Weekly Bulletin

<div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>The computation of the homotopy groups of the spheres has always been one of the fundamental and hardest problems in algebraic topology. The Freudenthal suspension theorem and the subsequent introduction of the stable category of spectra have quickly channeled much of the effort into computing the stable stems of the sphere spectrum. Between 1998 and 1999, Fabien Morel and Vladimir Voevodsky developed a homotopy theory for schemes, called $A^1$ or motivic homotopy theory. In this new algebro-geometric setting, the role of the classical sphere spectrum $1_{top}$ is replaced by a motivic sphere spectrum, $1_S$, encoding both the topological information and the algebraic properties of the chosen base scheme $S$. The computation of the stable homotopy groups of such a motivic spectrum is now a core problem in this new theory. Various authors have pursued this goal over the years, when $S$ is the spectrum of a field $k$.</p> <p>In this talk, we will explain different approaches to compute the stable homotopy groups of the motivic sphere spectrum over suitable Dedekind schemes $S$. We will determine the first stems of the sphere in terms of key algebraic-geometric invariants, such as the Milnor K-groups $K^M_*(S)$, the Picard group $Pic(S)$, and the Witt ring of quadratic forms $W(S)$. Specific attention is dedicated to the analysis of the class of rings of integers of number fields $O_{F,S}$.</p> </div> </div> </div>

Various conformally invariant energies for curves arise in geometry, physics and probability theory such as the Loewner energy for Jordan curves or the Möbius energy for knots. Even though these Lagrangians are defined through completely different tools, they are known to share similar analytical properties. They are nonlocal Lagrangians and are well-defined for the same class of Jordan curves, called Weil-Petersson curves. I will explain how these energies can be related to the notion of Grunsky operator and thus, compared together. This is a joint work with Yilin Wang.

<p>Diffeology was introduced by Souriau in 1980 in order to define a category of "differentiable groups" sufficiently large to contain all diffeomorphism groups while retaining most of the elementary properties of Lie groups. It turns out that, unlike manifolds, diffeological spaces have excellent categorical properties. In this talk, I will define the category of diffeological spaces and present two examples that first showed diffeology to be a useful notion: the diffeological group of diffeomorphisms of a manifold and the irrational torus.</p>

Ozsváth and Szabó developed an algebraic method for computing knot Floer homology: cut a knot diagram into simple pieces, assign to each piece a bimodule over certain explicit algebras, and tensor these together. Their method underlies a remarkably efficient computer program, and applying it to only part of a diagram yields invariants of tangles. I will explain how, for tangles with four endpoints, this invariant determines the Heegaard Floer tangle invariant HFT. HFT takes the form of a collection of immersed curves on the four-punctured sphere, or equivalently, a module over a certain quiver algebra. Combined with Ozsváth and Szabó’s program, this gives a practical way to compute HFT. It also points toward a potential "minus" version of HFT. This is joint work in progress with Chen Zhang.

<p>We study principally polarized superspecial abelian surfaces over finite fields through their refined Humbert invariants. Primitive representations by these invariants characterize the existence of (N, N )-isogenies from Jacobians to products of elliptic curves, leading to a study of splitting levels and minimum walks to the product locus in the corresponding isogeny graphs. Primitive representations also give information about embeddings of real quadratic orders into endomorphism rings, and hence on real multiplication. This is joint work with Gaurish Korpal.</p>

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Mixed finite element methods for solid mechanics approximate the stress tensor directly. One most decide how to impose the symmetry of this tensor. Exactly symmetric tensor elements are notoriously hard to construct; a popular alternative is therefore to impose symmetry weakly, with a Lagrange multiplier that approximates the rotation. We show that this choice comes at a price on thin domains. As beams, plates and shells become thin, the stress error of weakly symmetric discretisations can grow with the aspect ratio of the domain; strongly symmetric discretisations meanwhile remain unaffected. We illustrate this numerically on beams and plates, where weakly symmetric mixed discretisations of linear elasticity can be several times too stiff. A similar phenomenon arises in Cosserat media, causing failure in weakly symmetric discretisations both at a high coupling modulus, and on the same such thin domains. Both results are placed within a common framework of BGG and twisted complexes, giving a unifying condition under which such a discretisation remains robust. This is joint work with Charles Parker, Umberto Zerbinati and Kaibo Hu.

<p>Due to the insecurity of McEliece cryptosystems instantiated with Generalized Reed-Solomon codes, there have been several proposals of McEliece type systems that replace the permutation matrix by a matrix M with larger row and column weight. In many of them, the secret key is still a GRS code. Two proposals using M with row and column weight between 1 and 1 + R (where R is the rate of the code) have been broken previously. The case of weight 2 and higher has been left open open. We provide distinguishers for the public codes appearing in the cryptosystems in the high rate regime. In addition, we give a framework to turn a good enough distinguisher into a key recovery attack. In the case where the matrix M has row and column weight 2, we can successfully attack the scheme in the high rate regime using a cube code distinguisher.</p> <p> </p> <p>This is joint work with Julia Lieb and Michael Schaller. </p> <p> </p>

