## Jesse Zhang : Geodesic complexity of convex polyhedra

- Mathematical Biology ( 102 Views )Geodesic complexity of the d-dimensional boundary S of a convex polytope of dimension d+1 is intimately related to the combinatorics of nonoverlapping unfolding of S into a Euclidean space R^d following Miller and Pak (2008). This combinatorics is based on facet sequences, which are lists of adjacent facets traversed by geodesics in S. Our main result bounds the geodesic complexity of S from above by the number of distinct maximal facet sequences traversed by shortest paths in S. For d=2, results from the literature on nonoverlapping unfolding imply that this bound is polynomial in the number of facets. In arbitrary dimension d, a reinterpretation of conjectures by Miller and Pak (2008) leads to the conjecture that the geodesic complexity of S is polynomial in the number of facets. The theory and results developed here hold more generally for convex polyhedral complexes. This is joint work with Ezra Miller.

## Sean Lawley : Stochastics in medicine: Delaying menopause and missing drug doses

- Mathematical Biology ( 60 Views )Stochastic modeling and analysis can help answer pressing medical questions. In this talk, I will attempt to justify this claim by describing recent work on two problems in medicine. The first problem concerns ovarian tissue cryopreservation, which is a proven tool to preserve ovarian follicles prior to gonadotoxic treatments. Can this procedure be applied to healthy women to delay or eliminate menopause? How can it be optimized? The second problem concerns medication nonadherence. What should you do if you miss a dose of medication? How can physicians design dosing regimens that are robust to missed/late doses? I will describe (a) how stochastics theory offers insights into these questions and (b) the mathematical questions that emerge from this investigation. The first problem is based on joint work with Joshua Johnson (University of Colorado School of Medicine), John Emerson (Yale University), and Kutluk Oktay (Yale School of Medicine).

## Louis Fostier : A model of oocyte population dynamics for fish oogenesis

- Mathematical Biology ( 44 Views )We introduce and analyze a size-structured oocyte population model, with non local nonlinearities on recruitment, growth and mortality rates to take into account interactions between cells. We pay special attention to the form of the recruitment term, and its influence on the asymptotic behavior of the cell population.

This model is well-suited for representing oocyte population dynamics within the fish ovary. The nonlocal nonlinearities enable us to capture the diverse feedback mechanisms acting on the growth of oocytes of varying sizes and on the recruitment of new oocytes.

We firstly investigate the existence and uniqueness of global bounded solutions by transforming the partial differential equation into an equivalent system of integral equations, which can be solved using the Contraction Mapping Principle.

In a second step, we investigate the asymptotic behavior of the model. Under an additional assumption regarding the form of the growth rate, we can, with the use of a classical time-scaling transformation, reduce the study to that of a equation with linear growth speed and nonlinear inflow boundary condition. Using arguments from the theory of abstract semilinear Cauchy problems, we investigate the local stability of stationary solutions of this equation by reducing it to a characteristic equation involving the eigenvalues of the linearized problem around equilibrium states.

When the mortality rate is zero, the study of existence and stability of stationary solutions is simplified. Explicit calculations can be carried out in certain interesting cases.