Pristupačnost
Stranice Seminara za teoriju vjerojatnosti na PMF-u u Zagrebu.
Standardni termin održavanja seminara je utorak u 16 sati u učionici A002.
Stranice Seminara za teoriju vjerojatnosti na PMF-u u Zagrebu.
Standardni termin održavanja seminara je utorak u 16 sati u učionici A002.
Voditelji seminara: prof.dr.sc. Bojan Basrak, prof.dr.sc. Miljenko Huzak, prof.dr.sc. Hrvoje Šikić, prof.dr.sc. Zoran Vondraček
Tajnica seminara: Daniela Ivanković
Random temporal graphs are a version of the classical Erdos-Rényi random graph G(n,p) where additionally, each edge has a distinct random time stamp, and connectivity is constrained to sequences of edges with increasing time stamps. We are interested in the asymptotics for the distances in such graphs, mostly in the regime of interest where np is of order log n ('near' the phase transition). More specifically, we will discuss the asymptotic lengths of increasing paths: the lengths of the shortest and longest paths between typical vertices, as well as the maxima between any two vertices; this also covers the (temporal) diameter. In the regime np >> log n, longest increasing paths were studied by Angel, Ferber, Sudakov and Tassion. The talk contains joint work with Nicolas Broutin and Gábor Lugosi.
The study will also advance the analysis of distributional limits for the perimeter and diameter of convex hulls, drawing on the foundational work of Andrew Wade and his contemporaries. Techniques such as the Martingale Difference Sequence and Cauchy's formula will form the core methodological approach, aiming to model and understand the variance and behavior of geometric objects generated by random walks. The dissertation will explore the convex hull formed from the centers of mass points to demonstrate analogous results to those found for original convex hulls.
The model of randomly augmented graphs (also known as randomly perturbed graphs) combines two basic concepts studied in graph theory: Dirac type graphs and random graphs. Given an n-vertex graph G_α with minimum degree δ(G_α)>=αn, and a binomial random graph G_n,p, we call G_α U G_n,p a randomly augmented graph. This model has recently gained a lot of attention as a natural generalization of deterministic and random graphs. In my talk I will briefly survey the known results and methods used in studying randomly augmented graphs.
Razonodit ćemo slušatelja temom iz tzv. euklidske Ramseyeve teorije, koju su 70-tih godina prošlog stoljeća počeli sustavno proučavati Erdős, Graham, Montgomery, Rothschild, Spencer i Straus. Godine 1979. Erdős i Graham su pitali da li, za svako bojenje ravnine u konačno mnogo boja, nekoja boja sadrži vrhove pravokutnika bilo koje zadane površine. Taj je problem uvršten pod rednim brojem 189 na stranicu "Erdős problems": https://www.erdosproblems.com/189 Pokazat ćemo da je odgovor na ovo pitanje negativan, čak i za kvadre u više dimenzija, a jednostavnu konstrukciju izložit ćemo prema nedavnom preprintu: https://arxiv.org/abs/2309.09973 Potom ćemo spomenuti neke autorove povezane pozitivne rezultate koji koriste metode multilinearne Fourierove analize.
U ovom predavanju dat ću pregled nedavnih rezultata o Dirichletovim formama i pripadajućim Markovljevim procesima s jezgrom skokova degeneriranoj na granici. Objasnit ću opću teoriju kao i motivirajuće primjere, te opisati neke nove, neočekivane, značajke teorije potencijala i analize takvih Markovljevih procesa. Predavanje se temelji na nekoliko zajedničkih radova sa Soobin Choom, Panki Kimom i Renming Songom.
