Abstract: Un sistema de Steiner (n,k,t) (o t-design) es un hipergrafo k-uniforme en n vértices tal que todo subconjunto de t vértices está en una única arista. En 1847, Thomas Kirkman demostró que existen para k=3, t = 2, pero su existencia general fue un problema abierto por 167 años,
hasta Peter Keevash (2014).

En 1963, Erdős y Hanani conjeturaron que uno puede acercarse arbitrariamente a un sistema de Steiner en sentido asintótico exigiendo solo que cada t-subconjunto esté en al menos una arista (un cubrimiento).

En 1985 Vojtěch Rödl demostró la conjetura con un argumento probabilista sorprendentemente flexible: armar el cubrimiento de a “pequeños bocados”, eligiendo en cada paso un puñado de aristas al azar, de modo que lo que queda sigue siendo casi-regular y se puede volver a “morder”; iterando hasta que el residuo se vuelve despreciable.

El método pasó a llamarse Rödl Nibble y hoy es una herramienta importante en combinatoria probabilista: se ha usado para matchings casi-perfectos, índice cromático de hipergrafos y como ingrediente en la demostración general de la existencia de sistemas de Steiner.

Revisaremos el método a través de la formulación general de Pippenger y Spencer (1989) y deduciremos el teorema de Rödl como corolario.

Speaker: Raúl Astete (U.Chile)

Abstract:  Given 𝑛 jobs with known weights but unknown processing times, our goal is to minimize their total weighted completion time. It is well known that even ona single machine, no algorithm can be better tan 2-competitive, and this ratio is achieved by the (preemptive) Weighted Round-Robin rule. This initiated a line of work extending this positive result to more general scheduling settings involving parallel machines, release dates, or precedence constraints. Recently, the Polytope Scheduling Problem has received considerable attention. In this problem, there are finitely many resources, each with total availability 1, and processing a job 𝑗 at rate 𝑦_𝑗 requires 𝑏_{𝑑𝑗} 𝑦_𝑗 of resource 𝑑. In other words, at any time, the processing-rate vector must lie in the polytope {𝑦 ≥ 0 ∣ 𝐵𝑦 ≤ 1}. A natural generalization of the Weighted Round-Robin rule to this problem is the Proportional Fairness Algorithm.

We show that this algorithm achieves a competitive ratio of 2 on a large subclass of the Polytope Scheduling Problem that includes many classical scheduling problems, such as scheduling on unrelated machines or matroid scheduling. This subclass, which we call support-submodular, is characterized by the fact that, for all prices 𝛼_𝑗 ≥ 0 per unit of processing rate of job 𝑗, the maximum total price of a processing-rate vector for a subset 𝑈 of the jobs is a submodular function of 𝑈.

This is joint work with Alexander Lindermayr and Bart Zondervan.

Speakers: Sven Jäger, TU Berlin

Abstract: 
Consider the heat equation on a closed domain with C 2 boundary, subject to a random Wentzell boundary condition modeling stickiness and elastic killing, with coefficients undergoing Markovian switching. We derive the quenched and annealed evolution equations of these systems, inboth forward and backward directions. We show that as the switching rate tends to infinity, thesystem converges in the quenched L1 -sense to a heat equation with deterministic Wentzell condition,whose parameters are determined by the stationary law of the driving continuous-time Markov chain.

We further quantify the quenched uniform-in-time L1-error between the fast-switching process andits homogenized limit, establishing a rigorous error bound. In particular, for the time-dilatation t/ε,this error is o(ε−1/6 ), while the error between their time marginals is o(ε−1/2 ), as ε → 0. Finally, we apply our results to a model of neuronal receptor dynamics, where strong and weak ligand bindingcorrespond to distinct boundary regimes governing neurotransmitter clearance.

Speaker: Alexis Anagnostakis (CMM, U. de Chile)

Durante esta charla se presentará metodología avanzada para el mapeo de combustibles superficiales de la clasificación KITRAL. Para ello, se construyó un Data Cube integrando series temporales estacionales del programa Copernicus, combinando índices ópticos de Sentinel-2 y retrodispersión de radar de Sentinel-1, fusionados con variables topográficas y de estructura vegetal.

