STEM (Science, Technology, Engineering, and Mathematics) Panels in AUBG 30 Conference

Eugen Bronasco, University of Geneva

Hopf and pre Lie algebras in numerical analysis 


Consider the approximation by numerical integrators of systems of differential equations of the form: dy/dt = f(y), with y(0) = y_0,
where y(t) is in Rd and f : Rd → Rd is a smooth function. B-series were introduced in the 1960s to analyze numerical integrators using the Taylor expansion of the exact solution as a model. B-series give a straightforward way to express order conditions of numerical integrators and have a close connection to Hopf algebras on trees and pre-Lie algebras. This connection arose in the 1980s and has growing attention now. The algebraic structure of the B-series allows us to work with products of the B-series systematically. We will describe two kinds of products: composition of B-series and substitution of vector fields. We will look at the kinds of algebraic structures these two products entail and study the relations between them. In the end, we will outline some open questions, future projects, and applications of the algebraic results.

Elena Dimitrova, California Polytechnic State University

Vanishing ideals of points for algebraic design of experiments


Computation of vanishing ideals of points comes with surprising challenges even over a finite field. Groebner bases, a special type of generating sets, are the standard tool for computation. Although efficient techniques, such as the BM-algorithm, exist, for most sets of points the vanishing ideal has several equally "nice" generating sets. These sets yield multiple interpolating polynomials which is not optimal in the context of model selection. It is thus desirable to be able to find a unique or at least a small number of generating sets. In this talk, we will explore properties of vanishing ideals, particularly over finite fields, and identify properties of the points and their associated ideal that result in a unique reduced Groebner basis for the ideal. Additionally, we will see how these questions arise naturally in the design of experiments and selection of algebraic models of biological systems.

Mihaela Dimovska, University of Minnesota

Learning networks of dynamic systems via non invasive observations 


Learning the underlying structure of a networked dynamic system from observational data is an important problem in many domains, from climate studies to economics. One of the most well-known approaches to this problem is Granger causality, which relies on the premise that data are sampled at a frequency sufficient to capture the cause-to-effect delays, leading to strictly causal observed dynamics. For such strictly causal systems, it has been shown that Granger causality consistently reconstructs the underlying graph of the network. However, in many domains, such as finance, neuroscience or climate studies, the observed dynamics do not follow the strict causality assumption. Thus, many reconstruction methods that try to deal with non-strictly causal dynamics have been developed in the last decade. These methods, however, tend to put limiting assumptions on the underlying network structure of the systems. In this work, we develop a network reconstruction method for a large class of systems with non-strictly causal dynamics. We provide theoretical guarantees for the reconstruction, while posing no limitations on the underlying network structure. The only required assumption in the novel method is that at least one strictly causal operator is present in every feedback loop. We test the method on several benchmark examples and on random networks via simulations, and we also apply it on real-world datasets that show the effectiveness of the proposed algorithm.

Mariami Gachechiladze, University of Cologne

Quantifying causal influences in the presence of a quantum common cause 


Estimating relations of cause and effect are central and, yet, one of the most challenging goals of science. Since long ago, it has been realized that correlations do not imply causation. The reason is that any correlation observed between two or more random variables can, at least in the classical regime, be explained by a potentially unobserved common cause. Understanding under which conditions such confounding factors can be controlled, such that empirical data can be turned into a causal hypothesis, has found a firm theoretical basis for establishing the mathematical theory of causality. Today, concepts like interventions randomised controlled experiments, and instrumental variables are common work tools in the estimation of causal influences in a variety of fields.

Despite its success, all such ideas and applications rely on the classical notion of causality that we know cannot be applied to quantum phenomena since Bell's theorem. Quantum mechanics challenges our intuition on the cause-effect relations in nature. Some fundamental concepts, including Reichenbach's common cause principle or the notion of local realism, have to be reconsidered. Traditionally, this is witnessed by the violation of a Bell inequality. But are Bell inequalities the only signature of the incompatibility between quantum correlations and causality theory? Motivated by this question, we introduce a general framework that can estimate causal influences between two variables without the need for interventions and irrespectively of the classical, quantum, or even postquantum nature of a common cause. In particular, by considering the simplest instrumental scenario—for which violation of Bell inequalities is not possible—we show that every pure bipartite entangled state violates the classical bounds on causal influence, thus, answering in negative to the posed question and opening a new venue to explore the role of causality within quantum theory.

Euxhen Hasanaj, Carnegie Mellon University

Assigning cell types in single cell genomics data via Machine Learning 


Over the last few years, a number of methods have been developed for the assignment of cell types in single-cell data. In most cases, different groups from the same consortia, and even the same group when processing multiple types of single-cell data, rely on a different set of tools. This makes it hard to integrate and compare data from these groups since researchers often use different assignment techniques, markers, and even cell-type naming conventions. To enable large scale collaborations, integration, and comparisons across many different single-cell omics platforms and modalities, we present Cellar, an interactive and graphical cell-type assignment tool. Cellar implements a comprehensive set of methods, including methods for dimensionality reduction and representation, clustering, reference based alignment, identification of differentially expressed genes, and more. We demonstrate the advantages of Cellar by using it to annotate several datasets from multi-omics single-cell sequencing and spatial proteomics studies.

