# Interactions between algebra and the sciences

## Abstracts for the talks

**Daniel Bernstein***North Carolina State University***Tropical Linear Spaces in Phylogenetics**

One approach to phylogenetic tree reconstruction seeks an ultrametric (i.e. equidistant tree metric) that is l-infinity nearest to a given dissimilarity map. While the l-infinity nearest ultrametric is generally not unique, the set of all l-infinity nearest ultrametrics is a tropical polytope. We give an algorithm to compute a superset of its tropical vertices. Ardila and Klivans showed that the set of all ultrametrics on a finite set of size n is the Bergman fan associated to the matroid underlying the complete graph on n vertices. Therefore, we derive our results in the more general context of Bergman fans of matroids. This generality allows our algorithm to be applied to dissimilarity maps where only a subset of the entries are known.

**Carsten Conradi***Hochschule für Technik und Wirtschaft Berlin***Multistationarity in Biochemical Reaction Networks**

Multistationarity and switching have been recognized as important features of dynamical systems originating in Biology. Often these properties are established numerically. However, parameter uncertainty complicates numerical analysis. Hence techniques allowing the analytic computation of parameters where a given system exhibits either multistationarity or switching are desirable. We present conditions in form of polynomial systems that allow to determine parameter vectors where a mass action system exhibits multistationarity or switching. These polynomial systems arise from the structure of the reaction network and hence are not affected by parameter uncertainty.

**Robert Flassig***Max Planck Institute for Dynamics of Complex Technical System***Reactor Network Synthesis and Biochemical Process Design**

Under the hood we are dealing with optimal control problems in form of NLPs as well as dynamic flux balance modeling in form of an ODE system with embedded LP. We focus on the application (hydroformylation, apoptosis in HeLa cells, microalgal-based production) and tailoring of established optimization/solution methods. One of the challenges we address is the consideration of uncertainties (e.g. model parameters, design imperfections) during the phase of model identification but also model-based prediction and optimization.

**Alex Gavryushkin***ETH Zürich***Genetic Interactions and Fitness Landscapes**

Darwinian fitness of genotypes in a natural population is very hard to measure in practice. To address this challenge, we developed quantitative tools to make inference about epistatic gene interactions when the fitness landscape is only incompletely determined, for example, due to imprecise measurements or partial observations. We demonstrate that higher order genetic interactions can often be inferred for fitness rank orders, where all genotypes are ordered according to fitness, and even for partial fitness orders. In this talk, I will present a complete characterization of rank orders that imply higher order epistasis and an efficient algorithm for detecting such interactions, and show how this can be applied to detect interactions in various biological settings. I will also describe the progress we have made on the way to achieve a similar characterization for fitness graphs and partial fitness orders, and conclude with open problems.

**Nidhi Kaihnsa***Max Planck Institute for Mathematics in the Sciences***The Attainable Region in Linear Networks**

**Kaie Kubjas***Aalto University***Group-based Phylogenetic Models**

A phylogenetic model describes the evolution of genes. A group-based phylogenetic model assumes extra structure on the parameters of the model that is related to an abelian group. The joint probabilities of a group-based phylogenetic model are functions in the parameters of the model. We study polynomial inequalities that cut out the set of all joint probabilities for a time-reversible group-based model. We apply this knowledge to the study of boundaries and maximum likelihood estimation on group-based models. In particular, we use the degree of an algebraic variety and numerical algebraic geometry to obtain the maximum likelihood estimate exactly for small group-based models. This talk is based on joint work with Dimitra Kosta.

**Caitlin Lienkaemper***Pennsylvania State University***Using Algebra to Explore Neural Coding**

Our ability to perceive, process, and interact with our environment depends on a correspondence between patterns of neural activity and stimuli in the environment. This is known as neural coding. A combinatorial neural code describes a pattern of neural activity in terms of which subsets of a population of neurons fire together and which do not. Our canonical example is that of convex neural codes arising from place cells, neurons which fire whenever an animal is within the place cell's place field, a convex region of the environment. Convex neural codes have been studied from a variety of perspectives, including mathematical coding theory, geometry, and topology. Here, we present a summary of algebraic approaches to the study of neural coding, including the neural ideal and the neural toric ideal.

**Johannes Martini***Universität Göttingen***Binding Polynomials and Cooperativity**

**Ezra Miller***Duke University***Algebra for Topology in Biology and Statistics**

With "Big Data" comes a surge of geometric data sets and data

objects, particularly in biology and medicine, where shapes,

images, videos, networks, and small samples in high dimension

are common. One way to deal with such data is to summarize

topologically. That, in turn, leads to algebraic structures

which take their cue from graded polynomial rings and their

modules, although the theory is complicated by the passage

from integer exponent vectors to real exponent vectors. The

path to effective methods requires finitness conditions to

replace the usual ones from commutative algebra. Statistical

considerations interact with these finiteness conditions in

fundamental ways, some of them relying on an understanding of

how datasets of this nature relate to moduli of modules.

The talk will start with a general discussion of biological

and medical investigations that lend themselves to techniques

based on geometry, including concrete datasets and underlying

statistical questions. The primary focus will be on a

fundamental problem in evolutionary biology that drove the

genesis of the new algebraic ideas covered here; it concerns

evolution of changes to discrete morphological features, for

which we are doing statistics on a dataset comprising images

of fruit fly wing veins. The main material is joint work in

progress with David Houle (Biology, Florida State), Ashleigh

Thomas (grad student, Duke Math), Justin Curry (postdoc, Duke

Math), and Surabhi Beriwal (undergrad, Duke Math).

## Date and Location

**May 27, 2017**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Anna Seigal**

University of California

Berkeley (USA)**Jacinta Torres**

MPI für Mathematik in den Naturwissenschaften

Leipzig (Germany)

## Administrative Contact

**Saskia Gutzschebauch**

MPI für Mathematik in den Naturwissenschaften

Contact by Email