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This Week in Mathematics

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Current Week [Jan 16, 2022 - Jan 22, 2022]
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Entries for this week: 9
Tuesday January 18, 2022

Geometry and Topology Seminar [url]
Other persistence invariants: homotopy and the cohomology ring
    - Ling Zhou, Ohio State
Time: 3:05p Room: 201
Abstract/Desc: In topological data analysis, persistent homology has been a major tool used for measuring both geometric and topological features of datasets. In this talk, we discuss the notions of persistent homotopy groups and persistent cohomology rings. This is motivated by the fact that homotopy groups and cohomology rings are in general stronger invariants than homology. For persistent homotopy, we pay particular attention to persistent fundamental groups for which we obtained a precise structural description via dendrograms, which induces an ultrametric on the standard fundamental group. Also, we describe the notion of persistent rational homotopy groups, which is easier to handle but still contains extra information compared to persistent homology. In the case of persistent cohomology, we consider a certain persistent graded ring structure induced by the cup product. We then lift the standard cup-length to obtain a persistent invariant which can be computed efficiently and, in analogy with the case of persistent homotopy, also complements the information captured by persistent homology.

Wednesday January 19, 2022

Departmental Tea Time
C is for cookie, and shorthand for C[0,1] w/the sup norm
Time: 3: Room: 204 LOV

Colloquium [url]
Modeling across-scale feedbacks of infectious diseases
    - Hayriye Gulbudak, University of Louisiana, Lafayette.
Time: 3:05 Room: Love 101
Abstract/Desc: A current challenge for disease modeling and public health is to understand pathogen dynamics across infection scales from within-host to between-host. Viral and immune response kinetics upon infection impact transmission to other hosts and feedback into population-wide immunity, all of which influence the severity, trajectory, and evolution of a spreading pathogen. In this talk, I will introduce structured partial differential equation models linking immunology and epidemiology in order to investigate coevolution of virus and host, multi-scale data fitting, and impacts of dynamic host immunity from an individual to the whole population. We apply the models to vector-borne diseases, such as Rift Valley fever (RVF) and dengue (DENV), with immunological and epidemiological data. Using invasion dynamics analysis and multi-scale numerical methods, we characterize different scenarios of virus-host evolution and coexistence of viral strains under waning host cross-immunity. In the case of DENV, we recapitulate how intermediate levels of pre-existent antibodies enhance infection within a host, and how to scale up to distributions of antibody levels among epidemiological classes in the host population to determine risk of severe DENV prevalence. These results have implications for optimal vaccination policy, and the modeling framework developed here is currently being applied to examine the emergence of COVID-19 variants partially resistant to antibodies induced by host infection or vaccination.

Undergraduate Math Major Seminar [url]
Honors in the Major at FSU
    - Michael Franklin, FSU Honors Office
Time: 3:05pm Room: LOV 107
Abstract/Desc: Introduction to Honors in the Major at FSU. All math students are welcome!

Biomathematics Seminar
Gulbudak Colloquium
    - Hayriye Gulbudak,
Time: 3:05
Abstract/Desc: The usual biomath colloquium is cancelled. Please attend Hayriye Gulbudak's colloquium talk instead.

Thursday January 20, 2022

Financial Mathematics Seminar [url]
Optimal capital structure with stochastic variable costs
    - Yerkin Kitapbayev, North Carolina State University
Time: 3:05pm Room: zoom
Abstract/Desc: We examine the optimal capital structure of a firm with stochastic revenues, stochastic variable costs, and fixed costs. In this two-state variable setting with stochastic operating leverage, we establish an Early Default Premium (EDP) formula for the value of equity and derive an integral equation for the endogenous default boundary, a function of variable costs. The value of debt, the endogenous coupon, the optimal leverage ratio and the credit spread are solved for. The impact of taxes, fixed costs and bankruptcy costs is assessed. [This is joint work with Jerome Detemple and Kristoffer Glover.]

Friday January 21, 2022

Colloquium Tea
Time: 3:00 pm Room: 204 LOV

Colloquium [url]
Towards a mathematical understanding of scientific machine learning: theory, algorithms, and applications
    - Yeonjong Shin, Brown University
Time: 3:05 Room: Love 101
Abstract/Desc: Modern machine learning (ML) has achieved unprecedented empirical success in many application areas. However, much of this success involves trial and error and numerous tricks. These result in a lack of robustness and reliability in ML. Foundational research is needed for the development of robust and reliable ML. This talk consists of two parts. The first part will present the first mathematical theory of physics-informed neural networks (PINNs) - one of the most popular deep learning frameworks for solving PDEs. Linear second-order elliptic and parabolic PDEs are considered. I will show the consistency of PINNs by adapting the Schauder approach and the maximum principle. The second part will focus on some recent mathematical understanding and development of neural network training. Specifically, two ML phenomena are analyzed -- "Plateau Phenomenon" and "Dying ReLU." New algorithms are developed based on the insights gained from the mathematical analysis to improve neural network training.

Machine Learning and Data Science Seminar
Varifold representation of layered shapes and transport by diffeomorphisms
    - Thomas Pierron, Paris Saclay and FSU
Time: 1:20 Room: Love 102
Abstract/Desc: This talk will present a model for the representation of layered shapes across different scales using varifolds. In computational anatomy, varifolds have been introduced as it allows to represent non-oriented shapes, from both a discrete and continuous point of view. More particularly, we will see how varifolds can model the stratified geometry of some objects such as biological tissues. The shape space is constructed as a metric space with metric comparison provided by optimal deformation of the objects. We consider a group of deformations that are diffeomorphisms of the ambient space, which allows to compute registration between shapes in the context of Large Deformations Metric Mapping (LDDMM).

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