It is well-known that the spectral data of the Gaudin model associated to a finite semisimple Lie algebra is encoded by the differential data of certain flat connections associated to the Langlands dual Lie algebra on the projective line with regular singularities, known as oper/Gaudin correspondence. Recently, some progress has been made in understanding the correspondence associated with affine Lie algebras. I will present a physical perspective from Kondo line defects, physically describing a local impurity chirally coupled to the bulk 2d conformal field theory. The Kondo line defects exhibit interesting integrability properties and wall-crossing behaviors, which are encoded by the generalized monodromy data of affine opers. In the physics literature, this reproduces the known ODE/IM correspondence. I will explain how the recently proposed 4d Chern Simons theory provides a new perspective which suggests the possibility of a physicists’ proof. 

 In this talk I will discuss an algorithm to piecewise dualise linear quivers into their mirror duals. This applies to the 3d N=4 version of mirror symmetry as well as its recently introduced 4d counterpart, which I will review. The algorithm uses two basic duality moves, which mimic the local S-duality of the 5-branes in the brane set-up of the 3d theories, and the properties of the S-wall. The S-wall is known to correspond to the N=4 T[SU(N)] theory in 3d and I will argue that its 4d avatar corresponds to an N=1 theory called E[USp(2N)], which flows to T[SU(N)] in a suitable 3d limit. All the basic duality moves and S-wall properties needed in the algorithm are derived in terms of some more fundamental Seiberg-like duality, which is the Intriligator--Pouliot duality in 4d and the Aharony duality in 3d.

In this talk I will review some recent results obtained in collaboration with E. Runa and A. Kerschbaum on the one-dimensionality of the minimizers
of a family of continuous local/nonlocal interaction functionals in general dimension. Such functionals have a local term, typically the perimeter or its Modica-Mortola approximation, which penalizes interfaces, and a nonlocal term favouring oscillations which are high in frequency and in amplitude. The competition between the two terms is expected by experiments and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used  to model pattern formation, either in material science or in biology. The difficulty in proving the emergence of such structures is due to the fact that the functionals are symmetric with respect to permutation of coordinates, while in more than one space dimensions minimizers are one-dimesnional, thus losing the symmetry property of the functionals. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnetic systems, respectively  colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers are one-dimensional and periodic, in general dimension and also while imposing a nontrivial volume constraint.

I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed 
Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a "mirror P=W" conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting. This is joint work with Charles F. Doran.

Sela proved in 2006 that the (non abelian) free groups are stable. This implies the existence of a well-behaved forking independence relation, and raises the natural question of giving an algebraic description in the free group of this model-theoretic notion. In a joint work with Rizos Sklinos we give such a description (in a standard fg model F, over any set A of parameters) in terms of the JSJ decomposition of F over A, a geometric group theoretic tool giving a group presentation of F in terms of a graph of groups which encodes much information about its automorphism group relative to A. The main result states that two tuples of elements of F are forking independent over A if and only if they live in essentially disjoint parts of such a JSJ decomposition.

Model reduction is one of the most used tools to characterize real-world complex systems. A large realistic model is approximated by a simpler model on a smaller state space, capturing what is considered by the user as the most important features of the larger model. In this talk we will introduce a new information-theoretic criterion, called "autoinformation", that aggregates states of a Markov chain and provide a reduced model as Markovian (small memory of the past) and as predictable (small level of noise) as possible. We will discuss the connection of autoinformation to widely accepted model reduction techniques in network science such as modularity or degree-corrected stochastic block model inference. In addition to our theoretical results, we will validate such technique with didactic and real-life examples. When applied to the ocean surface currents, our technique, which is entirely data-driven, is able to identify the main global structures of the oceanic system when focusing on the appropriate time-scale of around 6 months.
arXiv link: https://arxiv.org/abs/2005.00337