Renormalization in condensed matter: Fermionic systems – from mathematics to materials.

*Manfred Salmhofer*, University of Heidelberg.

## Resumo

I review the role of renormalization theory in many-fermion systems, both from the point of view of mathematical physics and that of applications to models of correlated electrons in solids. The Wilsonian renormalization group method allows for an unbiased analysis of competing ordering tendencies, such as magnetism and superconductivity in effectively two-dimensional systems. As an example, I will consider ferromagnetism and superconductivity in the Hubbard model at Van Hove filling.

Minimal surfaces in hyperbolic manifolds.

*André Neves*, University of Chicago.

## Resumo

The study of geodesics in negatively curved manifolds is a rich subject which has been at the core of geometry and dynamical systems. Comparatively, much less is known about minimal surfaces on those spaces. I will survey some of the recent progress in that area.

Minimizing epidemic final size through social distancing.

*Pierre-Alexandre Bliman*, Inria and Laboratoire Jacques-Louis Lions, Paris.

## Resumo

How to apply partial or total containment measures during a given finite time interval, in order to minimize the final size of an epidemic - that is the cumulative number of cases infected during its course? We provide here a complete answer to this question for the SIR epidemic model. Existence and uniqueness of an optimal strategy is proved for the infinite-horizon problem corresponding to control on an interval $[0,T]$, $T\gt 0$ (1st problem), and then on any interval of length $T$ (2nd problem). For both problems, the best policy consists in applying the maximal allowed social distancing effort until the end of the interval $[0,T]$ (1st problem), or during a whole interval of length $T$ (2nd problem), starting at a date that is not systematically the closest date and that may be computed by a simple algorithm. These optimal interventions have to begin before the proportion of susceptible individuals crosses the herd immunity level, and lead to limit values of that proportion smaller than this threshold. More precisely, among all policies that stop at a given distance from the threshold, the optimal policies are the ones that realize this task with the minimal containment duration. Numerical results are exposed that provide the best possible performance for a large set of basic reproduction numbers and lockdown durations and intensities.

Details and proofs of the results are available in [BDPV,BD].

This is a joint work with Michel Duprez (Inria), Yannick Privat (Université de Strasbourg) and Nicolas Vauchelet (Université Sorbonne Paris Nord).

[BDPV] Bliman, P.-A., Duprez, M., Privat, Y., and Vauchelet, N. (2020). *Optimal immunity control by social distancing for the SIR epidemic model*. Journal of Optimization Theory and Applications. https://link.springer.com/article/10.1007/s10957-021-01830-1

[BD] Bliman, P. A., and Duprez, M. (2021). *How best can finite-time social distancing reduce epidemic final size?*. Journal of Theoretical Biology 511, 110557. https://www.sciencedirect.com/science/article/pii/S0022519320304124

Homotopy type of equivariant symplectomorphisms of rational ruled surfaces.

*Pranav Chakravarthy*, Hebrew University of Jerusalem.

## Resumo

In this talk, we present results on the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $CP^2$ blown up once, under the presence of Hamiltonian group actions of either $S^1$ or finite cyclic groups. For Hamiltonian circle actions, we prove that the centralizers are homotopy equivalent to either a torus or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions. We can show that the same holds for the centralizers of most finite cyclic groups in the Hamiltonian group. Our results rely on J-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on 4-manifolds, and on the Chen-Wilczynski smooth classification of $\mathbb Z_n$-actions on Hirzebruch surfaces.

Regularity of transition layers in Allen-Cahn equation.

*Kelei Wang*, Wuhan University.

## Resumo

In this talk I will survey the regularity theory for transition layers in singularly perturbed Allen-Cahn equation, from zeroth order regularity to second order one. Some applications of this regularity theory will also be discussed, including De Giorgi conjecture, classification of finite Morse index solutions and construction of minimal hypersurfaces by Allen-Cahn approximation.

A anunciar.

*George Em Karniadakis*, Brown University.

Mathematical structures of non-perturbative topological string theory: from GW to DT invariants.

*Joerg Teschner*, University of Hamburg.

## Resumo

We study the Borel summation of the Gromov-Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson-Thomas invariants of the resolved conifold, having a direct relation to the Riemann-Hilbert problem formulated by T. Bridgeland. There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann-Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by D. Jafferis and G. Moore.

A anunciar.

*Sergey Denisov*, Oslo Metropolitan University.

The geometry of commuting varieties of reductive groups.

*Carlos Florentino*, Faculty of Sciences - University of Lisbon.

## Resumo

Let $R_r(G)$ be the (connected component of the identity of the) variety of commuting $r$-tuples of elements of a complex reductive group $G$. We determine the mixed Hodge structure on the cohomology of the representation variety $R_r(G)$ and of the character variety $R_r(G)/G$, for general $r$ and $G$. We also obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomials, Poincaré polynomials and Euler characteristics of these representation and character varieties. In the character variety case, this gives the counting polynomial over finite fields, and some results also apply to character varieties of nilpotent groups.

This is joint work with S. Lawton and J. Silva (arXiv:2110.07060).

Annealed Flow Transport Monte Carlo.

*Michael Arbel*, Gatsby Computational Neuroscience Unit, University College London.

