In this talk we describe $T_{comp}$-equivariant topological K-ring of $T$-cellular varieties. We apply our results to obtain the description of the equivariant topological K-ring of cellular toric varieties and hence the description of the $G_{comp}\times G_{comp}$-equivariant $K$-ring of cellular toroidal embeddings $\bar{G}$ of a complex reductive algebraic group $G$.

We argue that quantum statistics can serve as a thermodynamic resource for heat to work conversion. To demonstrate this point we exploit the framework of Haldane exclusion statistics which proves a possibility of interpolating between fermions and bosons in a smooth fashion. We will show that such an exclusion statistics intrinsically breaks particle–hole symmetry, even in absence of interactions or band asymmetry, leading to enhanced thermoelectric response, strong violations of the Wiedemann–Franz law, and an increased thermoelectric figure of merit. Building on this mechanism, we will demonstrate how thermoelectric heat engines based on working mediums which follow fractional-exclusion statistics in general, described within a nonlinear Landauer framework, outperforms conventional fermionic systems in power generation, efficiency–power trade-offs, and refrigeration efficiency. The resulting thermodynamic advantages extend all the way to bosonic working media and suggest that quantum statistics itself can be harnessed to drive thermoelectric heat engines. Finally, we will discuss thermally driven magnon transport as a realistic route to realizing these ideas in spin caloritronics. Our results establish quantum statistics as a resource for thermoelectric energy conversion, heat engines, and refrigeration.

In this talk, we investigate the degree and cardinality of Grobner bases under various orderings. We identify a family of polynomials and a criterion for monomial orderings such that the reduced Grobner basis is double-exponential in both degree and cardinality. We also show that this criterion is satisfied by standard orderings such as lexicographic, degree lexicographic, and weighted orderings.