15.12.2025 15:00 Néstor Jara: Dichotomies respect to growth rates and their spectra
Since the works of M. Pinto [2], it has been studied that nonautonomous dynamics
can present different behaviors other than exponential, as polynomial, superexponential,
among others. More recently, C. Silva [4] presented a formalism to expand some classic
results [3] regarding the spectral structure that an equation can present when studied
under the lenses of a given growth rate.
In this talk, based on [1], we investigate the properties of the dichotomy spectrum of
nonautonomous linear systems under general growth behaviors. By introducing compar-
ison criteria we clarify how generalized dichotomies and bounded growth interact. We
also study the evolution of the dichotomy spectrum under these comparisons, revealing
that faster growth rates compress the spectrum, while slower growth rates expand it.
Joint work with:
Claudio A. Gallegos
References
[1] Gallegos, C. A.; Jara, N. The interplay of μ-dichotomy, bounded growth, and spectral
properties via growth rate comparisons (2025). arXiv:2507.21940.
[2] Naulin, R.; Pinto, M. Dichotomies and asymptotic solutions of nonlinear differential systems.
Nonlinear Anal. Theory Methods Appl. 23 (1994), No. 7, 871–882.
[3] Siegmund, S. Dichotomy spectrum for nonautonomous differential equations. J. Dynam. Dif-
ferential Equations 14 (2002), 243–258.
[4] Silva, C. M. Nonuniform μ-dichotomy spectrum and kinematic similarity. J. Differential Equa-
tions, 375, 618-652 (2023).
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17.12.2025 16:30 Tobias Ried (Georgia Tech): From optimal transport to branched microstructures: a journey through elliptic regularity theory
In this talk I will present a purely variational approach to the regularity theory of optimal transportation introduced by Goldman and Otto. The approach closely follows De Giorgi's strategy for the regularity theory of minimal surfaces: at its core lies a Campanato iteration, which allows one to transfer the scaling law of the local transport energy to small scales. In regularity theory, this typically leads to Schauder estimates; but the same idea can also be used to study the local energy scaling of minimizers of non-convex variational problems related to branching phenomena in strongly uniaxial ferromagnets and type-I superconductors in the intermediate state. I will highlight this connection and give a brief overview of further recent developments and point out some other interesting applications.
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Invited by Prof. Phan Thành Nam
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22.12.2025 16:30 Chiara Sabina Bariletto: TBA
TBA
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07.01.2026 14:30 Antrittsvorlesungen: Violetta Weger (TUM) und Murad Alim (TUM): Fakultätskolloquium
Quantum Geometry
Speaker: Murad Alim (TUM)
Abstract:
Quantum theory has not only reshaped our understanding of the physical world; it has also become a powerful source of ideas for modern mathematics. In this talk, I will introduce aspects of the emerging field of quantum geometry, where insights from quantum field theory and string theory interact with symplectic, complex, and algebraic geometry. I will explain how dualities in physical theories often reveal that seemingly different mathematical structures share common underlying principles, leading to deep new results and unexpected bridges between diverse areas. A central example is mirror symmetry, a duality relating symplectic and complex geometry with far-reaching consequences for enumerative geometry, representation theory and number theory.
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12.01.2026 14:15 Thomas Mikosch (University Copenhagen) : Modeling extremal clusters in time series
Real-life financial time series exhibit heavy tails and clusters of extreme values. In this talk we will address models that exhibit these stylized facts. This is the class of regularly varying time series, introduced by Davis and Hsing (1995, AoP) and further developed by Basrak and Segers
(2009, SPA). The marginal distribution of a regularly varying time series has tails of power-law type, and the dynamics caused by an extreme event in this time series is described by the spectral tail process. The perhaps best known financial time series models of this kind are Engle’s (1982) ARCH process, Bollerslev’s (1986) GARCH process and Engle’s and Russell’s (1998) Autoregressive Conditional Duration (ACD) model. The length and magnitude of extremal clusters in such a series can be described by an analog of the autocorrelation function for extreme events: the extremogram. The extremal index is another useful tool for describing expected extremal cluster sizes. Both objects can be expressed in terms of the spectral tail process and allow for statistical estimation. The probabilistic and statistical aspects of regularly varying time series are summarized in the recent monograph by Mikosch and Wintenberger (2024) “Extreme Value Theory for Time Series. Models with Power-Law Tails”. The talk is based on joint work with Olivier Wintenberger (Sorbonne).
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