## Resumo

Annealed Importance Sampling (AIS) and its Sequential Monte Carlo (SMC) extensions are state-of-the-art methods for estimating normalizing constants of probability distributions. We propose here a novel Monte Carlo algorithm, Annealed Flow Transport (AFT), that builds upon AIS and SMC and combines them with normalizing flows (NF) for improved performance. This method transports a set of particles using not only importance sampling (IS), Markov chain Monte Carlo (MCMC) and resampling steps - as in SMC, but also relies on NF which are learned sequentially to push particles towards the successive annealed targets. We provide limit theorems for the resulting Monte Carlo estimates of the normalizing constant and expectations with respect to the target distribution. Additionally, we show that a continuous-time scaling limit of the population version of AFT is given by a Feynman--Kac measure which simplifies to the law of a controlled diffusion for expressive NF. We demonstrate experimentally the benefits and limitations of our methodology on a variety of applications.

Geometric complexity in quantum matter: intrinsic sign problems in topological phases.

*Omri Golan*, QEDMA Quantum Computing, Israel.

## Resumo

The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers. Motivated by long standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. I will describe results establishing the existence, and the geometric origin, of intrinsic sign problems in a broad class of topological phases in 2+1 dimensions. Within this class, these results exclude the possibility of 'stoquastic' Hamiltonians for bosons, and of sign-problem-free determinantal Monte Carlo algorithms for fermions. The talk is based on Phys. Rev. Research 2, 043032 and 033515.

Revisiting and extending Poisson-Nijenhuis structures.

*Henrique Bursztyn*, Instituto Nacional de Matemática Pura e Aplicada.

A anunciar.

*John Nolan*, American University CAS- Math and Statistics.

A anunciar.

*John Nolan*, American University CAS- Math and Statistics.

Holographic Complexity and de Sitter Space.

*Shira Chapman*, Ben Gurion University of the Negev.

## Resumo

We compute the length of spacelike geodesics anchored at opposite sides of certain double-sided flow geometries in two dimensions. These geometries are asymptotically anti-de Sitter but they admit either a de Sitter or a black hole event horizon in the interior. While in the geometries with black hole horizons, the geodesic length always exhibit linear growth at late times, in the flow geometries with de Sitter horizons, geodesics with finite length only exist for short times of the order of the inverse temperature and they do not exhibit linear growth. We comment on the implications of these results towards understanding the holographic proposal for quantum complexity and the holographic nature of the de Sitter horizon.

A anunciar.

*Dung Xuan Nguyen*, Brown University.

A anunciar.

*Ely Kerman*, University of Illinois Urbana-Champaign.

A anunciar.

*Karol Życzkowski*, Jagiellonian University.

Equivariant machine learning structure like classical physics.

*Soledad Villar*, Mathematical Institute for Data Science at Johns Hopkins University.

## Resumo

There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincare groups, at any dimensionality $d$. The key observation is that nonlinear $O(d)$-equivariant (and related-group-equivariant) functions can be expressed in terms of a lightweight collection of scalars–scalar products and scalar contractions of the scalar, vector, and tensor inputs. These results demonstrate theoretically that gauge-invariant deep learning models for classical physics with good scaling for large problems are feasible right now.

Harmonic analysis of $2d$ CFT partition functions.

*Nathan Benjamin*, Princeton University.

## Resumo

We apply the theory of harmonic analysis on the fundamental domain of $SL(2,\mathbb{Z})$ to partition functions of two-dimensional conformal field theories. We decompose the partition function of $c$ free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space $H/SL(2,\mathbb{Z})$, and of target space moduli space $O(c, c; \mathbb{Z})\backslash O(c, c; \mathbb{R})/O(c) × O(c)$. This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to $AdS_3$ gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies.

Herding Cats: A Chaotic Field Theory.

*Predrag Cvitanović*, Georgia Tech.

## Resumo

Suppose you find yourself face-to-face with Young-Mills or Navier-Stokes or a nonlinear PDE or a funky metamaterial or a cloudy day. And you ask yourself, is this thing "turbulent" What does that even mean?

If you were had a serious course on 'chaos', as Professor Ribeiro had, you must have learned about the coin toss (Bernoulli map). I'll walk you through this basic example of deterministic chaos, than through the 'kicked rotor', a neat physical system that is chaotic, and then put infinity of these together to explain what `chaos' or `turbulence' looks like in the spacetime.

What emerges is a spacetime which is very much like a big spring mattress that obeys the familiar continuum versions of a harmonic oscillator, the Helmholtz and Poisson equations, but instead of being "springy", this metamaterial has an unstable rotor at every lattice site, that gives, rather than pushes back. We call this simplest of all chaotic field theories the `spatiotemporal cat'. There is a QM^3 version, ask Boris Gutkin or Tomaž Prosen to tell

you about it.

In the spatiotemporal formulation of turbulence there is no evolution in time, there are only a repertoires of admissible spatiotemporal patterns. In other words: throw away your integrators, and look for guidance in clouds' repeating patterns.

That's `turbulence'. And if you don't know, now you know.

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No actual cats, graduate or undergraduate, have shown interest in, or were harmed during this research.

A anunciar.

*Ciprian Manolescu*, Stanford University.

A anunciar.

*Eva Miranda*, Universitat Politècnica de Catalunya.

Climate Session.

*Jessica Silva Lomba*, Nova School of Business and Economics.

Climate Session.

*Jessica Silva Lomba*, Nova School of Business and Economics.

A anunciar.

*Cristiano Ciuti*, Université de Paris.

A anunciar.

*Thomas Opitz*, INRAE - Biostatistique et Processus Spatiaux.

A anunciar.

*Thomas Opitz*, INRAE - Biostatistique et Processus Spatiaux.

Instituto Superior Técnico
Av. Rovisco Pais,
Lisboa,
PT