Cancer stem cells (CSCs) have been identified in primary breast cancer tissues and cell lines. The size of CSC population varies a lot among cancer tissues and cell lines but is associated with aggressiveness of breast cancer. In this study, we develop a mathematical model to explore the key factors which control the size of CSC during tumor cell growth both in vitro and in vivo. Our mathematical model and experimental data suggest that there is a negative feedback mechanism to control the balance between CSC and non-stem cancer cells. We further calculate how feedback sensitivities and robustness can be regulated by different intrinsic and extrinsic factors.
We show how to efficiently count exactly the number of solutions of a system of n polynomials in n variables over certain local fields L, for a new class of polynomials systems. The fields we handle include the reals and the p-adic rationals. The polynomial systems amenable to our methods are made up of certain A-discriminant chambers, and our algorithms are the first to attain polynomial-time in this setting. We also discuss connections to Baker's refinement of the abc-Conjecture, Smale's 17th Problem, and tropical geometry. The results presented are, in various combinations, joint with Martin Avendano, Philippe Pebay, Korben Rusek, and David C. Thompson.
In 2001, Donaldson proved that the existence of cscK metrics on a
polarized manifold (X,L) with discrete automorphism group implies the
existence of balanced metrics on L^k for k large enough. We show that the
similar statement holds if one twists the line bundle L with a simple
stable vector bundle E. More precisely we show that if E is a simple
stable bundle over a polarized manifold (X,L), (X,L) admits cscK metric
and have discrete automorphism group, then (PE^*, \O(d) \otimes L^k)
admits a balanced metric for k large enough.
Directed cell motility is a process whereby the motility machinery of the cell (involving the interaction of actin with myosin) is organized spatially so as to cause directed motion. In Dictyostelium, this occurs as the cell responds to cAMP gradients during the aggregation process. In keratocytes, the cell spontaneously polarizes itself (without external cues). This talk will focus on spatially extended modeling of both the signaling system which encodes the directional information and the downstream mechanical response and the comparison of these models to detailed experimental studies of both of these systems.
What could be simpler than to study sums and products of integers? Well maybe it is not so simple since there is a major unsolved problem: for arbitrarily large numbers N, can there be sets of N positive integers where both the number of pairwise sums and pairwise product is less than N^{3/2}?
No one knows. This talk is directed at another problem concerning sums and products, namely how dense can a set of positive integers be if it contains none of its pairwise sums and products? For example, take the numbers that are 2 or 3 more than a multiple of 5, a set with density 2/5. Can you do better?
For an integer n > 2, the unit group modulo n has an even number
of elements, with half of them having representatives in (0,n/2)
and the other half having representatives in (n/2,n). It is
"balanced". Say a subgroup H of this unit group is "balanced"
if each coset of H is evenly split between the bottom half and
the top half. Suppose g>1 is a fixed integer. We are concerned with
the distribution of numbers n coprime to g for which the
cyclic subgroup in the unit group mod n is balanced.
This has an application to the statistical study of the rank
of the Legendre curve over function fields. (Joint work with
Douglas Ulmer.)
Institute for Mathematics, Free University of Berlin
Time:
Friday, January 13, 2012 - 2:00pm
Location:
RH 440R
The shadowing problem is related to the following question: under which condition, for any pseudotrajectory approximate trajectory) of a vector field there exists a close trajectory? It is known that in a neighbourhood of a hyperbolic set diffeomorphisms and vector fields have shadowing property. In fact more general statement is correct: structurally stable dynamical systems satisfy shadowing property.
We are interested if converse implication is correct. We consider notion of Lipschitz shadowing property and proved that it is equivalent to structural stability for the cases of diffeomorphisms and vector fields.
Talk is based on joint works with S. Pilyugin and K. Palmer
