Analysis of the PDE seminar
Thursday, 14 Dec, 4:00 PM
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Venue: Room 114
Host: Neela Nataraj

Speaker: Gopikrishnan Remesan
Affiliation: IIT Palakkad

Title: Two-phase model of compressive stress induced on a surrounding
hyperelastic medium by an expanding tumour.

Abstract: In vitro experiments in which tumour cells are seeded in a gelatinous medium or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally supplied, diffusible nutrient (e.g., oxygen), affect the tumour’s growth dynamics. In this article, we present a mathematical model that describes these in vitro experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour’s growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour’s growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour’s equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination.

Mathematics Colloquium

Thursday, 14 December  at 5.15 PM

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Venue: Ramanujan Hall

Host: Sudhir Ghorpade

Speaker: Mahir Bilen Can

Affiliation: Tulane University, New Orleans

Title: Symmetric spaces, Hessenberg varieties, and Wonderful Compactifications

Abstract: Symmetric spaces appear in various branches of mathematics and physics. Their origins go back to Cartan's influential work on Riemannian geometry. In this talk after briefly reviewing symmetric spaces and their origins, we will discuss a special family of Hessenberg varieties in relation to K-orbit closures in flag varieties, where K is the symmetric subgroup S(GL(k)xGL(n-k)) in SL(n). Our goal is to explain how K-orbits can be used for understanding the geometry of Hessenberg varieties of semisimple operators with two eigenvalues. If time permits, we will shift gears towards wonderful embeddings of Hermitian symmetric spaces. We will discuss some applications of regular SL(2) actions in this setting.  Parts of this talk are based on my joint work with Martha Precup, John Shareshian, and Ozlem Ugurlu.