Zlatko Joveski, Vanderbilt Univrsity

Interval Permutation Segment Graphs 


We introduce a model that generalizes the geometric intersection models of interval and permutation graphs. As a result, we obtain a new class of graphs called interval-permutation segment (IP-SEG) graphs, that generalizes interval and permutation graphs, but is not contained within the class of perfect graphs. We show that even though IP-SEG graphs may contain large chordless cycles, there are limitations in how such cycles may be represented in the corresponding model. This allows us to identify several families of forbidden subgraphs for the classes. Leveraging properties of the geometric intersection model, we design polynomial model-based algorithms for the clique, independent set, and longest chordless cycle problems on IP-SEG graphs. Further, we show that any IP-SEG graph has a special vertex elimination ordering that can be used to design a robust polynomial algorithm for the clique problem on IP-SEG graphs.

Mario Krali, EPFL Lausanne

Estimating an Extreme Bayesian Network via Scalings 


A recursive max-linear vector models causal dependence between its components by expressing each node variable as a max-linear function of its parental nodes in a directed acyclic graph and some exogenous innovation. Motivated by extreme value theory, innovations are assumed to have regularly varying distribution tails. We propose a scaling technique in order to determine a causal order of the node variables. All dependence parameters are then estimated from the estimated scalings. Furthermore, we prove asymptotic normality of the estimated scalings and dependence parameters based on asymptotic normality of the empirical spectral measure. Finally, we apply our structure learning and estimation algorithm to financial data and food dietary interview data. (joint work with Claudia Kluppelberg).

Kalina Mincheva, Tulane University

Tropical Algebra


Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences and what they remember about the geometry of a tropical variety.

Gheorghe Pupazan, Humboldt Universitat zu Berlin

Perfectoid fields and φ, Γ modules 


Presumably the most central objects of attention in modern algebraic number theory are the absolute Galois groups of local and global fields. Focusing on local fields of characteristic 0, we fix a prime number p and a finite extension L of Qp with absolute Galois group GL. Local class field theory establishes a canonical homomorphism, called the reciprocity map into the maximal abelian quotient of GL, explaining the connection between the external datum of abelian extensions of L and the internal datum of the multiplicative group of L. However, it appears to be extremely difficult to extend this reciprocity map to a natural homomorphism whose target is the entire Galois group GL. Instead, the Langlands philosophy proposes to understand GL through its representations. Among the various types of representations one can consider, we focus on continuous representations of GL with coefficients in the ring of integers of L.

Passing to local fields of characteristic p, let EL= kL ((X)) where kL is the residue field of L. A spectacular construction, whose original form for Qp is due to Fontaine and Wintenberger, shows that the absolute Galois group of EL is isomorphic to a normal subgroup of GL. This was the starting point of the proof of a deep theorem of Fontaine which says that the above mentioned Galois representations can equivalently be described in terms of the so-called (phi,Gamma)-modules when L= Qp. The latter are certain ring and module theoretic objects which are much more accessible than the Galois representations which remain fairly abstract due to the mysterious nature of GL. The equivalence of Fontaine was generalized later by Kisin, Ren and Schneider for arbitrary L in the framework of Lubin-Tate theory.

Vesna Stojanoska, University of Illinois Urbana Champaign

Homotopy and duality for solving polynomial equations 


The question of solving polynomial equations, or even deciding if solutions exist within a given number system, is a central one in number theory, and is notoriously difficult. To say that there is no exact procedure for finding solutions is an understatement. Sophisticated methods have been developed to at least try to answer the question of whether solutions to a given equation exist: these are called obstructions.

Suppose X is a variety over a number field K, for example given by polynomial equations over (a finite extension of) the rationals. The idea is that locally, solutions to equations can be found, at least in principle, using Hensel’s lemma: these form the adelic points X(A) of X. The question then is to detemine which adelic points come from a rational (i.e. global) point, and this is the question of whether the Hasse principle is satisfied. In the early 1970s, Manin found an obstruction to the Hasse principle which comes from the Brauer group of central simple algebras over K. If the obstruction is non-zero, the variety cannot have rational points. While undoubtedly powerful due to its computability, this Brauer-Manin obstruction is insufficient: there are examples of varieties with no rational points for which the obstruction or even its various enhancements vanish. The starting point of the project I will discuss is the observation that the Brauer-Manin obstruction can be produced naturally from Poitou-Tate duality in the cohomology of the number field K. If the original algebraic version was good but imperfect, our expectation is that enhancing the algebra with homotopy theory will yield more sophisticated invariants by using generalized rather than ordinary Galois cohomology. With this end in mind, I will explain how Poitou-Tate duality can be proved in this generalized setting.

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