Large scale data collection is now prevalent in almost all aspects of society. Availability of such data, and the analyses performed on such data, allow us to discover new, previously unseen statistical patterns, or possibly remind ourselves of patterns that were already familiar before. In many cases, statistical associations observed in large scale data give rise to new scientific questions. For example, analyses of university records demonstrate a disparity in admission rates between male and female university applicants; analyses of criminal justice data show that racial minorities in the US are more likely to be jailed than the majority group. In an entirely different and unrelated context, that of intensive care unit (ICU) medicine, a large body of evidence shows that critically ill individuals with a high body mass index have a better chance of surviving their illness when admitted to the ICU, compared to their leaner counterparts (known as obesity paradox). Seemingly disconnected, the above mentioned phenomena raise a basic question in common: how did the disparity observed in the data come about in the first place? Can we connect the observed disparity to the causal mechanisms that are present in the real world, and that generate the observed disparity? Providing causal explanations of this kind is key for the scientific understanding of the observed disparities. In fact, the discussed phenomena are amenable to almost the same methodological toolkit, despite the fact that they arise from entirely different scientific domains. In this talk, we discuss the issues of gender or racial bias in datasets, from a causal perspective. These topics are studied under the rubric of fair machine learning, but could also be seen as epidemiology of discrimination. In addition to this, we study some epidemiological questions in ICU medicine, such as the obesity paradox, and describe accompanying computational and statistical methods that are useful in tackling ICU research questions.
U ovom izlaganja promatrat ćemo označene točkovne procese kod kojih je moguća zavisnost između zadnjeg vremena međudolaska i oznake. Nadalje, korištenjem metoda i alata moderne teorije vjerojatnosti, kao što su teorija točkovnih procesa te metoda sparivanja, dokazat ćemo odgovarajuću verziju proširenog teorema obnavljanja.
We examine the almost-sure asymptotics of the solution to the stochastic heat equation driven by a Lévy space-time white noise. For fixed time and space we determine the exact tail behavior of the solution both for light-tailed and for heavy-tailed Lévy jump measures. When a spatial point is fixed and time tends to infinity, we show that the solution develops unusually high peaks over short time intervals, even in the case of additive noise, which leads to a breakdown of an intuitively expected strong law of large numbers. We also determine the almost-sure growth rate of the solution for any fixed time. This is joint work with Carsten Chong (Columbia).
We study semilinear problems in bounded $C^{1,1}$ domains for non-local operators with a boundary condition. The operators cover and extend the case of spectral fractional Laplacian. We also study harmonic functions related to the operator and boundary behaviour of Green and Poisson potentials.
In the theory of wavelets, the concept of maximal shift-invariant spaces plays an important role. Maximality is characterized by the strict positivity of the periodization function, the property which also appears in the characterization of several independence and basis related properties for the system of integer translates. In this talk, we consider the concept in a more general setting of LCA groups and unitary dual integrable representations. We describe the dual integrable triples which allow a decomposition of the Hilbert space into an orthogonal sum of n maximal cyclic subspaces and analyze how the questions concerned with maximality are reflected in redundancy and basis related properties of the generating orbit. Of particular interest is the case when n=1, i.e., when the generating orbit is complete in the whole space. The talk is based on a joint work with Hrvoje Šikić.
Teorija obnavljanja je važan dio moderne teorije vjerojatnosti sa širokom primjenom. Međutim, standardna teorija obnavljanja nije primjenjiva na događaje koji se pojavljuju u klasterima, što je čest slučaj u mnogim područjima primijenjene vjerojatnosti. Proučavat ćemo procese obnavljanja s klasterima na skupu realnih brojeva te planiramo, uz određene uvjete, proširiti klasične teoreme obnavljanja na procese ovog tipa (Blackwellov teorem obnavljanja i Ključni teorem obnavljanja), kao i opisati asimptotsku distribuciju pomaknutog točkovnog procesa s klasterima (Prošireni teorem obnavljanja).
In this talk, we discuss subgeometric ergodicity of a class of regime-switching diffusion processes. We derive conditions on the drift and diffusion coefficients which result in subgeometric ergodicity of the corresponding semigroup with respect to the total variation distance as well as a class of Wasserstein distances.