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These lecture notes for the 2013 CIME/CIRM summer school Combinatorial Algebraic Geometry deal with manifestly infinite-dimensional algebraic varieties with large symmetry groups. So large, in fact, that subvarieties stable under those symmetry groups are defined by finitely many orbits of equations—whence the title Noetherianity up to symmetry. It is not the purpose of these notes to give a systematic, exhaustive treatment of such varieties, but rather to discuss a few "personal favourites": exciting examples drawn from applications in algebraic statistics and multilinear algebra. My hope is that these notes will attract other mathematicians to this vibrant area at the crossroads of combinatorics, commutative algebra, algebraic geometry, statistics, and other applications.

We present the first spatial analysis of "fingerprints" of the El Nino/Southern Oscillation (ENSO) and atmospheric aerosol load after major volcanic eruptions (El Chichon and Mt. Pinatubo) in extreme low and high (termed ELOs and EHOs, respectively) and mean values of total ozone for the northern and southern mid-latitudes (defined as the region between 30 degrees and 60 degrees north and south, respectively). Significant influence on ozone extremes was found for the warm ENSO phase in both hemispheres during spring, especially towards low latitudes, indicating the enhanced ozone transport from the tropics to the extra-tropics. Further, the results confirm findings of recent work on the connection between the ENSO phase and the strength and extent of the southern ozone "collar". For the volcanic eruptions the analysis confirms findings of earlier studies for the northern mid-latitudes and gives new insights for the Southern Hemisphere. The results provide evidence that the negative effect of the eruption of El Chichon might be partly compensated by a strong warm ENSO phase in 1982–1983 at southern mid-latitudes. The strong west-east gradient in the coefficient estimates for the Mt. Pinatubo eruption and the analysis of the relationship between the AAO and ENSO phase, the extent and the position of the southern ozone "collar" and the polar vortex structure provide clear evidence for a dynamical "masking" of the volcanic signal at southern mid-latitudes. The paper also analyses the contribution of atmospheric dynamics and chemistry to long-term total ozone changes. Here, quite heterogeneous results have been found on spatial scales. In general the results show that EESC and the 11-yr solar cycle can be identified as major contributors to long-term ozone changes. However, a strong contribution of dynamical features (El Nino/Southern Oscillation (ENSO), North Atlantic Oscillation (NAO), Antarctic Oscillation (AAO), Quasi-Biennial Oscillation (QBO)) to ozone variability and trends is found at a regional level. For the QBO (at 30 and 50 hPa), strong influence on total ozone variability and trends is found over large parts of the northern and southern mid-latitudes, especially towards equatorial latitudes. Strong influence of ENSO is found over the Northern and Southern Pacific, Central Europe and central southern mid-latitudes. For the NAO, strong influence on column ozone is found over Labrador/Greenland, the Eastern United States, the Euro-Atlantic Sector, and Central Europe. For the NAO's southern counterpart, the AAO, strong influence on ozone variability and long-term changes is found at lower southern mid-latitudes, including the southern parts of South America and the Antarctic Peninsula, and central southern mid-latitudes.

International audience

International audience

We use statistical models for mean and extreme values of total column ozone to analyze "fingerprints" of atmospheric dynamics and chemistry on long-term ozone changes at northern and southern mid-latitudes on grid cell basis. At each grid cell, the r-largest order statistics method is used for the analysis of extreme events in low and high total ozone (termed ELOs and EHOs, respectively), and an autoregressive moving average (ARMA) model is used for the corresponding mean value analysis. In order to describe the dynamical and chemical state of the atmosphere, the statistical models include important atmospheric covariates: the solar cycle, the Quasi-Biennial Oscillation (QBO), ozone depleting substances (ODS) in terms of equivalent effective stratospheric chlorine (EESC), the North Atlantic Oscillation (NAO), the Antarctic Oscillation (AAO), the El Niño/Southern Oscillation (ENSO), and aerosol load after the volcanic eruptions of El Chichón and Mt. Pinatubo. The influence of the individual covariates on mean and extreme levels in total column ozone is derived on a grid cell basis. The results show that "fingerprints", i.e., significant influence, of dynamical and chemical features are captured in both the "bulk" and the tails of the statistical distribution of ozone, respectively described by mean values and EHOs/ELOs. While results for the solar cycle, QBO, and EESC are in good agreement with findings of earlier studies, unprecedented spatial fingerprints are retrieved for the dynamical covariates. Column ozone is enhanced over Labrador/Greenland, the North Atlantic sector and over the Norwegian Sea, but is reduced over Europe, Russia and the Eastern United States during the positive NAO phase, and vice-versa during the negative phase. The NAO's southern counterpart, the AAO, strongly influences column ozone at lower southern mid-latitudes, including the southern parts of South America and the Antarctic Peninsula, and the central southern mid-latitudes. Results for both NAO and AAO confirm the importance of atmospheric dynamics for ozone variability and changes from local/regional to global scales.

The Losev-Manin moduli space parametrizes pointed chains of projective lines. In this paper we study a possible generalization to families of pointed degenerate toric varieties. Geometric properties of these families, such as flatness and reducedness of the fibers, are explored via a combinatorial characterization. We show that such families are described by a specific type of polytope fibration which generalizes the twisted Cayley sums, originally introduced to characterize elementary extremal contractions of fiber type associated to projective $\mathbb{Q}$-factorial toric varieties with positive dual defect. The case of a one-dimensional simplex can be viewed as an alternative construction of the permutohedra., Comment: 21 pages, 5 figures. Final version. To appear in Mathematische Zeitschrift