Events
Department of Mathematics and Statistics
Texas Tech University
Reconnection is the process by which two approaching vortices cut and connect to each other. As a topological change event, it has been a subject of considerable fundamental interest for decades – not only in (classical) viscous and quantum fluids but also in many other fields, such as plasmas, polymers, DNAs, and macromolecules. For viscous fluid flows, reconnection is believed to play a significant role in various phenomena, such as turbulence cascade, fine-scale mixing, and aerodynamic noise generation. We first delineate the fundamental processes involved in vortex reconnection and its apparent role in turbulence cascade – a feature of turbulent flows that has eluded even the topmost brains. Vortex reconnection is also a fruitful avenue for understanding the long-standing and highly debated mathematical question regarding the occurrence of finite-time singularity of Navier-Stokes and Euler equations. We also show that reconnection is one of the main mechanisms for aeroacoustic noise generation. In addition, we address the helicity dynamics related to reconnection, including core dynamics, polarized vortex reconnection, and helicity conservation among different forms, e.g., link, writhe, and twist. Finally, the similarity between classical (viscous) and quantum reconnections are discussed.
This work is done with Dr. Jie Yao, TTU Mechanical Engineering PostDoc Researcher
Presented jointly with the Applied Math seminar group, watch this Colloquium at 3 PM CST via this zoom link.When considering the semi-discrete and fully discrete versions of the original "smooth" mKdV equation, particular choices must be made, and justifications for those choices are needed in order to believe the discretizations are correct. What makes a discretization "correct" is that it has a rich mathematical structure, often imitating, and sometimes expanding upon, the structure found in the original smooth equation. In the case at hand, such justifications have been established (by Inoguchi, Kaji, Kajiwara, Ohta, Matsuura, Park and others), and here we (joint work with Joseph Cho and Tomoya Seno) add another structural layer by connecting discretized mKdV equations with the transformation theory of curves in the plane. We will see how transformation theory of surfaces in 3-space has provided elegant approaches for discretizing surfaces, and curves as well, giving us a backdrop for describing the tools we need -- isoperimetric flows, and both discrete and infinitesimal Darboux transformations, of planar curves. The permutability characteristics of these transformations provide a computationally quick avenue to the discretized mKdV equations, bringing more geometry into the underlying mathematical structure, and additionally showing these discretizations are compatible with transformation theory of geometric objects.
This week's Biomath seminar details available at this pdf
It is known that the Betti numbers for any finitely generated module
over a local complete intersection ring grow on the order of a
polynomial. Further, it can be shown that, for large enough degree,
there are two polynomials of interest: one explicitly giving the even
Betti numbers and one giving the odd Betti numbers. The aim of this
talk is to show a bound on the discrepancy of these two polynomials
for every finitely generated module over a complete intersection with
respect to an invariant of the ring called its "quadratic
codimension". This is joint work with Lucho Avramov and Mark Walker.
Join Zoom Meeting https://zoom.us/j/96217128540?pwd=NjU5dzE2RjZvV0prejhOOWVjVENadz09
Meeting ID: 962 1712 8540
Passcode: 474170
We discuss equilibrium surfaces for an energy which is a linear combination of the classical bending energy for curves and a surface energy containing the squared L2 norm of the difference of the mean curvature and the spontaneous curvature, i.e. the Helfrich energy.
Please virtually attend this week's PDGMP seminar via this zoom link Wednesday the 17th at 4 PM.We will introduce C^∞-rings, which play the same role for smooth manifolds as commutative rings do for schemes. Then we will define differential graded C^∞-rings, introduce a model structure on them, and perform some computations of derived intersections.Systemic risk is the risk that the distress of one or more institutions triggers a collapse of the entire financial system. We extend CoVaR (value-at-risk conditioned on an institution) and CoCVaR (conditional value-at-risk conditioned on an institution) systemic risk contribution measures and propose a new CoCDaR (conditional drawdown-at-risk conditioned on an institution) measure based on drawdowns. This new measure accounts for consecutive negative returns of a security, while CoVaR and CoCVaR combine together negative returns from different time periods. For instance, ten 2% consecutive losses resulting in 20% drawdown will be noticed by CoCDaR, while CoVaR and CoCVaR are not sensitive to relatively small one period losses. The proposed measure provides insights for systemic risks under extreme stresses related to drawdowns. CoCDaR and its multivariate version, mCoCDaR, estimate an impact on big cumulative losses of the entire financial system caused by an individual firm’s distress. It can be used for ranking individual systemic risk contributions of financial institutions (banks). CoCDaR and mCoCDaR are computed with CVaR regression of drawdowns. Moreover, mCoCDaR can be used to estimate drawdowns of a security as a function of some other factors. For instance, we show how to perform fund drawdown style classification depending on drawdowns of indices. Case study results, data, and codes are posted on the web.
Please attend the Mathematical Finance seminar this Friday, March 19th via this zoom link at 2 PM, passcode 455846.