Daniel Silver - US grants

The investigators have developed a new method for applying concepts of symbolic dynamical systems to the study of knot and link groups. The set of representations into a finite group of the commutator subgroup of an oriented knot group or the augementation subgroup of an oriented link group has the structure of a shift of finite type, a special type of dynamical system that can be completely described by a finite graph. The set of representations into a finite abelian group of the commutator subgroup of an oriented link group is a higher dimensional shift of finite type. Dynamical invariants of the shift such as topological entropy, directional entropy and the zeta function produce new, computable knot invariants, and give useful information about branched cyclic covering spaces. Elementary and effective obstructions to conjugacy for free group automorphisms result as a byproduct. The project will expand the investigators' previous work, using these new techniques to address several open questions in knot theory. DNA, solar plasma filaments and garden hoses have a common feature: each exhibits knotting and linking. The mathematical theory of knots and links began in the mid-nineteenth century, and today its importance is recognized in diverse areas of physics, chemistry, biology and engineering. Until recently, the methods employed in this field have come from topology and algebra. The two principal investigators have discovered tools from the field of symbolic dynamics that are novel and effective for studying knots and links. Symbolic dynamics, which studies arrays of data, is central to information and communication theory and has important applications in the analysis of chaotic systems and the science of materials. This project will extend the investigators' techniques, address unanswered questions in knot theory and establish new bridges between disciplines.

DMS-0071004
Daniel S. Silver and Susan G. Williams


Knot theory applications from a newly emerging area
of dynamics, the study of dynamical systems of algebraic
origin, will be developed. The set of representations into
a compact group of the derived group of a link group will
be studied as a dynamical system. New results about knot
symmetries and branched covers of links are expected.
A recently discovered relationship between symbolic
dynamical systems and topological quantum field theory
will be explored. The research will address open questions
in knot theory and dynamics and expand on previous results
in these fields. It will continue a new program of research
in knot theory that incorporates ideas from symbolic
dynamical systems.

The investigation of knots and links has attracted great
attention not only in the mathematical world but also
in the larger scientific community. Knotting and linking
phenomena are found in molecular biology and plasma physics,
for example, while striking similarites in computational
methods arise frequently in theoretical physics. The proposed
research will provide new understanding of knots and links
by using techniques from symbolic dynamical systems,
a mathematical branch of information theory. The research
will also strengthen interaction between researchers in
the fields of topology and dynamics.

The investigators will combine their two areas of expertise,
topology and dynamical systems, to investigate open questions
in knot theory. In particular, they will investigate Lehmer's
Question. Seventy years ago D.H. Lehmer began constructing large prime
numbers using polynomials that are small in a precise sense:
integral polynomials with Mahler measure close to but
different from 1. Lehmer could do no better than 1.17628...,
a value that he achieved with a remarkable polynomial of
degree 10. He then asked if that value could improved.
Lehmer's Question remains open despite the best efforts of
many. Lehmer's polynomial continues to appear in surprisingly
separated fields. The investigators will attack Lehmer's Question from the
perspective of topology and geometry. They will investigate
applications of Mahler measure to the study of knots and
links. Their methods will include traditional ones from
commutative algebra, group theory, geometry and topology, as
well as new techniques, many from dynamical systems. Computer
methods will be used to develop examples.


The mathematical theory of knots arose from
physical theories of the nineteenth century. Since then,
the field has expanded greatly, attracting the interests
of scientists in many fields, including biology, chemistry
and physics. Some reasons for the attraction are not hard to
see. DNA, solar plasma filaments and fluid flow, for example,
all exhibit knotting or linking behaviour. The investigators
will combine their two areas of expertise, topology and
dynamical systems, to investigate open questions in knot
theory. The proposed research will provide new understanding
of knots and links by using techniques from symbolic
dynamical systems, a mathematical branch of information
theory. It will promote and strengthen interaction between
researchers in different fields. Undergraduate and graduate
students will participate in the project.

The PIs will continue their program to understand the algebra arising from knots and links by using new techniques of algebraic dynamics.
Modules over Laurent polynomial rings in d variables are replaced by their Pontryagin duals, which are compact abelian groups. Actions by the variables become commuting homeomorphisms, and corresponding dynamical invariants (e.g., numbers of periodic points, topological entropy, etc.) provide new topologicial invariants.
The focus will be on Mahler measure, surface dynamics and twisted homology. The project will strengthen newly emerging connections between knot theory, dynamical systems and number theory.


Although knots and links arise in a host of scientific phenomena such as DNA, solar flares and fluid flows, their mathematical study is fairly new. Fueled by ideas from many areas of mathematics, progress in knot theory has exploded during the past two decades, and the subject now attracts wide interest from physicists, biologists and engineers as well as mathematicians. This project will continue the investigators' program of applying ideas from another mathematical field, dynamical systems, in order to gain a new perspective.
Computer methods will be used to develop hypotheses. The PIs will also write a monograph, the first survey of this interdisciplinary study, designed for graduate students and researchers.

BREAST CANCER PROGRAM (BC) ABSTRACT: The purpose of BC is to decrease the incidence and morbidity of breast cancer through discovery and translation into practice emphasizing the need of the SKCC catchment area. Overarching goals are to: 1) Address gaps in our understanding of modifiable risk factors, and disparities in outcomes that are prevalent in the populations in our catchment area; 2) Utilize novel imaging and new therapeutic interventions to improve cancer screening, to detect progression early, and to increase the efficacy of pre-existing therapies; and 3) Develop and translate new therapies into the clinic by making impactful discoveries that address gaps in understanding of breast cancer biology. To address these goals, current aims of the Program are to: Aim 1: Develop innovative strategies and technologies for prevention and control Aim 2: Develop novel imaging and therapeutic interventions to improve detection and clinical outcome Aim 3: Elucidate molecular mechanisms of development/progression, and catalyze clinical translation BC was completely revitalized after the Director change, now comprised of 26 basic, population, and clinical researchers. BC members generated 617 publications, an increase of +76.2% over the prior period. Of these, 79 (12.8%) were intra-programmatic, a slight decrease since the last renewal, but reflective of new hires and complete re-organization of BC; 174 were inter-programmatic (28.2%). Overall impact was uneven prior to the leadership change, but now has an average impact factor of 4.9, with 5.0% appearing in journals with an impact factor >10. In 2016, SKCC also began to track collaborations with authors from other NCI-designated Cancer Centers; at present, 44.3% of BC publications were in collaboration with other NCI-designated Centers. Overall impact is illustrated by high-impact discoveries in journals including Mol Cell, Canc Disc, PNAS, Nat Comm, and NEJM. BC members have been increasingly productive in securing funding since the Program restructure in 2015 and BC leadership change in 2016. BC was identified in the strategic planning process under Dr. Knudsen as requiring a restructure and new leadership. BC funding hit a nadir in 2015, when peer-reviewed funding had dropped precipitously to $2.3M (total)/$1.5M (direct). After the restructure and leadership change, new BC leadership facilitated impactful discoveries and resurgence of peer-reviewed, cancer focused grant funding. At the time of reporting, total cancer relevant funding is $7.3M total/$5.2M direct, with peer-reviewed funding now $5.4M (total) and $3.5M (direct). Currently, 51.9% of BC peer-reviewed funding is derived from NCI, and 64.4% from combined federal cancer-dedicated peer review sources (NCI + DOD Cancer Programs).