We will discuss some new questions about rational points on curves, and give some partial answers.

In this series of three lectures I will consider SRB measures (after Sinai, Ruelle and Bowen) which arguably form one of the most important classes of invariant measures with “chaotic” behavior in dynamics. This ensures a crucial role they play in applications of dynamical systems to science (this is why they are often called “physical measures”).

In the first lecture I introduce these measures and describe their ergodic properties. I also outline a construction of SRB measures for Anosov systems.

In the second lecture I consider the case of partially hyperbolic dynamical systems and outline a construction of SRB measures in this case. I also discuss an important role SRB measures play in the Pugh-Shub Stable Ergodicity problem (and I will also discuss this problem in a general setting).

The third lecture deals with the most general situation of the so-called “chaotic” attractors (or attractors with non-zero Lyapunov exponents), which appear in many models in physics, biology, etc. I will present a general rigorous definition of chaotic attractors and outline a construction to build SRB measures for this attractors.

While the first lecture will serve as an introduction to the subject and will be accessible for students, the other two lectures are more advanced.

In this series of three lectures I will consider SRB measures (after Sinai, Ruelle and Bowen) which arguably form one of the most important classes of invariant measures with “chaotic” behavior in dynamics. This ensures a crucial role they play in applications of dynamical systems to science (this is why they are often called “physical measures”).

In the first lecture I introduce these measures and describe their ergodic properties. I also outline a construction of SRB measures for Anosov systems.

In the second lecture I consider the case of partially hyperbolic dynamical systems and outline a construction of SRB measures in this case. I also discuss an important role SRB measures play in the Pugh-Shub Stable Ergodicity problem (and I will also discuss this problem in a general setting).

The third lecture deals with the most general situation of the so-called “chaotic” attractors (or attractors with non-zero Lyapunov exponents), which appear in many models in physics, biology, etc. I will present a general rigorous definition of chaotic attractors and outline a construction to build SRB measures for this attractors.

While the first lecture will serve as an introduction to the subject and will be accessible for students, the other two lectures are more advanced.

Normal somatic cells loose the ability to divide after a limited number of cell divisions. This phenomenon, known as replicative senescence, is an important barrier to tumor progression. Essentially all human cancers acquire mechanisms that allow them to escape replicative senescence, most often through high levels of telomerase expression (∼ 90%). In this talk we will discuss how replicative senescence can protect against early non-neoplastic mutations that are potentially cancer precursors. We will also discuss various mathematical approaches to calculate the probability of escaping replicative senescence through a mutation that activates telomerase.

Until recently, the bacteria Chlamydia trachomatis was the principal cause worldwide of eye infection that leads to blindness (and possibly death).  For an on-going organized effort for its eradication, it is useful to know how rapid this infectious bacteria can multiply.  This problem provides us with an opportunity to demonstrate 1) the process of formulating a mathematical model for a biological phenomenon, and  2) an unfamiliar use of elementary calculus techniques to analyze the model to extract the desired information about the most rapid growth of the bacterial population.  The mathematical modeling and analysis involved show the type of research activities available to undergraduate Mathematics majors and bio-medical sciences students through the Mathematical and Computational Biology for Undergraduate (MCBU) summer research program each summer since 2011.  Math majors, especially those who have taken or are currently taking any one of Math 112A,B, Math 113A, Math 113B, Math 115, are encouraged to apply to participate this summer.  So are majors of any of the bio-medical fields.  For information about this National Science Foundation supported program that offers eligible undergraduate participants a summer stipend as well as room and board for the duration of the eight weeks program, interested students should visit the program website http://www.math.uci.edu/~mcbu/

 

Pizza served!