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General Important on site information to ensure that your stay at CMO is positive and productive. Potential Participants If you are interested in attending a CMO workshop, please refer to the guidelines below. Using Quantum Invariants to do Interesting Topology. Modeling and Engineering of the Mammalian Embryo. Mathematical and Conceptual Aspects of Quantum Theory. Approximation Algorithms and the Hardness of Approximation. Geometric and Variational Methods in Celestial Mechanics. Locality and Functoriality in Symplectic Geometry.

Algebraic Methods in Coding Theory and Communication. On Exchangeability in Network Models. Abstract: We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts.

1.1 M. Gromov : Geometry as the art of asking questions

Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size. We also show that, for finitely exchangeable network models, the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts.

We then characterize all possible conditional independence structures for finitely exchangeable random graphs. Conditional independence ideals with hidden variables. Abstract: We study a class of determinantal ideals that are related to conditional independence CI statements with hidden variables. Such CI statements correspond to determinantal conditions on flattenings of marginals of the tensor P of joint probabilities of the observed random variables.

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Workshop on Recent Developments in the Geometry and Combinatorics of Hessenberg Varieties

We focus on an example that generalizes the CI ideals of the intersection axiom. In this example, the minimal primes are again determinantal ideals, which is not true in general. This is a joint work with Oliver Clarke and Fatemeh Mohammadi. Abstract: Computing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed.

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Two if the monomial curve is complete intersection and three otherwise. The same result is true even if we restrict to complete intersections. Abstract: We discuss recent developments in computational algebraic geometry that were motivated by the study of rough paths in stochastic analysis.

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Every path in a real vector space is encoded in a signature tensor whose entries are iterated integrals. As the path varies over a nice family we obtain an algebraic variety with interesting properties. Nondegenerate multistationarity in small reaction networks. Abstract: Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with mass-action kinetics, admit multiple positive steady states? The interest behind this question is in steady states that are stable.

As a step toward this difficult question, here we address the question of multiple nondegenerate positive steady states. Mathematically, this asks whether certain families of parametrized, real, sparse polynomial systems ever admit multiple positive real roots that are simple.

Our main results settle this problem for certain types of small networks, and our techniques point the way forward for larger networks. This is joint work with Anne Shiu. Abstract: Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action.

Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their signature tensors of order three. We establish identifiability results and recovery algorithms for piecewise linear paths, polynomial paths, and generic dictionaries.

This is joint work with Max Pfeffer and Bernd Sturmfels.


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Maximum likelihood under total positivity.