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We prove some cases of the Zilber-Pink conjecture on unlikely intersections in Shimura varieties. Firstly, we prove that the Zilber-Pink conjecture holds for intersections between a curve and the union of the Hecke translates of a fixed…
Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate…
This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions.…
A method how to construct Boolean-valued models of some fragments of arithmetic was developed in Krajicek (2011), with the intended applications in bounded arithmetic and proof complexity. Such a model is formed by a family of random…
While intersections of convex sets are convex, their unions have rather complicated behavior. Some natural contexts where they appear include duality arguments involving boundaries of convex sets and valuations, which have an Euler…
Lattices induced by coverings arise naturally in matroid theory and combinatorial optimization, providing a structured framework for analyzing relationships between independent sets and closures. In this paper, we explore the structural…
We address the striking coexistence of localized waves (`pulses') of different lengths which was observed in recent experiments and full numerical simulations of binary-mixture convection. Using a set of extended Ginzburg-Landau equations,…
We prove results concerning the behavior of Hodge ideals under restriction to hypersurfaces or fibers of morphisms, and addition. The main tool is the description of restriction functors for mixed Hodge modules by means of the…
A common objection to the definition of intuitionistic implication in the Proof Interpretation is that it is impredicative. I discuss the history of that objection, argue that in Brouwer's writings predicativity of implication is ensured…
We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial…
It has become increasingly common nowadays to collect observations of feature and response pairs from different environments. As a consequence, one has to apply learned predictors to data with a different distribution due to distribution…
We isolate two combinatorial properties, each expressible by a $\Pi_2$-sentence over the structure $(H(\omega_3),\in,\omega_1,\omega_2,\text{NS}_{\omega_2})$, such that each property is consistent with CH, and their conjunction together…
For a class of competition-diffusion nonlinear systems involving the square root of the Laplacian, including the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies Holder boundedness for every exponent less than…
I prove preservation theorems for countable support iteration of proper forcing concerning certain classes of capacities and submeasures. New examples of forcing notions and connections with measure theory are included.
In this paper we discuss what kind of constrains combinatorial covering properties of Menger, Scheepers, and Hurewicz impose on remainders of topological groups. For instance, we show that such a remainder is Hurewicz if and only it is…
In this paper, we present the Cantor Intersection Theorem and a formulation of Baire Theorem in complete PM spaces. In addition, the Heine-Borel property for PM spaces is considered in detail.
We use categorical method and birational geometry to study moduli spaces of quiver representations. From certain "representable" functor, we construct a birational transformation from the moduli space of representations of one quiver to…
This work addresses the problem of conducting valid inference for additive and linear mixed models after model selection. One possible solution to overcome overconfident inference results after model selection is selective inference, which…
In this paper, we introduced $\alpha$-Hurewicz $\&$ $\theta$-Hurewicz properties in a topological space $X$ and investigated their relationship with other selective covering properties. We have shown that for an extremally disconnected…
A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a…