Description: Details about the plenary lecture will be announced soon.
Description: Certain mathematical objects have achieved Internet fame, among them the M¨obius strip, the Klein bottle, and the fourth dimension. In this age of short attention spans, brief descriptions are repeated everywhere, but their frequency is often inversely proportional to their accuracy. For instance, we may read that the Klein bottle has no inside or outside and can only exist in the fourth dimension—wrong on both counts. This talk will explore the precise meanings of “M¨obius strip” and “Klein bottle,” how these objects were discovered, and some of their lesser-known representations..
Description: Since Russell (1834), mathematicians have tried to understand why solitons, breathers and kinks are stable, and travel unchanged while everything around them disperses. In this talk I will first review what we can rigorously prove about this long-time behavior, including asymptotic stability of kinks in scalar field theories, and nonexistence of breathers. In a second part I will review how physics-informed networks can nowadays ”learn” solitons and their behavior, and provide new insights on unsolved questions. I will show how the classical energy estimates also yield error bounds on the approximation procedure and how, in return, networks become exploratory laboratories for the analysis and discovery of Strichartz extremizers..
Description: After a full description of the (1,1)-Aeppli cohomology of a complex manifold that admits a transitive compact Lie group of biholomorphisms, called C-spaces, we give three geometric applications:
1. Classification of pluriclosed C-spaces: products of flag manifolds and compact Lie groups.
2. Explicit description of all pluriclosed left-invariant metrics on a compact Lie group.
3. Global stability of Bismut flat metrics under the pluriclosed flow on compact Lie groups.
Description: Many equations arising in mathematics and the sciences have a remarkable smoothing effect. The heat equation provides a striking example: even when starting from very irregular initial data, its solutions become smooth instantaneously. Classical regularity theory seeks to explain this phenomenon for broad classes of elliptic and parabolic equations. In many important problems, however, the regularizing mechanism is only partially present. It may operate only in certain regions or above a distinguished scale, as in problems arising from homogenization and numerical schemes.
In this talk, I will revisit some results from the classical theory and present recent work on multiscale Schauder estimates. Applications include higher-order regularity for nonlocal equations with rough kernels, further time regularity for parabolic equations, and problems whose elliptic structure degenerates below a scale-dependent threshold.
Description: Methods and concepts of algebraic geometry, particularly real and computational algebraic geometry, have been used in numerous applied fields. In this talk, I will review their applications in molecular biology, where the goal is to analyze standard systems biology models to predict dynamic behavior in regions of parameter space based on the reaction network structure, without the need for simulations. I will also present two recent results obtained with different collaborators..
Description: In this talk I will present the motivation to study the Equivariant K-theory of Magnetic Crystallographic groups in order to understand some topological phases of matter in Magnetic Crystals.The I will present some results on the Magnetic Equivariant K-theory groups and an application to certain magnetic symmetry in Magnetic Crystals. The subject of the talk is in collaboration with Higinio Serrano, Miguel Xicotencalt and Rafael Gonzalez.
Description: Let f be a pseudo-Anosov homeomorphism of a closed surface and let g be a surface diffeomorphism isotopic to f. The pseudo-Anosov representative minimizes topological entropy in its isotopy class, so htop(g) ≥ htop(f). In the rigidity case in which the two entropies coincide, Handel’s work provides a semiconjugacy from g to f: there is a continuous surjection h, homotopic to the identity, such that
h ∘ g = f ∘ h
Thus, equality of entropy forces the dynamics of g to admit the pseudo-Anosov map as a global topological factor. The remaining freedom is encoded in the fibers of h, that is, in the sets that the semiconjugacy collapses to individual points. In this context, Handel suggested that these fibers need not be connected. I will discuss results that provide evidence that this expectation may be incorrect, at least under natural dynamical assumptions. More precisely, in two different settings—Axiom A diffeomorphisms and area-preserving diffeomorphisms- we show that entropy equality forces the semiconjugacy to be monotone, meaning that all its fibers are connected. I will explain the geometric ideas behind these results and describe how entropy rigidity, hyperbolicity, and invariant continua interact to control the topology of the fibers.
Description: Details about the plenary lecture will be announced soon.
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