Resumen: En esta charla estudiamos el operador de Dirac tridimensional con condiciones de borde de tipo MIT en un tubo delgado retorcido, donde la sección transversal es transportada y rotada a lo largo de una recta según una función de torsión. Comenzaremos estableciendo la auto-adjunción del operador y describiendo su espectro esencial. El resultado central es la construcción de un operador efectivo unidimensional que rige la dinámica cuando el grosor del tubo tiende a cero.
We started in L1 by reviewing the algebraic geometry of toric varieties and the structure of their derived categories of coherent sheaves.
In this L2, we will introduce constructible sheaves. We will define singular supports and sheaf categories with singular support conditions, and briefly review its relation with the wrapped Fukaya categories.
La medición constituye uno de los pilares fundamentales de la ciencia psicológica, pero también uno de sus conceptos más debatidos. Aunque las prácticas psicométricas se han sofisticado notablemente durante el último siglo, persisten desacuerdos fundamentales respecto de qué significa realmente medir, qué condiciones deben cumplirse para justificar una afirmación de medición y en qué medida la medición psicológica es comparable con la medición en las ciencias físicas. Esta presentación ofrece una actualización del estado actual de estos debates, revisando el desarrollo histórico del concepto de medición desde la psicofísica y la redefinición operacionalista de Stevens hasta los enfoques contemporáneos provenientes de la metrología, la filosofía de la ciencia y la psicometría. Se discutirán los principales desafíos conceptuales, incluyendo la objeción de la cantidad, la naturaleza de los atributos psicológicos, la justificación epistemológica de las inferencias de medición, el papel de los valores y del contexto sociocultural, y las oportunidades y desafíos que plantean la inteligencia artificial y los nuevos enfoques dinámicos de medición. Finalmente, se presentarán marcos conceptuales recientes que buscan integrar distintas tradiciones filosóficas y ofrecer una comprensión más amplia y rigurosa de la medición en las ciencias humanas.
In this series of 5 lectures, we will develop the coherent–constructible correspondence for toric varieties, which in combination with a result of Ganatra-Pardon-Shende establishes Homological Mirror Symmetry for toric varieties. The goal is to explain the mechanism of this correspondence so that participants can use it as a practical tool for studying toric mirror symmetry and its potential applications.We will begin by reviewing the algebraic geometry of toric varieties and the structure of their derived categories of coherent sheaves. (L1)
Next, we will introduce constructible sheaves. We will define singular supports and sheaf categories with singular support conditions, and briefly review its relation with the wrapped Fukaya categories. (L2)
Next, we will specify to toric varieties and introduce the main ideas in the works of Bondal, Fang-Liu-Treumann-Zaslow and Kuwagaki, as well as the correspondence between line bundles and twisted polytope sheaves studied by Zhou. We will review the approaches taken by these authors to prove the Coherent-Constructible Correspondence through explicit examples. In particular, we will calculate some examples by matching the Cech resolution of line bundles with the twisted polytope sheaves. We will also explain the correspondence between tensoring with a line bundle and convolution by the corresponding twisted polytope sheaf studied recently by Bose-Williams. (L3-L4)
Finally, we will place the coherent–constructible correspondence in the broader context of homological mirror symmetry and explain the connection to the approaches taken by Abouzaid and Hanlon-Hicks. We will discuss a few recent advances and open problems as an invitation. (L5)
To derive an equation describing the distribution of a dilute gas of identical particles, Boltzmann assumed that the many-particle probability density function
stays close to a product for all time, provided it is a product initially; this'ld imply that in a gas of many particles, the position-velocity of the particles stay independent.
Mark Kac introduced a probabilistic space-homogeneous model for a gas of N particles, for which Boltzmann’s assumption holds in the limit as $N\to\infty$:
For fixed time $t$, and any $k$ fixed, the $k$ marginal distribution of the $N$ particle density functions becomes a product for large $N$. This is known as chaos (or molecular chaos), its propagation in time is known as propagation of chaos.
We retake the question, first asked in [CCLLV] For which probability distibutions g on R can we produce a sequence $\{fN\}_{N\geq 1}$, of probability distributions, with fN supported on the energy sphere $\{v1^2+v2^2+...+vN^2=N\}$ which are chaotic to $g$ ?
