Consider the moduli space of stable, n-marked curves M and the tautological subring R^*(M) of its Chow ring. The standard calculus for R^*(M) is based on the ''strata algebra" SA(M), which is constructed via the inductive structure of the boundary of M and the excess intersection formula, and in which calculations are expressed in terms of ''graph sums". In this talk I will discuss a new calculus for R^*(M), based on the introduction of a new ring L^*(M), built out of tropical geometry, and in which several standard calculations simplify significantly. I will explain how the comparison between SA and L is analogous to the comparison between equivariant cohomology and equivariant cohomology of the fixed locus in GKM theory. Finally, I will sketch how this idea can be used to give explicit formulas for the Brill-Noether cycles -- informally, the cycles on M parametrizing curves on which a line bundle of the form omega^k(sum a_ix_i) has at least r+1 linearly independent sections. This is a joint work with M. Abreu and N. Pagani.

The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of the Balaban's approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyse this step for the O(4) non-linear sigma model in two dimensions and demonstrate in this case how various ingredients of the Balaban approach play together. First, using variational calculus on Lie groups, the equation for the minimum is derived. Then this non-linear equation is solved by the Banach fixed point theorem. This step requires a detailed control of lattice Green functions and their integral kernels via random walk expansions.

We discuss the embezzlement of entanglement and its relation to the classification of the latter, as well as its application to relativistic quantum field theory. Embezzlement (of entanglement), introduced by van Dam and Hayden, denotes the task of producing any entangled state to arbitrary precision from a shared entangled resource state, the embezzling state, using local operations without communication while perturbing the resource arbitrarily little. We show that Connes' classification of type III von Neumann algebras can be given a quantitative operational interpretation in terms of embezzlement. In particular, this quantification implies that all type III factors, apart from some type III_0 factors, host embezzling states. In contrast, semifinite factors (type I or II) cannot host embezzling states. Specifically, type III_1 factors are characterized as 'universal embezzlers', meaning every normal state is embezzling. The latter observation provides a simple explanation as to why relativistic quantum field theories maximally violate Bell inequalities. To understand the connection between embezzlement of entanglement and the classification of von Neumann algebras, we use a technique introduced by Haagerup and Størmer that associates to each normal state on a von Neumann algebra a state on the flow of weights. Our results then follow by quantifying the invariance of states on the flow of weights on the restriction of the dual modular flow. If time permits, we will also discuss the connection between embezzling states and embezzling families, as used by van Dam and Hayden.
This is joint work with Lauritz van Luijk, Reinhard F. Werner, and Henrik Wilming.

  The classical Pieri rule is a multiplication formula for Schubert class and Chern classes of the tautological bundle. Combinatorially, it is given by adding a chain of boxes on partitions. In this talk, we will discuss its generalization to equivariant Motivic Chern classes and its dual basis Segre motivic classes. Our formula is in terms of ribbon Schubert operators, which is roughly speaking adding ribbons on partitions. As an application, we have found a little surprising relation between motivic Chern classes and Segre motivic classes, extending the relation between ideal sheaves and structure sheaves over Grassmannians.

  Pairs of cointeracting bialgebras appear recently in the literature of combinatorial Hopf algebras, with examples based on formal series, on trees (Calaque, Ebrahimi-Fard, Manchon and Bruned, Hairer, Zambotti), graphs (Manchon), posets... These objects have one product (a way to combine two elements in a single one) and two coproducts (the first one reflecting a way to decompose a single element into two parts, maybe into several ways, the second one reflecting a way to contract parts of an element in order to obtain a new one). All these structures are related by convenient compatibilities.
  We will give several results obtained on pairs of cointeracting bialgebras: actions on the group of characters, antipode, polynomial invariants... and we will give applications to a Hopf algebra of graphs, including the Fortuin and Kasteleyn's random cluster model, a variation of the Tutte polynomial.

This seminar aims to introduce the "crossing map", a transformation of operators in Hilbert spaces defined in terms of modular theory and inspired by "crossing symmetry" from elementary particle physics, and discuss the strong connection between crossing-symmetric twists and endomorphisms of standard subspaces. Crossing symmetry has many interesting connections, including T-twisted Araki-Woods algebras, q-Systems, and algebraic Fourier transforms.

Given a dense open immersion U -> S, some smooth and proper families X_U -> U do not extend to smooth proper families over S. More often (but still not always), there is a "best smooth extension", the Neron model. I will talk about how to construct Neron models for families of smooth curves and their Jacobians. Neron models are not compatible with base change, so there are no "moduli spaces of Neron models", but we will see that they relate to some logarithmic moduli functors, and that their base change behaviour can be understood tropically (in terms of combinatorics of dual graphs).Given a dense open immersion U -> S, some smooth and proper families X_U -> U do not extend to smooth proper families over S. More often (but still not always), there is a "best smooth extension", the Neron model. I will talk about how to construct Neron models for families of smooth curves and their Jacobians. Neron models are not compatible with base change, so there are no "moduli spaces of Neron models", but we will see that they relate to some logarithmic moduli functors, and that their base change behaviour can be understood tropically (in terms of combinatorics of dual graphs).

In this talk we will introduce Ulrich sheaves on projective algebraic varieties. The corresponding notion for modules over rings originated from the work of Ulrich, but only after a paper by Eisenbud and Schreyer its geometric side received many attention due to the connection with determinantal and Pfaffian representations of (Chow forms of) varieties. The main questions are existence of such sheaves and, if so, their minimal rank. Positive answer to the first question is known for curves, surfaces (up to change of polarization), Veronese varieties, Segre varieties and complete intersections. Outside those examples, very few is known in dimension at least 3; therefore we will focus on Fano 3-folds.

We consider the Microcanonical Variational Principle for the vortex system in a bounded domain. In particular, we are interested in the thermodynamic properties of the system in domains of second kind, i.e. for which the equivalence of ensembles does not hold. For connected domains close to the union of disconnected disks (dumbbell domains), we show that the system may exhibit first-order phase transitions, while the entropy is convex for large energy (joint work with Dario Benedetto and Emanuele Caglioti).

Note: This talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV.

  Let k be a field, G be a reductive group over k, B be a Borel subgroup of G and C be a smooth curve over k. The B-orbit stratification of the flag variety G/B induces a natural stratify-cation on the moduli space of maps from C to G/B . The open stratum is strictly related to the so-called Zastava spaces.
  In this talk, we give an overview of the main properties of these spaces. If time permits, we also present some consequences on the moduli space of G-bundles.