Events
Department of Mathematics and Statistics
Texas Tech University
We apply ideas from algebraic topology to study distributions on object spaces. We present a framework for using persistence landscapes to vectorize persistence diagrams as in Bubenik (2015) and Patrangenaru et al. (2018). We apply these methods to analyze data from The Cancer Imaging Archive (TCIA), using a technique developed earlier for regular types of digital images. This talk includes brain images from CPTAC and breast images from CBIS-DDSM. Examples of applications are provided via analyzing Glioblastoma patients with similar images from clinically normal individuals and analyzing mammogram images for patients with benign and malignant masses. Results show persistence landscape may capture topological features distinguishing the two groups.
Watch online Monday the 16th at 4 PM via this Zoom link -- passcode 0q3jJzIn 1950s, J.-L. Lions generalized the deterministic Navier-Stokes equations when the space is n-dimensional by adding a diffusive term with a fractional Laplacian and claimed the uniqueness of its solution when the exponent of the fractional Laplacian is equal to or bigger than 1/2 + n/4. E.g., it has to be no smaller than 5/4 when n = 3. In 2020, Luo and Titi (also Buckmaster, Colombo and Vicol in a pre-print) proved the non-uniqueness of weak solution to such generalized Navier-Stokes equations when this exponent is strictly smaller than 5/4, complementing the work of Lions, by relying on the convex integration technique due to [Buckmaster and Vicol, 2019]. We prove an analogous result in the stochastic case; i.e., non-uniqueness in law of three-dimensional generalized Navier-Stokes equations with an exponent strictly less than 5/4 that is forced by random noise. An analogous result holds in the two-dimensional case as well.  | Tuesday
Nov. 17
3:30 PM
MATH 017
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| Real-Algebraic Geometry
Schemes
David Weinberg
Department of Mathematics and Statistics, Texas Tech University
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Immunity following natural infection or immunization may wane, increasing susceptibility to infection with time since infection or vaccination. Symptoms, and concomitantly infectiousness, depend on residual immunity. We quantify these phenomena in a model population composed of individuals whose susceptibility, infectiousness, and symptoms all vary with immune status. We also model age, which affects contact, vaccination and possibly waning rates. The resurgences of pertussis that have been observed wherever effective vaccination programs have reduced typical disease among young children follow from these processes. As one example, we compare simulations with the experience of Sweden following resumption of pertussis vaccination after the hiatus from 1979 to 1996, reproducing the observations leading health authorities to introduce booster doses among school-aged children and adolescents in 2007 and 2014, respectively. Because pertussis comprises a spectrum of symptoms, only the most severe of which are medically attended, accurate models are needed to design optimal vaccination programs where surveillance is less effective.
Watch online Tuesday the 17th at 3:30 PM via this zoom link
A construction of Tate shows that every algebra over a ring $R$ possesses a DG-algebra resolution over $R$. These resolutions are not always minimal and Avramov even shows that certain algebras cannot have a minimal resolution with a DG-algebra structure. In this talk, I will explicitly construct the minimal free resolution of $k [\![ \underline{x} ]\!] /\mathcal{I} \times_k k [\![ \underline{y} ]\!] / \mathcal{J}$ over $k [\![ \underline{x}, \underline{y} ]\!]$. From there, I will discuss when this minimal resolution exhibits a DG-structure.
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Symplectic integrators are commonly used to integrate Hamiltonian systems due to its desired property of exact energy conservation. However, symplectic integrators can be less efficient than conventional Runge-Kutta integrators. In this work, we consider a different approach based on deep learning to integrate Hamiltonian systems, and propose a novel symplectic neural network (HenonNet) to identify underlying flow maps from a finite dataset of discrete time maps. Our network architecture consists of layers that are symplectic maps due to its construction. Therefore, the network is a structure-preserving approximation to Hamiltonian systems and enjoy the same property symplectic integrators have. We further prove a universal approximation theorem of HenonNet to any C^r symplectic diffeomorphism.
As an example, we consider the Poincare maps for toroidal magnetic fields that are routinely employed to study gross confinement properties in fusion devices. In practice, evaluating a Poincare map requires numerical integration along a magnetic field line, a process that can be slow and that cannot be easily accelerated using parallel computations. We show that our network architecture is capable of accurately learning realistic Poincare maps from observations. After training, such learned Poincare maps evaluate orders of magnitude faster than non-symplectic Runge-Kutta integrators. Moreover, the learned network exactly reproduces the primary physics constraint imposed on field-line Poincare maps: flux preservation, which indicates its long-time stability and accuracy.
Watch online Wednesday the 19th at 4 PM via this Zoom link