Using rescaled states, we expand the the class of admissible g, obtained in
[CCLLV]. We also mention some new ideas in this direction.
References:
Carlen, Eric A., et al. Entropy and chaos in the Kac model. Kinetic and Related Models 3.1 (2010): 85-122.
Cortez, Roberto, and Hagop Tossounian. Chaos for rescaled measures on Kac’s sphere. Electronic Journal of Probability 28 (2023): 1-29.
We consider the Dirac operator with mass perturbed by a nonnegative potential in some Lebesgue space, and study the problem of minimizing the lowest eigenvalue in the spectral gap under a constraint on the corresponding norm. This is the Dirac analogue of the classical Keller problem for Schrödinger operators. When the potential is of electrostatic type, we showed that solutions exist and are characterized by a nonlinear Dirac equation with Kerr-type nonlinearity. In the talk, I will outline the main strategies and differences for the electrostatic case and work in progress for potentials of mass type, which give rise to quite different challenges but open a road to a variational characterization of solitary waves for the Soler model.
This is joint work Jean Dolbeault, David Gontier and Fabio Pizzichillo.
This study analyzes several dice games that originated in North and South America, including Tasholiwe, Chuncana, and Yole, as described by anthropologists Stewart Culin, Erland Nordenskiold, and Raúl Martín Crovetto. The study focuses on the relationship between the point values assigned to the outcomes and their probabilities. The author argues that the rules of these games require an understanding of probability principles and statistical evaluation based on relative frequencies.
Este estudio analiza varios juegos de dados originarios de América del Norte y del Sur, entre ellos el Tasholiwe, el Chuncana y el Yole, tal como los describen los antropólogos Stewart Culin, Erland Nordenskiold y Raúl Martín Crovetto. El estudio se centra en la relación entre los valores asignados a los resultados y sus probabilidades. El autor sostiene que las reglas de estos juegos requieren una comprensión de los principios de la probabilidad y una evaluación estadística basada en frecuencias relativas.
In this study we considered five generalizations of the standard Weibull distribution with three parameters to describe the baseline hazard in the survival part of the joint model framework. The proposed distributions were the gamma-Weibull, exponentiated Weibull, generalized Weibull, Marshall-Olkin-Weibull and generalized power Weibull distribution, which can handle with complex forms like constant, increasing, decreasing, bathtub and upside-down shapes. We used these distributions to study the first stage of spontaneous labour aiming to evaluate the time to c-section delivery. Particularly, were considered information about cervical dilatation, as time-dependent in the longitudinal part of the joint model, and others clinical variables as well as previous parity, comorbidity or pre pregnancy complications, if the woman was referred from another health facility and membranes status. As we are considering non nested models, we selected the best distribution using information based criterion methods like AIC and BIC, and with the appropriated model we make predictions about the survival probability, which is the complementary probability of c-section, and investigate the impact of the cervical dilatation evolution and others variables in the C-section outcome.
Keywords: joint modelling; labour modeling; longitudinal process; model selection; survival process; weibull generalizations.
Resolvent estimates play a fundamental role in scattering and resonance theory, where they provide quantitative information on wave propagation, local energy decay, and the distribution of resonances. In the semiclassical regime, the growth of the resolvent is closely related to the properties of the underlying classical dynamics and has been the subject of extensive study for Schrödinger operators.
In this talk, I will present new results on the semiclassical resolvent of magnetic Dirac operators. First, I will establish an exponential upper bound for the resolvent norm, without imposing any assumption on the underlying classical dynamics. This estimate provides a general control of the resolvent in the semiclassical regime. I will then discuss the influence of trapping on the cutoff resolvent. More precisely, I will show that the presence of trapped classical trajectories implies a logarithmic lower bound on the cutoff resolvent. This extends to the relativistic Dirac setting a classical phenomenon known for semiclassical Schrödinger operators, namely the existence of a logarithmic gap between non-trapping and trapping regimes. The proof of this result is based on the long-time propagation of coherent states (up to the Ehrenfest time).
