Immanant varieties

We introduce immanant varieties, associated to simple characters of a finite group. They include well studied classes of varieties, as Segre embeddings, Grassmannians and some other Chow varieties. For a one-dimensional character $\chi$, we define $\chi$-matroids by a maximality property. For trivial characters, by exploring the combinatorics of incidence stratifications, we provide a basis for the Chow vector spaces of the corresponding immanant varieties

On the intelligent management of sepsis in the intensive care unit

The management of the Intensive Care Unit (ICU) in a hospital has its own, very specific requirements that involve, amongst others, issues of risk-adjusted mortality and average length of stay; nurse turnover and communication with physicians; technical quality of care; the ability to meet patient's family needs; and avoid medical error due rapidly changing circumstances and work overload. In the end, good ICU management should lead to an improvement in patient outcomes. Decision making at the ICU environment is a real-time challenge that works according to very tight guidelines, which relate to often complex and sensitive research ethics issues. Clinicians in this context must act upon as much available information as possible, and could therefore, in general, benefit from at least partially automated computer-based decision support based on qualitative and quantitative information. Those taking executive decisions at ICUs will require methods that are not only reliable, but also, and this is a key issue, readily interpretable. Otherwise, any decision tool, regardless its sophistication and accuracy, risks being rendered useless. This thesis addresses this through the design and development of computer based decision making tools to assist clinicians at the ICU. It focuses on one of the main problems that they must face: the management of the Sepsis pathology. Sepsis is one of the main causes of death for non-coronary ICU patients. Its mortality rate can reach almost up to one out of two patients for septic shock, its most acute manifestation. It is a transversal condition affecting people of all ages. Surprisingly, its definition has only been standardized two decades ago as a systemic inflammatory response syndrome with confirmed infection. The research reported in this document deals with the problem of Sepsis data analysis in general and, more specifically, with the problem of survival prediction for patients affected with Severe Sepsis. The tools at the core of the investigated data analysis procedures stem from the fields of multivariate and algebraic statistics, algebraic geometry, machine learning and computational intelligence. Beyond data analysis itself, the current thesis makes contributions from a clinical point of view, as it provides substantial evidence to the debate about the impact of the preadmission use of statin drugs in the ICU outcome. It also sheds light into the dependence between Septic Shock and Multi Organic Dysfunction Syndrome. Moreover, it defines a latent set of Sepsis descriptors to be used as prognostic factors for the prediction of mortality and achieves an improvement on predictive capability over indicators currently in use.La gestió d'una Unitat de Cures Intensives (UCI) hospitalària presenta uns requisits força específics incloent, entre altres, la disminució de la taxa de mortalitat, la durada de l'ingrès, la rotació d'infermeres i la comunicació entre metges amb al finalitad de donar una atenció de qualitat atenent als requisits tant dels malalts com dels familiars. També és força important controlar i minimitzar els error mèdics deguts a canvis sobtats i a la presa ràpida de deicisions assistencials. Al cap i a la fi, la bona gestió de la UCI hauria de resultar en una reducció de la mortalitat i durada d'estada. La presa de decisions en un entorn de crítics suposa un repte de presa de decisions en temps real d'acord a unes guies clíniques molt restrictives i que, pel que fa a la recerca, poden resultar en problemes ètics força sensibles i complexos. Per tant, el personal sanitari que ha de prendre decisions sobre la gestió de malalts crítics no només requereix eines de suport a la decisió que siguin fiables sinó que, a més a més, han de ser interpretables. Altrament qualsevol eina de decisió que no presenti aquests trets no és considerarà d'utilitat clínica. Aquesta tesi doctoral adreça aquests requisits mitjançant el desenvolupament d'eines de suport a la decisió per als intensivistes i es focalitza en un dels principals problemes als que s'han denfrontar: el maneig del malalt sèptic. La Sèpsia és una de les principals causes de mortalitats a les UCIS no-coronàries i la seva taxa de mortalitat pot arribar fins a la meitat dels malalts amb xoc sèptic, la seva manifestació més severa. La Sèpsia és un síndrome transversal, que afecta a persones de totes les edats. Sorprenentment, la seva definició ha estat estandaritzada, fa només vint anys, com a la resposta inflamatòria sistèmica a una infecció corfimada. La recerca presentada en aquest document fa referència a l'anàlisi de dades de la Sèpsia en general i, de forma més específica, al problema de la predicció de la supervivència de malalts afectats amb Sèpsia Greu. Les eines i mètodes que formen la clau de bòveda d'aquest treball provenen de diversos camps com l'estadística multivariant i algebràica, geometria algebraica, aprenentatge automàtic i inteligència computacional. Més enllà de l'anàlisi per-se, aquesta tesi també presenta una contribució des de el punt de vista clínic atès que presenta evidència substancial en el debat sobre l'impacte de l'administració d'estatines previ a l'ingrès a la UCI en els malalts sèptics. També s'aclareix la forta dependència entre el xoc sèptic i el Síndrome de Disfunció Multiorgànica. Finalment, també es defineix un conjunt de descriptors latents de la Sèpsia com a factors de pronòstic per a la predicció de la mortalitat, que millora sobre els mètodes actualment més utilitzats en la UCI

