Matrix models for classical groups and Toeplitz±\pm Hankel minors with applications to Chern-Simons theory and fermionic models

We study matrix integration over the classical Lie groups $U(N),Sp(2N),O(2N)$ and $O(2N+1)$, using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz$\pm$Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large $N$ the partition functions, Wilson loops and Hopf links of Chern-Simons theory on $S^{3}$ with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern-Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom.Comment: 32 pages, v2: Several improvements, including a Conclusions and Outlook section, added. 36 page

Strongly nonlinear thermovoltage and heat dissipation in interacting quantum dots

We investigate the nonlinear regime of charge and energy transport through Coulomb-blockaded quantum dots. We discuss crossed effects that arise when electrons move in response to thermal gradients (Seebeck effect) or energy flows in reaction to voltage differences (Peltier effect). We find that the differential thermoelectric conductance shows a characteristic Coulomb butterfly structure due to charging effects. Importantly, we show that experimentally observed thermovoltage zeros are caused by the activation of Coulomb resonances at large thermal shifts. Furthermore, the power dissipation asymmetry between the two attached electrodes can be manipulated with the applied voltage, which has implications for the efficient design of nanoscale coolers.Comment: 6 pages, 4 figure

Forcing consequences of PFA together with the continuum large

We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a large continuum.Comment: 35 page

Toeplitz minors and specializations of skew Schur polynomials

We express minors of Toeplitz matrices of finite and large dimension in terms of symmetric functions. Comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.Comment: v2: Added new results on specializations of skew Schur polynomials, abstract and title modified accordingly and references added; v3: final, published version; 18 page

From graphs to tensegrity structures: Geometric and symbolic approaches

A form-finding problem for tensegrity structures is studied; given an abstract graph, we show an algorithm to provide a necessary condition for it to be the underlying graph of a tensegrity in $\mathbb{R}^d$ (typically $d=2,3$) with vertices in general position. Furthermore, for a certain class of graphs our algorithm allows to obtain necessary and sufficient conditions on the relative position of the vertices in order to underlie a tensegrity, for what we propose both a geometric and a symbolic approach.Comment: 17 pages, 8 figures; final versio

Few new reals

We introduce a new method for building models of CH, together with $\Pi_2$ statements over $H(\omega_2)$, by forcing over a model of CH. Unlike similar constructions in the literature, our construction adds new reals, but only $\aleph_1$-many of them. Using this approach, we prove that a very strong form of the negation of Club Guessing at $\omega_1$ known as Measuring is consistent together with CH, thereby answering a well-known question of Moore. The construction works over any model of ZFC + CH and can be described as a finite support forcing construction with finite systems of countable models with markers as side conditions and with strong symmetry constraints on both side conditions and working parts