We present some recent results on PDE systems modeling the dynamics of suspension bridges. The structure is represented as a fish-bone plate with intermediate piers, consisting of a degenerate rectangular planar domain where a central beam moves vertically and a continuum of cross-sections rotate around their midpoints; the piers are modeled as rigidly fixed sections. The dynamics is then governed by a nonlinear, nonlocal coupled system of beam-wave equations describing the evolution of vertical and torsional oscillations, possibly accounting for damping effects and external forces. After establishing a rigorous functional framework and introducing a suitable notion of weak solutions, we address the long-term behavior of the system. Specifically, we discuss the dissipativity of the underlying dynamical system and investigate the existence, uniqueness, and global stability of stationary and periodic solutions, both for purely vertical and fully coupled vertical/torsional motions. We show in particular that the system may exhibit a rich and unpredictable dynamics, characterized by a global attractor with a nontrivial structure.
The talk is based on joint work with Maurizio Garrione and Filippo Gazzola (Politecnico di Milano).
Los modelos quantum double de Kitaev forman una familia prototípica de modelos exactamente solubles que exhiben orden topológico y excitaciones tipo anyón. En esta charla me centraré en describir cómo estos modelos pueden formularse de manera natural dentro del marco de las C*-álgebras, lo que permite una descripción algebraica robusta de sus observables y de las excitaciones. Finalmente, calcularé el conjunto completo de estados KMS del modelo quantum double para el caso abeliano.
Este modelo describe el movimiento de un fluido incompresible e irrotacional bajo la influencia de la gravedad, confinado por un fondo rígido en la parte inferior y una superficie libre en la parte superior. Aunque las ondas solitarias están bien comprendidas en el caso de un fondo plano, nos interesa analizar qué ocurre cuando la topografía del fondo cambia—por ejemplo, cuando la onda encuentra una elevación o depresión repentina en el lecho marino. Esto da lugar a un régimen de interacción entre la onda y el fondo variable, en el que tanto la velocidad como la forma de la onda evolucionan de manera dinámica.
Para entender mejor esta interacción compleja, también estudiamos modelos simplificados o “de juguete”, como la ecuación de Korteweg–de Vries (KdV), la ecuación de Whitham y los sistemas de Boussinesq tipo abcd. Estos modelos capturan aspectos esenciales de la dinámica en formas más manejables, y nos ayudan a identificar los mecanismos clave que rigen la respuesta de la onda.
We study the spectral properties of the Dirac operator $L_0$ obtained by linearizing the one-dimensional Soler model around standing waves with power nonlinearity $f(s)=s|s|^{p-1}$, $p>0$. We give a sharp characterization of the spectral gap. If $p\ge1$, the gap contains no eigenvalues other than the symmetry-induced energies $-2\omega$ and $0$. If $0<p<1$, additional eigenvalues bifurcate from the thresholds of the essential spectrum and enter the gap. We further prove that the thresholds are never eigenvalues for any $p>0$ and that the thresholds are not resonances for $p>1$.
Atomic inertial sensors constitute highly controllable and scalable platforms for precision measurements of inertial effects, particularly acceleration. In this talk, I will present and analyze a theoretical proposal for an atomtronic angular accelerometer based on an angularly ac?shaken ring lattice. I will show how supercurrents of ultracold atoms circulating in the ring can be harnessed to achieve high?precision measurements of angular acceleration.
In this system, a significant net atomic current emerges when the lattice driving frequency is tuned to integer fractions of the system’s Bloch frequency. These resonances give rise to directed transport, with the resulting supercurrents encoding information about the external angular acceleration.
Within the Bose–Hubbard model, I will discuss two main regimes: the single-particle regime and the weakly interacting many-body regime. In the single-particle limit, I will demonstrate analytically that the resonance width scales inversely with the measurement time, thereby imposing a Fourier limited bound on the achievable precision in angular-acceleration estimation. By contrast, weak onsite interactions modify this behavior. Numerical simulations will show that interactions can lead to a pronounced sharpening of the resonances. As a result, the sensitivity to the angular acceleration can surpass the Fourier limited scaling of the non-interacting case, improving the measurement precision by several orders of magnitude.
La reciente segunda vuelta de las elecciones presidenciales en Perú (2026) constituye una motivación concreta para revisar críticamente las herramientas estadísticas utilizadas en la detección de fraude electoral. Este trabajo se inscribe en ese contexto, no con el objetivo de emitir un veredicto, sino de problematizar los marcos analíticos desde los cuales tales veredictos suelen construirse.