Algorithms in Intersection Theory in the Plane

This thesis presents an algorithm to find the local structure of intersections of plane curves. More precisely, we address the question of describing the scheme of the quotient ring of a bivariate zero-dimensional ideal $I\subseteq \mathbb K[x,y]$, \textit{i.e.} finding the points (maximal ideals of $\mathbb K[x,y]/I$) and describing the regular functions on those points. A natural way to address this problem is via Gr\"obner bases as they reduce the problem of finding the points to a problem of factorisation, and the sheaf of rings of regular functions can be studied with those bases through the division algorithm and localisation. Let $I\subseteq \mathbb K[x,y]$ be an ideal generated by $\mathcal F$, a subset of $\mathbb A[x,y]$ with $\mathbb A\hookrightarrow\mathbb K$ and $\mathbb K$ a field. We present an algorithm that features a quadratic convergence to find a Gr\"obner basis of $I$ or its primary component at the origin. We introduce an $\mathfrak m$-adic Newton iteration to lift the lexicographic Gr\"obner basis of any finite intersection of zero-dimensional primary components of $I$ if $\mathfrak m\subseteq \mathbb A$ is a \textit{good} maximal ideal. It relies on a structural result about the syzygies in such a basis due to Conca \textit{\&} Valla (2008), from which arises an explicit map between ideals in a stratum (or Gr\"obner cell) and points in the associated moduli space. We also qualify what makes a maximal ideal $\mathfrak m$ suitable for our filtration. When the field $\mathbb K$ is \textit{large enough}, endowed with an Archimedean or ultrametric valuation, and admits a fraction reconstruction algorithm, we use this result to give a complete $\mathfrak m$-adic algorithm to recover $\mathcal G$, the Gr\"obner basis of $I$. We observe that previous results of Lazard that use Hermite normal forms to compute Gr\"obner bases for ideals with two generators can be generalised to a set of $n$ generators. We use this result to obtain a bound on the height of the coefficients of $\mathcal G$ and to control the probability of choosing a \textit{good} maximal ideal $\mathfrak m\subseteq\mathbb A$ to build the $\mathfrak m$-adic expansion of $\mathcal G$. Inspired by Pardue (1994), we also give a constructive proof to characterise a Zariski open set of $\mathrm{GL}_2(\mathbb K)$ (with action on $\mathbb K[x,y]$) that changes coordinates in such a way as to ensure the initial term ideal of a zero-dimensional $I$ becomes Borel-fixed when $|\mathbb K|$ is sufficiently large. This sharpens our analysis to obtain, when $\mathbb A=\mathbb Z$ or $\mathbb A=k[t]$, a complexity less than cubic in terms of the dimension of $\mathbb Q[x,y]/\langle \mathcal G\rangle$ and softly linear in the height of the coefficients of $\mathcal G$. We adapt the resulting method and present the analysis to find the $\langle x,y\rangle$-primary component of $I$. We also discuss the transition towards other primary components via linear mappings, called \emph{untangling} and \emph{tangling}, introduced by van der Hoeven and Lecerf (2017). The two maps form one isomorphism to find points with an isomorphic local structure and, at the origin, bind them. We give a slightly faster tangling algorithm and discuss new applications of these techniques. We show how to extend these ideas to bivariate settings and give a bound on the arithmetic complexity for certain algebras

Bernstein-Sato polynomials in commutative algebra

This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.Comment: 64 page

BERNSTEIN-SATO POLYNOMIALS IN COMMUTATIVE ALGEBRA

This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra

Multipartite Quantum States and their Marginals

Subsystems of composite quantum systems are described by reduced density matrices, or quantum marginals. Important physical properties often do not depend on the whole wave function but rather only on the marginals. Not every collection of reduced density matrices can arise as the marginals of a quantum state. Instead, there are profound compatibility conditions -- such as Pauli's exclusion principle or the monogamy of quantum entanglement -- which fundamentally influence the physics of many-body quantum systems and the structure of quantum information. The aim of this thesis is a systematic and rigorous study of the general relation between multipartite quantum states, i.e., states of quantum systems that are composed of several subsystems, and their marginals. In the first part, we focus on the one-body marginals of multipartite quantum states; in the second part, we study general quantum marginals from the perspective of entropy.Comment: PhD thesis, ETH Zurich. The first part contains material from arXiv:1208.0365, arXiv:1204.0741, and arXiv:1204.4379. The second part is based on arXiv:1302.6990 and arXiv:1210.046

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

The Genetic Architecture of Structural Renal and Urinary Tract Malformations

Structural renal and urinary tract malformations are the most common cause of kidney failure in children. These congenital anomalies of the kidneys and urinary tract (CAKUT) are a phenotypically diverse group of malformations that result from defects in embryonic kidney, ureter, and bladder development. A genetic basis for CAKUT has been proposed, with over 50 monogenic causes reported, however, a molecular diagnosis is detected in less than 20% of patients. In this thesis, I used bioinformatics and statistical genetics methodology to investigate the genetic architecture of structural renal and urinary tract malformations using whole-genome sequencing (WGS) data from the 100,000 Genomes Project. Population-based rare and common variant association testing was performed in over 800 cases and 20,000 controls of diverse ancestry seeking enrichment of single-nucleotide/indel and structural variation on a genome-wide, per-gene, and cis-regulatory element basis. Using a sequencing-based genome-wide association study (GWAS) I identified the first robust genetic associations of posterior urethral valves (PUV), the most common cause of kidney failure in boys. Bayesian fine-mapping and functional annotation mapped these two loci to the transcription factor TBX5 and planar cell polarity gene PTK7, with both signals replicated in an independent cohort. Significant enrichment of rare structural variation affecting cis-regulatory elements was also detected providing novel insights into the pathogenesis of this poorly understood disorder. I also demonstrated that the contribution of known monogenic disease to CAKUT has been overestimated and that common and low-frequency variation plays an important role in phenotypic variability. These findings support an omnigenic rather than monogenic model of inheritance for CAKUT and are consistent with the extensive genotypic-phenotypic heterogeneity, variable expressivity, and incomplete penetrance observed in this condition. Finally, this work demonstrates the value of sequencing-based GWAS methodology in rare disease, beyond conventional monogenic gene discovery, and provides strong support for an inclusive diverse-ancestry approach