Title: On simplicity of the Cremona groups
Abstract: The Cremona group or rank n is the group of birational transformations of the projective space of dimension n. One of the central questions regarding these groups, dating all the way back to the 19th century and settled only recently, is whether they are simple. In this talk I will review the history of the problem, as well as the modern machinery that led to the proof of their non-simplicity. I will also demonstrate how to use this machinery to obtain an explicit free product structure in dimension 3 which, among other consequences, recovers non-simplicity in an effective way.
Title: On the number of pairwise touching cylinders in R^d.
Abstract: John E. Littlewood posted the question ``Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants.'' Bozoki, Lee, and Ronyai constructed a configuration of 7 mutually touching unit cylinders. The best-known upper bounds show that at most 10 unit cylinders in R^3 can touch one another. We consider this problem in higher dimensions, and obtain exponential (in d) upper bounds on the number of mutually touching cylinders in R^d. Our method is fairly flexible, and it makes use of the fact that cylinder touching can be expressed as a combination of polynomial equalities and non-equalities. (Joint work with Josh Zahl)
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Title: Evolution Equations: A new perspective on direct and inverse problems
Abstract: We consider initial-boundary value problems (IBVPs) for linear evolution partial differential equations (PDEs) with time-dependent coefficients. We present integral representations of solutions for a broad class of such problems, derived through extensions of the unified transform (UT) method. Using these representations, we investigate the well-posedness of the associated IBVPs for certain linear and nonlinear PDEs. Moreover, we show that, for a particular class of evolution PDEs, the solution of an IBVP posed on a finite interval can be reconstructed as a superposition of solutions to two associated IBVPs posed on the half-line. This result provides a fundamentally new perspective for the analysis of this broad class of problems. Finally, we formulate the inverse problem characterizing the controllability of such systems and develop a hybrid approach for determining the corresponding controller, with applications to physical problems arising in agriculture.
This is joint work with T. Ozsari, D. Mantzavinos and L. Mindrinos.
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Title: Classification theory and certain singular Calabi-Yau varieties
Abstract: The Minimal Model Program (MMP) plays a fundamental role in the classification theory of complex projective varieties up to birational equivalence. It predicts that Fano, Calabi-Yau and canonically polarized varieties, which are characterized by the “positivity” of their canonical class, constitute – in an appropriate sense – the three “building blocks” of all mildly singular complex projective varieties. In this talk, after giving an overview of the MMP, I will focus on two special kinds of Calabi-Yau varieties, namely primitive symplectic and primitive Enriques varieties, which constitute generalizations of irreducible holomorphic symplectic and Enriques manifolds, respectively, to the singular setting. In particular, I will present some results from recent joint works with Francesco Denisi, Ángel David Ríos Ortiz and Zhixin Xie regarding the behaviour of primitive Enriques varieties under certain birational operations and deformations, aiming to motivate thus their definition.
Title: Trace-based multidimensional reduction methods and applications
Abstract: In this talk, we present a set of new multidimensional reduction methods, with particular emphasis on both linear and nonlinear approaches for processing high-order tensor data. We first review linear techniques, including Multidimensional Principal Component Analysis (MPCA) and Multilinear Orthogonal Neighborhood Preserving Projections (MONPP), which extend classical matrix-based dimensionality-reduction models to tensor representations while preserving structural information. We then discuss nonlinear methods, such as Multi-Laplacian Eigenmaps, capable of capturing complex manifold structures inherent in real-world datasets.
All these techniques are formulated within a unified framework based on trace optimization, where dimensionality reduction is cast as the maximization or minimization of appropriate trace criteria. This perspective not only simplifies the derivation of individual algorithms but also highlights their conceptual relationships and shared mathematical foundations. Beyond their standard (linear) formulations, we also introduce the kernel-based extensions of MPCA, MONPP, and related methods, which enable the exploitation of nonlinear correlations in high-dimensional or multimodal data by implicitly mapping them into more expressive feature spaces.
A comparative analysis is provided to explore the theoretical underpinnings, statistical assumptions, computational complexity, and robustness characteristics of each method. Special attention is given to scalability issues, parameter sensitivity, and the preservation of local and global geometric structures. Practical insights are offered through examples and use-case discussions, illustrating how these dimensionality-reduction techniques can be effectively deployed in domains such as image processing, signal processing, and large-scale data analytics.
Overall, the study provides a comprehensive overview and a set of guidelines for selecting the most appropriate multidimensional reduction method depending on data characteristics, computational constraints, and application objectives.
Title: l^p asymptotic behavior of isotropic transition densities on homogeneous trees
Abstract: We study the large-time l^p behavior of transition densities of an isotropic ran- dom walk in homogeneous trees, which are infinite, connected, acyclic graphs in which every vertex has the same degree, and can be thought as discrete counter- parts of hyperbolic space. Caloric functions of interest are then convolutions of these transition densities with a finitely supported initial condition, and we are interested in their large time behavior in l^p norm.
For each p ∈ [1, ∞], we introduce a notion of a p-mass function and prove that caloric functions with compactly supported initial data, asymptotically decouple as the product of this mass function the transition density. Using tools of Fourier analysis available on such graphs, we show that this function even boils down to a constant, still depending on p, if the initial condition is radial, that is, depends only on the distance to the origin. Determining the spatial concentration of the densities in p-norm plays an important role, in turn clarifying the interplay between the exponential volume growth of the graph and heat diffusion. The results extend to affine buildings, even exotic ones beyond the Bruhat–Tits framework.
Joint work with B. Trojan.
Title: Controlling oscillation by positive quantities
Abstract: It is a nontrivial task to understand the interference of waves travelling in multiple directions in a quantitative way. Various conjectures in this vein were made in the 1970's by Stein and (implicitly) by Mizohata and Takeuchi, but until recently they have remained largely unresolved. Some of these conjectures propose that the interference should be controlled by positive quantities. We shall describe some of these conjectures and the motivations for their formulations, and give a high-level review of progress made towards them. However, it is still not completely clear what are the right questions to ask; we hope to shed some light on this.
Title: Metric characterizations in the Ribe program
Abstract: A key goal of the Ribe program is to provide metric characterizations of classical linear properties of norms in an attempt to create a nonlinear analogue of the local theory of Banach spaces. In this talk, we shall explore some recent advances in this direction, including the development of new bi-Lipschitz invariants for metric spaces, the refutation of conjectured metric analogues of classical results from the linear theory, and the emergence of new, purely metric phenomena within the Ribe program dictionary. The talk is based on joint works with Manor Mendel and Assaf Naor.
Τίτλος: Η εικασία αθροίσματος-γινομένου για ρητούς με λίγους πρώτους διαιρέτες.
Περίληψη: Μία αρκετά γνωστή εικασία στην συνδυαστική θεωρεία αριθμών είναι η εικασία του αθροίσματος-γινομένου που δηλώνει ότι για κάθε περιττό σύνολο ακέραιων (ή ρητών ή πραγματικών) αριθμών, ο αριθμός αθροισμάτων ή ο αριθμός γινομένων είναι κοντά στο θεωρητικό μέγιστο. Ξέρουμε σχετικά λίγα για αυτό το ερώτημα αλλά η κατάσταση βελτιώνεται αν υποθέσουμε ότι οι ακέραιοι (ή οι ρητοί) έχουν σχετικά λίγους πρώτους διαιρέτες. Στην γενικού ενδιαφέροντος ομιλία θα παρουσιάσουμε μία ιστορική περίληψη και πρόσφατα αποτελέσματα με την Rishika Agrawal και τον Thomas Bloom.