Notiziario Scientifico

Notiziario dei seminari di carattere matematico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma

Settimana dal 04-12-2023 al 10-12-2023

Lunedì 04 dicembre 2023
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Analisi Matematica
Antonio Tribuzio (Univ. Bonn)
Energy scaling of singular-perturbation models involving higher-order laminates
Motivated by the appearance of complex microstructures in the modelling of shape-memory alloys, we study the energy scaling behaviour of some N-well problems with surface energy given by a singular higher-order term. In the case of absence of gauge invariances (e.g. with respect to the action of SO(n) or Skew(n)), we provide an ansatz-free lower bound which relies on a bootstrap argument in Fourier space and gives evidence of the higher order of lamination involved. The upper bound is provided by iterated branching constructions. In the end, we show how a similar approach can be used in the determination of a lower bound for a more realistic model, namely the geometrically linearized cubic-to-tetragonal phase transition, in which a second order lamination is forced by the presence of affine boundary conditions. This is a joint work with Angkana Rüland. This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001
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Lunedì 04 dicembre 2023
Ore 17:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Daniel Parisi (Sapienza Università di Roma)
Entropy and mixing time of non-local Markov chains.
We discuss the convergence to the stationary distribution for non-local Markov chains on general spin systems on arbitrary graphs. We show that the relative entropy functional of the corresponding Gibbs measure satisfies the block factorization of entropy, an inequality that controls the entropy on a given region V in terms of a weighted sum of the entropies on blocks A ⊂ V when each A is given an arbitrary nonnegative weight α_A. This inequality generalizes the approximate tensorization of entropy and provides a natural extension of the classical Shearer inequality satisfied by the Shannon entropy. As a consequence of block factorization, we obtain optimal bounds on the mixing time of a large class of sampling algorithms for the ferromagnetic Ising/Potts models, including non-local Markov chains such as the heat-bath block dynamics and the Swendsen-Wang dynamics. The methods also apply to spin systems with hard constraints such as q-colorings and the hard-core gas model. First, we consider spin systems on the d-dimensional lattice Z^d satisfying strong spatial mixing. Then we extend our analysis to spin systems on an arbitrary graph satisfying spectral independence. Finally, we show that the existence of a contractive coupling for any local Markov chain implies spectral independence.

Martedì 05 dicembre 2023
Ore 14:30, Aula d'Antoni, Dipartimento di Matematica, Tor Vergata
seminario di Geometria
Ruadhaí Dervan (University of Glasgow)
Stability conditions for varieties
Stability conditions in algebraic geometry are used to construct moduli spaces. Experience from the theory of vector bundles (and coherent sheaves) suggests it is useful to have many stability conditions, so that one can geometrically understand the birational behaviour of resulting moduli spaces by varying the stability condition. Motivated by this, I will describe a mostly conjectural analogous story for projective varieties with an ample line bundle. Here the classical notion of stability is K-stability, which aims to construct higher dimensional analogues of the moduli space of stable curves, and the main point will be to introduce variants of K-stability defined using extra topological choices. The main results will link these new stability conditions with differential geometry, through the solvability of certain geometric PDEs, and I will try to explain how these links come about and what the general picture should be.
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Martedì 05 dicembre 2023
Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, Università di Roma "Tor Vergata"
Seminario di Analisi Matematica
Liangjun Weng (Università di Roma "Tor Vergata")
The capillary Minkowski problem
The classical Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere to be the surface area measure of a convex body. In a smooth setting, it reduces to the study of a Monge-Ampere equation on the unit sphere. This problem has been completely solved through the seminal works of Nirenberg, Pogorelov, Cheng-Yau, etc. In this talk, a new Minkowski-type problem will be introduced. The problem asks for the existence of a convex hypersurface with prescribed Gauss-Kronecker curvature and capillary boundary supported on an obstacle, which can be deduced as a Monge-Ampere equation with a Robin (or Neumann) boundary value condition on the spherical cap. Then obtain a necessary and sufficient condition for solving this problem.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006
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Mercoledì 06 dicembre 2023
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Algebra e Geometria
Bruno Klingler (Humboldt Universität zu Berlin)
Abelian differentials and their periods
An abelian differential is a smooth projective curve endowed with an algebraic one-form. I will discuss the uniformization of the moduli of abelian differentials provided by their periods, its arithmetic and functional transcendence properties. Joint work with Leonardo Lerer.

