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The lack of multi-omics cancer datasets with extensive follow-up information hinders the identification of accurate biomarkers of clinical outcome. In this cohort study, we performed comprehensive genomic analyses on fresh-frozen samples from 348 patients affected by primary colon cancer, encompassing RNA, whole-exome, deep T cell receptor and 16S bacterial rRNA gene sequencing on tumor and matched healthy colon tissue, complemented with tumor whole-genome sequencing for further microbiome characterization. A type 1 helper T cell, cytotoxic, gene expression signature, called Immunologic Constant of Rejection, captured the presence of clonally expanded, tumor-enriched T cell clones and outperformed conventional prognostic molecular biomarkers, such as the consensus molecular subtype and the microsatellite instability classifications. Quantification of genetic immunoediting, defined as a lower number of neoantigens than expected, further refined its prognostic value. We identified a microbiome signature, driven by Ruminococcus   bromii , associated with a favorable outcome. By combining microbiome signature and Immunologic Constant of Rejection, we developed and validated a composite score (mICRoScore), which identifies a group of patients with excellent survival probability. The publicly available multi-omics dataset provides a resource for better understanding colon cancer biology that could facilitate the discovery of personalized therapeutic approaches. A large, publicly available dataset integrating RNA, whole-exome, T cell receptor and 16S rRNA sequencing from patients with colon cancer enables the discovery of a prognostic score consisting of tumor, immune and microbial features.

Hypertension is the leading cause of death in developed countries and reduction of salt intake is recommended as a key preventive measure. To assess the dietary sodium and potassium intakes in a national sample of Italian children and adolescents and to examine their relationships with BMI and blood pressure (BP) in the framework of the MINISAL survey, a program supported by the Italian Ministry of Health. The study population included 1424 healthy subjects (766 boys, 658 girls) aged 6-18 years (mean age: 10.1±2.9) who were consecutively recruited in participating National Health Service centers in 10 Italian regions. Electrolyte intake was estimated from 24 hour urine collections tested for completeness by the concomitant measurement of creatinine content. Anthropometric indices and BP were measured with standardized procedures. The average estimated sodium intake was 129 mmol (7.4 g of salt) per day among boys and 117 mmol (6.7 g of salt) among girls. Ninety-three percent of the boys and 89% of the girls had a consumption higher than the recommended age-specific standard dietary target. The estimated average daily potassium intakes were 39 mmol (1.53 g) and 36 mmol (1.40 g), respectively, over 96% of the boys and 98% of the girls having a potassium intake lower than the recommended adequate intake. The mean sodium/potassium ratio was similar among boys and girls (3.5 and 3.4, respectively) and over 3-fold greater than the desirable level. Sodium intake was directly related to age, body mass and BP in the whole population. The Italian pediatric population is characterized by excessive sodium and deficient potassium intake. These data suggest that future campaigns should focus on children and adolescents as a major target in the framework of a population strategy of cardiovascular prevention.

In this work, we prove that if a graded, commutative algebra RR over a field kk is not Koszul, then, denoting by m\\mathfrak {m} the maximal homogeneous ideal of RR and by MM a finitely generated graded RR-module, the nonzero modules of the form mM\\mathfrak {m} M have infinite Castelnuovo-Mumford regularity. We also prove that over complete intersections which are not Koszul, a nonzero direct summand of a syzygy of kk has infinite regularity. Finally, we relate the vanishing of the graded deviations of RR to having a nonzero direct summand of a syzygy of kk of finite regularity.

Let $\\mathfrak{p}$ be a prime ideal in a commutative noetherian ring R and denote by $k(\\mathfrak{p})$ the residue field of the local ring $R_\\mathfrak{p}$. We prove that if an R-module M satisfies $\\operatorname{Ext}_R^{n}(k(\\mathfrak{p}),M)=0$ for some $n\\geqslant\\dim R$, then $\\operatorname{Ext}_R^i(k(\\mathfrak{p}),M)=0$ holds for all $i \\geqslant n$. This improves a result of Christensen, Iyengar and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.