La detección de fraude electoral se ha desarrollado a partir de una amplia gama de propuestas estadísticas que, pese a su diversidad, comparten un supuesto central: la existencia de una clean election, caracterizada por un proceso generador de datos ideal frente al cual se evalúan desviaciones de elecciones particulares. Este marco no solo orienta el análisis, sino que también tiende a encerrar la discusión dentro de los límites del propio modelamiento estadístico, dificultando la incorporación de otras perspectivas y formas de escrutinio.
Tanto las principales críticas a este enfoque como un replanteamiento de la pregunta central provienen, significativamente, desde fuera de esta tradición. En particular, el informe de la Comisión de Venecia (2018) propone abandonar la pregunta “¿es esta elección fraudulenta?” y reemplazarla por una más operativa: “¿cuál es la tasa mínima de fraude necesaria para cambiar el resultado electoral?”. Este desplazamiento evita la dependencia de un modelo ideal previo y abre el análisis a escenarios concretos de intervención.
A partir de lo que denominamos Estadística Constructivista, proponemos una metodología que construye explícitamente los eventos de interés para caracterizar esta pregunta y responderla sin recurrir a supuestos estructurales sobre cómo “debería” comportarse una elección. En este sentido, el enfoque es agnóstico: no representa conocimiento experto ni información contextual específica del país en términos de supuestos no empíricos.
En la segunda parte de la discusión, evaluamos la incorporación de supuestos adicionales —habitualmente defendidos en nombre del “realismo” o del conocimiento experto—. Mostramos que estos no mejoran los resultados obtenidos en la fase agnóstica y que, en algunos casos, los debilitan.
Este seminario/simposio busca abrir la discusión más allá de sus fronteras habituales. Está diseñado como un espacio de intercambio y confrontación, donde distintos actores —académicos y no académicos— puedan cuestionar tanto los resultados como los supuestos que los sostienen.
Trabajo conjunto con Matthis Dartevelle y Sebastien Van Bellegem (UC Louvain, Bélgica).
How did endogeneity problems arise? What is exogeneity? Is exogeneity really important for statistical modeling? In this joint seminar–symposium, I will offer some answers to these questions in order to discuss them with you.
En esta charla introduciremos nociones de controlabilidad y observabilidad de EDPs. Se muestra el resultado para conjuntos medibles de dimensión de Hausdorff mayor a $n-1$ y se explora la posibilidad de observar desde un número irracional algebraico para $n=1$.
2.- Aplicaciones birracionales.
3.- Superficies regladas.
4.- Teorema de Castelnuovo.
5.- Superficies con p_g=0 y q>=1
6.- Dimensión de Kodaira.
7.- Cubrimientos ramificados.
8.- Superficies con k=0.
9.- Superficies con k=1 y superficies elípticas.
10.- Superficies de tipo general.
The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called minimal and maximal words.
We formulate the conjecture that every minimal and maximal word is normal over an appropriate subalphabet.
The aim of the talk is to convince the audience that the conjecture seems true and of considerable difficulty. In particular, we shall discuss its connections with several older conjectures, including the existence of Z-numbers (Mahler, 1968) and Z_p/q-numbers (Flatto, 1992), the existence of triple expansions in rational base p/q (Akiyama, 2008), and the Collatz-inspired '4/3 problem' (Dubickas and Mossinghoff, 2009).
The talk is based on a joint work with Shalom Eliahou and Léo Vivion.
About twenty years ago, Peres and Weiss generalised the classical Poisson limit theorem for appearances of words of increasing length in a sequence x. They showed that the theorem holds for almost every x with respect to the infinite uniform product measure. A natural question is whether this Poisson behaviour persists when the sequence is sampled according to a different product measure.
In our first result, we consider non-stationary product measures and show that there exists a quantitative threshold above which the Poisson limit theorem holds for almost every x, while below this threshold it may fail. In contrast, our second result shows that for a biased infinite product measure (a non-fair coin) the limiting behaviour is almost surely non-Poisson. This shows that the Poisson regime is specific to the equiprobable case and to small deviations from it.
This talk is based on works with Mike Hochman and Jon V. Kogan.