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11:00

12:00

MA A1 10

Evolution of triangulations: Hausdorff and spectral dimensions

How complex networks formed by triangulations and higher-dimensional simplicial complexes can represent closed evolving manifolds [1]. In particular, for triangulations, the set of possible transformations of these networks is restricted by the condition that at each step, all the faces must be triangles, which is the key constraint in this theory. Stochastic application of these operations leads to random networks with different architectures. I will show how geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties can be described. This characterisation includes the Hausdorff and spectral dimensions of the resulting networks, their degree distributions, and various structural correlations. The results reveal a rich zoo of architectures and geometries of these networks, some of which appear to be small worlds while others are finite-dimensional with a wide spectrum of Hausdorff and spectral dimensions. 

[1] D. C. da Silva, G. Bianconi, R. A. da Costa, S. N. Dorogovtsev, and J. F. F. Mendes, Complex network view of evolving manifolds, Phys. Rev. E 97, 032316 (2018).

By: José Fernando Mendes (University of Aveiro)

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14:00

15:00

PH H3 33

Reductions of non-lc-ideals and non-$F$-pure ideals assuming weak ordinarity

Abstract: It has been known for several decades that there are close connections between certain classes of singularities in the Minimal Model Program over $\mathbb{C}$ and so-called $F$-singularities which are defined in positive characteristic via Frobenius. There is a more refined conjecture which relates multiplier ideal filtrations and test ideal filtrations. The triviality of certain parts of this filtration may be used to define some of the singularities mentioned previously. It is known that this conjecture is equivalent to the so-called weak ordinarity conjecture from arithmetic geometry (which roughly speaking asserts that if $X$ is a smooth projective varietiy of dimension $d$ defined over $\Spec \mathbb{Z}$ then its reductions mod $p$ admit bijective Frobenius action on $H^{d-1}(X_p,\mathcal{O}_{X_p})$ for inifinitely many $p$) by work of Srinivas, Mustata, Bhatt, Schwede, Takagi. I will survey these results and time permitting talk about a similar conjectural relation between maximal non-lc ideal filtrations and non-$F$-pure ideal filtrations which in some cases is also equivalent to weak ordinarity.

By: Axel Stabler (Johannes Gutenberg Universität Mainz)

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