Least number of n-periodic points of self-maps of $$PSU(2)\times PSU(2)$$

Let $$f:M\rightarrow M$$ f : M → M be a self-map of a compact manifold and $$n\in {\mathbb {N}}$$ n ∈ N . In general, the least number of n-periodic points in the smooth homotopy class of f may be much bigger than in the continuous homotopy class. For a class of spaces, including compact Lie groups, a necessary condition for the equality of the above two numbers, for each iteration $$f^n$$ f n , appears. Here we give the explicit form of the graph of orbits of Reidemeister classes $$\mathcal {GOR}(f^*)$$ GOR ( f ∗ ) for self-maps of projective unitary group PSU(2) and of $$PSU(2)\times PSU(2)$$ P S U ( 2 ) × P S U ( 2 ) satisfying the necessary condition. The structure of the graphs implies that for self-maps of the above spaces the necessary condition is also sufficient for the smooth minimal realization of n-periodic points for all iterations.

Bodily, emotional, and public sphere at the time of COVID-19. An investigation on concrete and abstract concepts

The outbreak of Covid-19 pandemics has dramatically affected people’s lives. Among newly established practices, it has likely enriched our conceptual representations with new components. We tested this asking Italian participants during the first lockdown to rate a set of diverse words on several crucial dimensions. We found concepts are organized along a main axis opposing internal and external grounding, with fine-grained distinctions within the two categories underlining the role of emotions. We also show through a comparison with existing data that Covid-19 impacted the organization of conceptual representations. For instance, subclasses of abstract concepts that are usually distinct converge into a unitary group, characterized by emotions and internal grounding. Additionally, we found institutional and Covid-19 related concepts, for which participants felt more the need for others to understand the meaning, clustered together. Our results show that the spread of Covid-19 has simultaneously changed our lives and shaped our conceptual representations.

Large N behaviour of the two-dimensional Yang–Mills partition function

We compute the large N limit of the partition function of the Euclidean Yang–Mills measure on orientable compact surfaces with genus $g\geqslant 1$ and non-orientable compact surfaces with genus $g\geqslant 2$ , with structure group the unitary group ${\mathrm U}(N)$ or special unitary group ${\mathrm{SU}}(N)$ . Our proofs are based on asymptotic representation theory: more specifically, we control the dimension and Casimir number of irreducible representations of ${\mathrm U}(N)$ and ${\mathrm{SU}}(N)$ when N tends to infinity. Our main technical tool, involving ‘almost flat’ Young diagram, makes rigorous the arguments used by Gross and Taylor (1993, Nuclear Phys. B400(1–3) 181–208) in the setting of QCD, and in some cases, we recover formulae given by Douglas (1995, Quantum Field Theory and String Theory (Cargèse, 1993), Vol. 328 of NATO Advanced Science Institutes Series B: Physics, Plenum, New York, pp. 119–135) and Rusakov (1993, Phys. Lett. B303(1) 95–98).

Bootstrapping the $$ \mathcal{N} $$ = 1 Wess-Zumino models in three dimensions

Using numerical bootstrap method, we determine the critical exponents of the minimal three-dimensional $$ \mathcal{N} $$ N = 1 Wess-Zumino models with cubic superpotetential $$ \mathcal{W}\sim {d}_{ijk}{\Phi}^i{\Phi}^j{\Phi}^k $$ W ∼ d ijk Φ i Φ j Φ k . The tensor dijk is taken to be the invariant tensor of either permutation group SN, special unitary group SU(N), or a series of groups called F4 family of Lie groups. Due to the equation of motion, at the Wess-Zumino fixed point, the operator dijkΦjΦk is a (super)descendant of Φi. We observe such super-multiplet recombination in numerical bootstrap, which allows us to determine the scaling dimension of the super-field ∆Φ.

Base change for ramified unitary groups: The strongly ramified case

We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.

ON A CERTAIN LOCAL IDENTITY FOR LAPID–MAO'S CONJECTURE AND FORMAL DEGREE CONJECTURE : EVEN UNITARY GROUP CASE

Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.