More and more, traveling waves are observed inside individual cells. These waves can be pulses of biochemical factors (diffusing proteins or metabolites) but also mechanical factors (such as the cell cortex). One example of mechanical traveling wave is offered by cellular blebs, pressure-driven “bubbles” on the cell surface implicated in cell division, apoptosis and cell motility. Blebs exhibit a range of behaviors including contracting in place, travel around the cell’s periphery, or repeated blebbing, making them biophysically interesting. Mechanical traveling waves are naturally modeled using “non-local” integro-PDEs, which lack the theoretical tools available for reaction-diffusion waves. This lack obfuscates simple questions such as what determines if a bleb will travel or not, and, if it travels, what determines its velocity? We present results in two parts: First, we develop a simple model of the cell surface describing the membrane, cortex, and adhesions, including the slow timescale cortical healing (treating implicitly the fast timescale of fluid motion). We find traveling and stationary blebs, which we characterize through numerical simulation. In the second part, we review the so-called Maxwell condition for reaction-diffusion systems that determines whether an excitation will travel or recover in place. We present our progress in deriving an analogue of the Maxwell condition for non-local integro-PDEs suitable for our cell surface model. This condition allows the theoretical (simulation-free) elucidation of blebbing including bleb travel.
Up until the mid 70s the kind of spectra most people had in mind in the
context of theory of Schrodinger operators were spectra occurring for
periodic potentials and for atomic and molecular Hamiltonians. Then
evidence started to build up that "exotic" spectral phenomena such as
singular continuous, Cantor, and dense point spectrum do occur in
mathematical models that are of substantial interest to theoretical
physics. One area where such exotic phenomena are particularly abundant is
quasiperiodic operators. They feature a competition between
randomness (ergodicity) and order (periodicity), which is often resolved
at a deep arithmetic level. Mathematically, the methods involved include a
mixture of ergodic theory, dynamical systems, probability, functional and
harmonic analysis and analytic number theory. The interest in those models was enhanced by strong
connections with some major discoveries in physics, such as integer
quantum Hall effect, experimental quasicrystals, and quantum chaos theory,
in all of which quasiperiodic operators provide central or important
models.
We will give a general overview concentrating on aspects where the
competition and/or collaboration between order and chaos plays an
important role
Stem cells are an important component of tissue architecture. Identifying the exact regulatory circuits that can stably maintain tissue homeostasis (that is, approximately constant size) is critical for our basic understanding of multicellular organisms. It is equally critical for figuring out how tumors circumvent this regulation, thus providing targets for treatment. Despite great strides in the understanding of the molecular components of stem-cell regulation, the overall mechanisms orchestrating tissue homeostasis are still far from being understood. Typically, tissue contains the stem cells, transit amplifying cells, and terminally differentiated cells. Each of these cell types can potentially secrete regulatory factors and/or respond to factors secreted by other types. The feedback can be positive or negative in nature. This gives rise to a bewildering array of possible mechanisms that drive tissue regulation. In this talk I describe a novel stochastic method of studying stem cell lineage regulation, which is based on population dynamics and ecological approaches. The method allows to identify possible numbers, types, and directions of control loops that are compatible with stability, keep the variance low, and possess a certain degree of robustness. I will also discuss evolutionary optimization and cancer-delaying role of stem cells.
For a sparse polynomial f(x) of high degree and few terms
over a non-algebraically closed field F, the number of F-rational
roots is often very small. In the case F is the real numbers, this
is the famous Descartes's rule. In the case that F is a finite field,
the situation is much more complicated. In this lecture, we discuss
some recent results and conjectures in this direction, both
theoretical and numerical.
In 1932 von Neumann proposed classifying the statistical behavior of physical systems. The idea was to take a diffeomorphism of a compact manifold and describe what one might observe as random (as in coin flipping) or predictable (as in a translation on a compact group), or even better have a dictionary in which one could look up the precise behavior.
Remarkable progress was made on this problem; benchmarks include the Halmos-von Neumann theorem on discrete spectrum and the work of Kolmogorov on Entropy that culminated in the Ornstein classification of Bernoulli shifts. One genre of applications of this theory were the results of Furstenberg on Szemeredi’s theorem and eventually the work of Green and Tao.
Still the problem resisted a complete solution. Strange examples of completely determined systems that showed completely random statistical behavior began to surface. Starting in the 1990’s anti-classification theorems began to appear. These results showed, in a rigorous way, that complete invariants for measure preserving systems cannot exist. Moreover the isomorphism relation itself is completely intractable. Very recently these results were extended to measure preserving diffeomorphisms of the 2-torus.
What is the maximum number of rational points on a curve of genus g over a finite field of size q? What is the distribution of rational point counts for degree d plane curves over a fixed finite field? We discuss these and several related questions and show how to use curves over finite fields to construct interesting error-correcting codes.
This week's graduate seminar will feature a question-and-answer session with experienced TAs. Come prepared with your own questions, and try to make them as specific as possible. For example, "How do you teach effectively?" vs "How do you encourage participation?" vs "Have you ever had a class where students will not speak, no matter how hard you try?" Do you agree that the last one is so much easier to answer?