El expositor será Camilo Matus-Olivares, Doctor en Ciencias Agroalimentarias y Medioambiente de la Universidad de La Frontera (Chile).

El evento  también se transmitirá en vivo por el canal de YouTube de CopernicusLAC Chile en este enlace: https://bit.ly/seminario28agosto

Abstract: Given a $k$-uniform hypergraph $H$, the Ramsey number $R(H)$ is the smallest $N$ such that every $2$-colouring of the edges of $K_N^{(k)}$ contains a monochromatic copy of $H$. For graphs, Burr gave two constructions bounding $R(T)$ from below for a tree $T$, and his formula is now known to be exact for trees of small maximum degree. In the hypergraph setting, almost nothing is known beyond loose paths and cycles.

We study $R(T^{(k)})$, where $T^{(k)}$ is the $k$-expansion of a tree $T$, obtained by adding $k-2$ new vertices to each edge. We show that for bounded-degree trees the Ramsey number is at most $(1+\eta)\frac{1}{2}(2k-1)n$, and we prove a general lower bound $R(H) \geq |V(H)| + \tau(H) – 1$ for connected $k$-graphs $H$, where $\tau$ is the vertex cover number. These bounds are asymptotically tight for loose paths. In this talk, we will see how Burr's constructions generalise to the hypergraph setting, what the resulting lower bound looks like for expansions, and discuss the conjectures this work suggests.

Speaker: Vicente Sandoval (DIM, U. Chile)

Abstract: In this talk, we will consider the massless Boltzmann equation on FLRW spacetime with spatial topology $\mathbb{T}^3$ in the linear and decelerated expanding regimes, where the scale factor is $t^q$ with $q\in (0,1]$. In these backgrounds, the massless Boltzmann equation admits a non-stationary Maxwell-Jüttner equilibrium that maximises the entropy of the system. In this work, this equilibrium is shown to be non-linearly stable for the massless Boltzmann equation in the case of hard ball interaction. We will also discuss the non-linear stability of the vacuum solution when $q\in (1/3,1]$. This is joint work with Robert M. Strain (University of Pennsylvania) and Martin Taylor (Imperial College London).

For those unable to attend in person, the talk will also be available via Zoom at the following link: https://uchile.zoom.us/j/97781230158?pwd=MDIJ5OSVvuObCde6BEes5HcHwxeSa8.1

Speaker: Renato Velozo Ruiz (Universidad de Chile)

Durante el seminario se presentará Harpía Watch Forest, un sistema automatizado que es usado para el monitoreo de la cobertura forestal en Panamá, mediante imágenes satelitales y técnicas de teledetección. Se abordará la metodología empleada para la detección de cambios en la cobertura vegetal utilizando imágenes Sentinel-2, índices espectrales como el NDVI y algoritmos de análisis temporal. Además, se mostrarán los principales resultados obtenidos, las capacidades del sistema para apoyar la vigilancia ambiental y su contribución a la toma de decisiones para la conservación y el manejo sostenible de los recursos forestales.

Los expositores serán Yubrany González y Elvis García, profesionales del Departamento de Teledetección de la Dirección de Información Ambiental del Ministerio de Ambiente de Panamá.

El evento se realizará en el Centro de Modelamiento Matemático (CMM) albergado en la Universidad de Chile y se transmitirá en vivo por el canal de YouTube de CopernicusLAC Chile a través del siguiente enlace: https://bit.ly/seminario21agosto

Abstract:  In Lorentzian geometry, the light-ray transform is the operator that integrates functions over lightlike geodesics. It appears naturally in the study of wave equations, geometric scattering rigidity, and inverse problems. In this talk, I will discuss some work in progress with L. Busch and L. Oksanen, about the stability of this operator. First, I will present an instability result: the modulus of continuity cannot be smaller than certain polynomial. This can be upgraded to a logarithmic bound in the Gevrey category. Secondly, on stationary geometries, I will show how to obtain a stability result for moments of order k, using moments of the light-ray transform from 0 to k.

Speaker: Sebastián Muñoz Thon, Université Paris-Saclay