Cells are complex systems whose behavior emerges from a huge number of reactions taking place within and among different molecular districts. The availability of bulk and single-cell omics data fueled the creation of multi-omics systems biology models capturing the dynamics within and between omics layers. Powerful modeling strategies are needed to cope with the increased amount of data to be interrogated and the relative research questions. Here, we present MultiOmics Network Embedding for SubType Analysis (MoNETA) for fast and scalable identification of relevant multi-omics relationships between biological entities at the bulk and single-cells level. We apply MoNETA to show how glioma subtypes previously described naturally emerge with our approach. We also show how MoNETA can be used to identify cell types in five multi-omic single-cell datasets.

This thesis consists of two parts: 1) A bimodule structure on the bounded cohomology of a local ring (Chapter 1), 2) Modules of infinite regularity over graded commutative rings (Chapter 2). Chapter 1 deals with the structure of stable cohomology and bounded cohomology. Stable cohomology is a Z-graded algebra generalizing Tate cohomology and first defined by Pierre Vogel. It is connected to absolute cohomology and bounded cohomology. We investigate the structure of the bounded cohomology as a graded bimodule. We use the information on the bimodule structure of bounded cohomology to study the stable cohomology algebra as a trivial extension algebra and to study its commutativity. In Chapter 2 it is proved that if a graded, commutative algebra R over a field k is not Koszul, then the nonzero modules mM, where M is a finitely generated R-module and m is the maximal homogeneous ideal of R, have infinite Castelnuovo-Mumford regularity.

Free resolutions of ideals in commutative rings provide valuable insights into the complexity of these ideals. In 1966, Taylor constructed a free resolution for monomial ideals in polynomial rings, which Gemeda later showed admits a differential graded (DG) algebra structure. In 2002, Avramov proved that the Taylor resolution admits a DG algebra structure with divided powers. In 2007, Herzog introduced the generalized Taylor resolution, which is usually smaller than the original Taylor resolution. Recently, in 2023, VandeBogert showed that the generalized Taylor resolution also admits a DG algebra structure. In this paper, we extend Avramov's result by proving that the generalized Taylor resolution admits a DG algebra structure with divided powers. Along the way, we correct a sign error in VandeBogert's product formula. We provide applications concerning homotopy Lie algebras and resolutions of squarefree monomial ideals.

Let \\(\\Bbbk\\) be a field, and let \\(I\\) be a monomial ideal in the polynomial ring \\(R=\\Bbbk[x_1,\\ldots,x_n]\\). In her thesis, Taylor introduced a complex that provides a finite free resolution of \\(R/I\\) as an \\(R\\)-module. Building on this, Ferraro, Martin and Moore extended this construction to monomial ideals in skew polynomial rings. Since the Taylor resolution is generally not minimal, significant effort has been devoted to identifying classes of ideals with minimal free resolutions that are relatively straightforward to construct. In a 1987 paper, Eliahou and Kervaire developed a minimal free resolution for a class of monomial ideals in \\(R\\) known as stable ideals. This result was later generalized to stable ideals in skew polynomial rings by Ferraro and Hardesty. In a 2002 paper, Herzog and Takayama constructed a minimal free resolution for monomial ideals with linear quotients, a broader class of ideals containing stable ideals. Their resolution reduces to the Eliahou-Kervaire resolution in the stable case. In this paper, we generalize the Herzog-Takayama resolution to skew polynomial rings.

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings. Finally we show that the resolution we construct admits a product generalizing the one given by Eliahou and Kervaire in the commutative case.

Let \\((R,\\mathfrak{m},\\Bbbk)\\) be a regular local ring of dimension 3. Let \\(I\\) be a Gorenstein ideal of \\(R\\) of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that \\(I\\) is generated by the sub-maximal pfaffians of this matrix. Let \\(J\\) be the ideal obtained by multiplying some of the pfaffian generators of \\(I\\) by \\(\\mathfrak{m}\\); we say that \\(J\\) is a trimming of \\(I\\). Building on a recent paper of Vandebogert, we construct an explicit free resolution of \\(R/J\\) and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class \\(\\mathbf{G}\\) hold true in our context. Furthermore, we address the realizability question for ideals of class \\(\\mathbf{G}\\).