Mercoledì 06 dicembre 2023
Ore 16:00, Aula dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata
Seminario Algebre di Operatori
Christopher J Fewster (University of York)
Exact measurement schemes for local observables and the preparation of physical local product states
For a long time, quantum field theory (QFT) lacked a clear and consistent measurement framework, a gap that was described as "a major scandal in the foundations of quantum physics" [1]. I will review the recent framework put forward by Verch and myself [2], which is consistent with relativity in flat and curved spacetimes and has resolved the long-standing problem of "impossible measurements" put forward by Sorkin [3]. The central idea in this framework is that the "system" QFT of interest is measured by coupling it to a "probe" QFT, in which the system, probe, and their coupled variant, all obey axioms of AQFT in curved spacetime. It has been shown that every local observable of the free scalar field has an asymptotic measurement scheme, i.e., can be measured to arbitrary accuracy by a sequence of probes and couplings [4]. I will describe new results that (a) show that there are exact measurement schemes for all local observables in a class of free theories, (b) provide a protocol for the construction of Hadamard local product states in curved spacetime. The latter is complementary to a recent existence result of Sanders [5]. [1] Earman, J., and Valente, G. Relativistic Causality in Algebraic Quantum Field Theory, International Studies in the Philosophy of Science, 28:1, 1-48, (2014) [2] Fewster, C.J., Verch, R. Quantum Fields and Local Measurements. Commun. Math. Phys. 378, 851–889 (2020). [3] Bostelmann, H., Fewster, C.J., and Ruep, M.H. Impossible measurements require impossible apparatus Phys. Rev. D 103, 025017 (2021) [4] Fewster, C.J., Jubb, I. & Ruep, M.H. Asymptotic Measurement Schemes for Every Observable of a Quantum Field Theory. Ann. Henri Poincaré 24, 1137–1184 (2023). [5] Sanders, K. On separable states in relativistic quantum field theory, J. Phys. A: Math. Theor. 56 505201 (2023) This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)
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Giovedì 07 dicembre 2023
Ore 14:15, Aula M1, Dipartimento di Matematica e Fisica, Università Roma Tre
Seminario di Geometria
Concettina Galati (Calabria)
Degenerations of singularities and Severi varieties of surfaces with nodes and triple points
This talk will be devoted to a joint work in progress with Ciro Ciliberto where we study Severi varieties of surfaces in threefolds with nodes and triple points. In particular, we will report about certain results of degenerations of singularities of surfaces, obtained by limit linear systems techniques.
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Giovedì 07 dicembre 2023
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
P(n)/N(p) : Problemi differenziali nonlineari/Nonlinear differential problems
Danilo Gregorin Afonso (Sapienza Università di Roma)
Semilinear elliptic equations under geometric constraints: energy instability and overdetermined problems
We will analyze semilinear elliptic problems of the type −Δu = f(u) in domains that are constrained to be inside a fixed unbounded open set C, with appropriate boundary conditions. Our aim is to understand how the geometry of C selects domains in which positive solutions of the equation have special properties, mainly related to notions of symmetry. Our arguments are primarily based on analyzing how the energy of a positive solution in a domain varies when the domain moves inside C. We first consider the case where C is generic. We show how to define an energy functional T when the equation possesses more than one solution and compute the domain derivative of T. In the case when C is a cone or a cylinder, we show that some special domains may be unstable as critical points to the energy shape functional. This is done by analyzing the sign of the second derivative of the energy functional. Time allowing, we will show that in a special class of domains, namely bounded cylinders, solutions other than the one-dimensional ones do exist, under fairly general assumptions on the nonlinearity f. This is accomplished by means of bifurcation theory and Morse index comparison. The seminar is based on joint work with Alessandro Iacopetti and Filomena Pacella.
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