tremendous success, it does not always provide sharp lower bounds. Frequently, algebraically and geometrically defined graphs outperform random graphs. We will show how historically, geometry over finite fields has been a rich source of such graphs. I will show a broad class of graphs defined from the geometry of finite fields, which has found several recent applications in extremal graph theory. Often, certain interesting families of graphs had in fact already been discovered and studied, years before their value in extremal graph theory was realized. I will demonstrate some instances of this phenomenon as well, which indicates that there might still be uncharted territory to explore. |
|
1:00pm to 2:00pm - RH 510R - Algebra Be'eri Greenfeld - (UC San Diego) Growth of infinite-dimensional algebras, symbolic dynamics and amenability The growth of an infinite-dimensional algebra is a fundamental tool to measure its infinitude. Growth of noncommutative algebras plays an important role in noncommutative geometry, representation theory, differential algebraic geometry, symbolic dynamics, homological stability results and more. We analyze the space of growth functions of algebras, answering a question of Zelmanov (2017) on the existence of certain 'holes' in this space, and prove a strong quantitative version of the Kurosh Problem on algebraic algebras. We use minimal subshifts with highly correlated oscillating complexities to resolve a question posed by Krempa-Okninski (1987) and Krause-Lenagan (2000) on the GK-dimensions of tensor products. An important property implied by subexponential growth (both for groups and for algebras) is amenability. We show that minimal subshifts of positive entropy give rise to graded algebras of exponential growth all of whose representations are amenable, thereby answering a conjecture of Bartholdi (2007; extending an open question of Vershik). Finally, we discuss amenability-type properties of associative and Lie algebras related to this question. This talk is partially based on joint works with J. Bell and with E. Zelmanov. |
|
2:00pm - RH 306 - Mathematical Physics Xiaowen Zhu - (U Washington) Cantor spectrum for a moire model |
|
1:00pm - RH 114 - Graduate Seminar Jeff Streets - (UCI) TBA |