Mathematical Physics (MP)
http://hdl.handle.net/20.500.11824/16
2021-08-05T21:48:15ZClassification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units
http://hdl.handle.net/20.500.11824/1314
Classification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units
Pełka, T.; Raźny, P.
We give a corrected statement of (Gurjar and Miyanishi 1988, Theorem 2), which classifies smooth affine surfaces of Kodaira dimension zero, whose coordinate ring is factorial and has trivial units. Denote the class of such surfaces by $\mathcal{S}_0$. An infinite series of surfaces in $\mathcal{S}_0$, not listed in loc. cit., was recently obtained by Freudenburg, Kojima and Nagamine (2019) as affine modifications of the plane. We complete their list to a series containing arbitrarily high-dimensional families of pairwise nonisomorphic surfaces in $\mathcal{S}_0$. Moreover, we classify them up to a diffeomorphism, showing that each occurs as an interior of a 4-manifold whose boundary is an exceptional surgery on a 2-bridge knot. In particular, we show that $\mathcal{S}_0$ contains countably many pairwise nonhomeomorphic surfaces.
2021-07-31T00:00:00ZSome contributions to the theory of singularities and their characteristic classes
http://hdl.handle.net/20.500.11824/1294
Some contributions to the theory of singularities and their characteristic classes
Pallarés Torres, I.
In this Ph.D. thesis, we give some contributions to the theory of singularities, as well as to the theory of characteristic classes of singular spaces. The first part of this thesis is devoted to the theory of singularities of mappings. We obtain formulas for an important analytical invariant, the image Milnor number of a map-germ. The first contribution is a version of the Lê-Greuel formula for the image Milnor number for corank 1 map-germs, which gives a recursive method to compute it. This work is in collaboration with my Ph.D. supervisor J. J. Nuño. In the second work of this thesis, we obtain two formulas for the image Milnor number for weighted-homogeneous map-germs for dimensions four and five. These formulas are obtained by combining the theory of characteristic classes of singular spaces with a recursive method through examples of singularities. This work is in collaboration with Prof. G. Peñafort. The last part which composes the main work of this Ph.D. thesis is the proof for projective varieties of the Brasselet-Schürmann-Yokura conjecture. This conjecture is within the theory of characteristic classes of singular spaces. Characteristic classes of singular varieties are homology classes generalizing the classical cohomological characteristic classes of manifolds. The conjecture states that two different characteristic classes coincide for compact complex algebraic varieties that are rational homology manifolds. This work is in collaboration with my Ph.D. supervisor J. Fernández de Bobadilla. This conjecture is the characteristic class version for rational homology manifolds of the famous Hodge Index Theorem which computes the signature of a compact complex manifold in terms of Hodge numbers.
2021-06-02T00:00:00ZStochastic resetting by a random amplitude
http://hdl.handle.net/20.500.11824/1293
Stochastic resetting by a random amplitude
Dahlenburg, M.; Chechkin, A. V.; Schumer, R.; Metzler, R.
Stochastic resetting, a diffusive process whose amplitude is reset to the origin at random times, is a vividly studied strategy to optimize encounter dynamics, e.g., in chemical reactions. Here we generalize the resetting step by introducing a random resetting amplitude such that the diffusing particle may be only partially reset towards the trajectory origin or even overshoot the origin in a resetting step. We introduce different scenarios for the random-amplitude stochastic resetting process and discuss the resulting dynamics. Direct applications are geophysical layering (stratigraphy) and population dynamics or financial markets, as well as generic search processes.
2021-05-18T00:00:00ZMacroscopic Dynamics of the Strong-Coupling BCS-Hubbard Model
http://hdl.handle.net/20.500.11824/1292
Macroscopic Dynamics of the Strong-Coupling BCS-Hubbard Model
Bru, J.-B.; de Siqueira Pedra, W.
The aim of the current paper is to illustrate, in a simple example, our
recent, very general, rigorous results
on the dynamical properties of fermions and quantum-spin systems with
long-range, or mean-field, interactions, in infinite volume. We consider
here the strong-coupling BCS-Hubbard model, because this example is very pedagogical and,
at the same time, physically relevant for it highlights the impact of the
(screened) Coulomb repulsion on ($s$-wave) superconductivity.
2020-01-01T00:00:00Z