Let G(n,p) be the binomial random graph of density p and let X_H be the number of (unlabelled) copies of a fixed graph H in G(n,p). The study of these random variables goes back to the early works of Erdős and Rényi, and their typical behaviour is now well understood. Let G(n,p) be the binomial random graph of density p, and let X_H be the number of (unlabelled) copies of a fixed graph H. While the typical behaviour of X_H is well understood, its large deviations, particularly the upper tail probability, have proved much more challenging. Even for triangle counts, precise logarithmic asymptotics for the upper tail probability were obtained only relatively recently, in breakthrough work of Harel, Mousset, and Samotij. Together with subsequent work by Basak and Basu, this essentially settled the upper tail problem for connected regular graphs. In this talk, we will discuss the difficulties that arise for irregular graphs and present new results extending the work of Basak and Karmakar to sparser edge probabilities. Our results account for an additional entropy contribution and cover an essentially optimal range for certain classes of irregular graphs. The talk is based on joint work with Matan Harel, Frank Mousset, and Wojciech Samotij.

The plane Cremona group Bir(P^2) is an interesting object that has been studied for over 150 years, and offers a rich structure that can be studied from different perspectives and point of views. We focus on a dynamical perspective: this point of view allows to study and classify elements in Bir(P^2) up to birational conjugacy. In this talk, we give an overview of the construction of the group, then we define and study various length functions and dynamical invariants that allow us to give a classification of the Cremona maps.

<p style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;">The Bardeen–Cooper–Schrieffer (BCS) theory provides a remarkably successful description of superconductivity in terms of Cooper pairs. A fundamental mathematical question is whether the BCS approximation can be justified directly from the underlying many-body Hamiltonian.</p> <p style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;">In this talk, I will discuss recent joint work with Riccardo Panza in which we prove that the ground-state energy of the reduced BCS Hamiltonian agrees with the minimum of the BCS functional up to an error that remains bounded in the thermodynamic limit. I will explain the main ideas of the proof, which rely on a completion-of-squares argument and a localization method for almost-commuting operators.</p> <p style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;">Finally, I will discuss how one can go beyond the BCS approximation by applying an additional RPA-type unitary transformation to account for quantum fluctuations around the BCS state.</p>

In this talk, we revisit the resolution of continuous-time principal–agent problems with drift and volatility control, originally addressed by Cvitanić, Possamaï, and Touzi (2018) through dynamic programming and second-order backward stochastic differential equations (2BSDEs), and develop new results in this framework. We begin by introducing an alternative problem in which the principal is allowed to directly control the quadratic variation of the output process. On the one hand, the resolution of this contractible-volatility problem follows the classical methodology of Sannikov (2008), thus relying on standard (first-order) BSDEs only. On the other hand, we introduce a new form of contracts allowing the principal to achieve her contractible-volatility value, thereby ensuring both the optimality of this contract form and the equivalence between the original and the alternative problems. At the same time, this alternative approach reveals that the optimality of the contract form introduced by Cvitanić, Possamaï, and Touzi (2018) implicitly relies on an additional duality assumption, which was not identified before. This observation motivates the construction of new families of contracts that remain optimal even when the duality assumption fails. Altogether, this line of work both simplifies and strengthens the existing theory of continuous-time principal–agent problems with volatility control and opens new directions for further extensions and applications in economics and finance. Talk based on joint works with Alessandro Chiusolo, Dylan Possamaï, and Nizar Touzi.

Using the theory of rigid meromorphic cocycles for orthogonal groups, we give a new construction of Darmon and Vonk's p-adic singular moduli for real quadratic number fields. These p-adic numbers are conjectured to be algebraic and satisfy prime factorizations similar to the factorizations of differences of singular moduli. Mysteriously, they are only defined up to multiplication by units of certain orders in real quadratic number fields. Our new approach sheds some light on this ambiguity up to units. Moreover, it immediately implies the antisymmetry conjecture of Darmon and Vonk. This is joint work with Mike Daas, Håvard Damm-Johnsen and Sören Sprehe.

Howmany binary strings of length n can one pick so that every two differ in at least δn coordinates? This packing question in the Hamming cube is among the oldest in coding theory and combinatorics, and its asymptotic answer is still unknown. The best lower bound (Gilbert–Varshamov, 1950s) comes from greedy packing, and the longstanding best upper bound (McEliece–Rodemich–Rumsey–Welch, 1977) from Delsarte's linear program. The former remains unimproved, while the latter stood for nearly fifty years. We give a new, unified route to upper bounds by bridging two classical viewpoints on coding: Hamming's, which concerns worst-case errors, and Shannon's, which bounds communication rates over noisy channels by their capacity. The key is that minimum distance is oblivious to the channel: a code of distance δn can be decoded with constant success probability on any suitable channel whose single-bit error rate is below δ. This bounds the code's rate by the channel's capacity. We get to choose the channel, and the twist is that its outputs may even be quantum states. In this framework, classical channels fall short of the 1977 MRRW bounds. A channel with pure-state qubit outputs reproduces the first MRRW bound, and slightly mixing those states gives a strictly better bound for every 0 < δ < ½. A related construction recovers and improves the second MRRW bound as well. In the end, a longstanding combinatorial problem about binary strings comes down to entropy calculations for one or two qubits. No background in quantum information or coding theory will be assumed. Joint work with Omar Alrabiah (arXiv:2608.09347).

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