[geometry-ml:02871] Tropical geometry - basic and applications のご案内 (1/21)

[Yoshihisa Miyanishi]

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Tropical geometry - basic and applications, Osaka University, Toyonaka Campus
日　時  ：2017年1月21日（土）10:00-15:20
場　所　：大阪大学 基礎工学研究科 J棟6階 J617
Program：
10:00-10:50  T. Kato (Kyoto Univ.)
Dynamical scale transform in tropical geometry

11:00-11:50  T. Kato (Kyoto Univ.) with S.Tsujimoto and A.Zuk Spectral coincidence between lamplighter and BBS automata

13:20-14:10  A. Nobe (Chiba Univ.)
Tropical elliptic curves and discrete integrable systems

14:30-15:20  Oscar Ortega, Shawn Garbett, and Carlos F. Lopez* (Vanderbilt Univ.)
TroPy: A framework to analyze dynamic cue-response execution in reaction networks

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Abstract: 
1000-1050 T. Kato (Kyoto Univ.)
Dynamical scale transform in tropical geometry

Tropical geometry is a dynamical scaling limit which extracts the dynamical framework.
In my first talk, I will describe basic subjects in tropical geometry. In particular we induce uniform estimates of rational
orbits by comparison to the piecewise linear dynamics which appear at infinity of the scaling limit. We apply it to the solution
to Burnside problem in theory of automata groups, and construct an infinite quasi-recursive rational dynamics. This is the first
example of rational dynamics which posses infinitely many orbits which are essentially different from each other but each of them
satisfies almost periodicity.

1100-1150 T. Kato (Kyoto Univ.) with S.Tsujimoto and A.Zuk Spectral coincidence between lamplighter and BBS automata

BBS is an integrable automaton which appear as a tropical limit of the KdV equation  in mathematical physics. Lamplighter group is
a finitely generated and infinite group which is the unique non trivial group in (2,2) automata groups. 
They arise from different mathematical fields, but we discovered complete coincidence of their spectra of the Markov operators.
They consist of point spectra and dense in the closed interval. This unexpected encounter between mathematical physics and group
theory promotes classification project of Mealy automata. By numerical computations, it turns out that there are surprisingly few
cases which possesses both integrable and stochastic properties. 

1320-1410 A. Nobe (Chiba Univ.)
Tropical elliptic curves and discrete integrable systems

We present an eight-parameter family of 2-dimensional discrete integrable systems each member of which is called the ultradiscrete
QRT map. The family of map dynamical systems is arising from the addition of points on a pencil of tropical elliptic curves, and
the time evolution of each map is linearized on the tropical Jacobian of the curve. We also present a solvable chaotic system
derived from the duplication of points on the tropical Hesse pencil, and give the general solutions to the system by using
tropical analogues of the level-three theta functions.

1430-1520 scar O. Ortega, Shawn Garbett, and Carlos F. Lopez* (Vanderbilt Univ.)
TroPy: A framework to analyze dynamic cue-response execution in reaction networks

The advent of quantitative techniques to probe biomolecular-signaling processes yielded a need to build mathematical models to
extract mechanistic insight from complex datasets. These complex mathematical models can yield useful insights about intracellular
signal execution but the task to identify key molecular drivers in signal execution, within a complex network, remains a central
challenge in quantitative biology. This challenge is compounded by the fact that cell-to-cell variability within a cell population
could yield multiple signal execution modes and thus multiple potential drivers in signal execution. Here we present a novel
approach to identify signaling drivers and characterize dynamic signal processes within a network. Our method, TroPy, combines
physical chemistry, statistical clustering, and tropical algebra formalisms to identify interactions that drives time-dependent
behavior in signaling pathways. We use our algorithm to study the effect of different protein concentration levels in cell-to-cell
variability in extrinsic apoptosis execution. We independently sample 10,000 initial conditions from experimentally obtained
protein distributions to generate simulated protein time-courses. These signaling trajectories are then analyzed with quasi-steady
state and quasi-equilibrium approximations to characterize the time-domain of the reaction network. A transformation to tropical
space is then carried out to create dynamic signatures of signal execution that are subsequently clustered using the Partitioning
Around Medoids (PAM) algorithm. With this approach, we identify groups of initial conditions that yield trajectories with common
protein time-course properties within the signaling network. For example, we study the mitochondrial protein Bid as its
accumulation rate determines the time of Mitochondrial Outer Membrane Permeabilization (MOMP). The timing of MOMP itself is
determined by the rate at which tBID accumulates to a threshold set by the levels of BCL2 family proteins. We find that reaction
flux through Bid clusters into four groups that define different distributions for time of death in HeLa cells. 
We discuss the obtained results in the context of biological applications, such as cancer-targeted drug resistance, with the goal
of guiding experiments to improve targeted treatments.

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  宮西吉久
  E-mail: miyanishi ＠ sigmath.es.osaka-u.ac.jp