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How does the shape of a body determine its capacity? Classical results say that balls _minimize_ the Newton capacity among bodies of given volume, but _maximize_ it among bodies of given diameter. While the lower bound on capacity in terms of volume is a cornerstone of potential theory, the upper bound in terms of diameter (proved by Szego in 1930) was largely relegated to a footnote. Recently isodiametric shape optimization for Riesz capacities has become interesting in the context of aggregation problems with pair interactions of attractive-repulsive type. In this talk I will discuss situations where the capacity-maximizer is not a ball. When does symmetry-breaking occur, and what are good candidates for non-radial maximizers.

Abstract: I will report on recent progress in joint works with Pax Kivimae (University of Colorado) on the Elastic Manifold, a model of disorder coupled with elastic interactions, introduced by Marc Mezard and Giorgio Parisi in the 90’s, and abundantly studied since then, in particular in recent works by Fyodorov, Le Doussal, Texier and Rosso in the context of the study of depinning.

I will start by a zero-temperature question, i.e. the study of the topological complexity defined by this model, as given in the work with Benjamin Mc Kenna and Paul Bourgade (CPAM 2024) before proceeding to a new Mezard-Parisi type formula at positive temperature and our (still partial) understanding of the Replica Symmetry Breaking phase for this model

Conformal Field Theory (CFT) represents a class of quantum field theories that have profound applications across various physics domains, from critical phenomena in condensed matter to high energy physics. In this talk, I will talk about our recently progress on applying fuzzy (non-commutative) sphere approach to 3D CFTs, including deconfined phase transition, 5 SU(2) QCD3 and Chern-Simons-matter theory, which are relevant to the gapless spin liquid, phase transitions in magnets and fractional quantum Hall states.

Small divisor problems appeared first in celestial mechanics at the end of XIX century but they play now a very important role in statistical physics, as they appear in several problems ranging from quasi-crystals to graphene. We present two results obtained generalizing methods introduced for the convergence of the Lindstedt series and assuming suitable Diophantine conditions: a)the proof of the validity of the Harris-Luck criterion for the 2d Ising model with quasi-periodic potential; b)the proof of the quantization of the conductance in Haldanelike topological insulators on the cylinder with weak quasi-periodic disorder.

Abstract: I will discuss how the merger of two classes of models - the Kondo lattice model and the Kitaev spin liquid model has opened up a new vein of solvable Many Body physics models which, like BCS theory or the Peierls instability, are solvable to logarithmic accuracy[1,2].

Kitaev's spin liquid technology has led to the discovery of solvable two and three dimensional spin liquids - in which the Heisenberg spins fracitonalize into spinons, in some cases, with Fermi surfaces of gapless spin excitations. Remarkably, when these fractionalized spin fluids interact with electrons new forms of metal and superconductor are formed. I will give a short overview of these theoretical developments.

• [1] Solvable 3D Kondo Lattice Exhibiting Pair Density Wave, Odd-Frequency Pairing, and Order Fractionalization, Piers Coleman, Aaditya Panigrahi and Alexei Tsvelik, Phys. Rev. Lett. 129, 177601 (2022).
• [2]Tractable model for a fractionalized Fermi liquid on a square lattice, Piers Coleman, Elio Koenig, Aaditya Panigrahi and Alexei Tsvelik, arXiv April 2026.

I will discuss the concept of fractional order and provide examples of tractable models of condensed matter physics where existence of such order can be established.

I will discuss the concept of fractional order and provide examples of tractable models of condensed matter physics where existence of such order can be established.

I will discuss two striking examples of "boundary superfluids" -- novel 1D states of bosons coupled to correlated 2D bulk -- instantiated by an edge dislocation in a He-4 crystal and in correlated bosons hopping on a "comb" lattice. I will describe corresponding boundary anomalous superfluid phases -- a new class of 1D phases of bosons qualitatively distinct from a Luttinger liquid -- their stability to instanton deconfinement under a superflow and the associated quantum phase transitions.

The toric code, when deformed in a way that preserves the self-duality Z_2 symmetry exchanging the electric and magnetic excitations, admits a transition to a topologically trivial state that spontaneously breaks the Z_2 symmetry. Numerically, this transition was found to be continuous, which makes it particularly enigmatic given the longstanding absence of a continuum field-theoretic description. We propose an SO(4) Chern-Simons-Higgs (CSH) theory at level k=2 and argue that it serves as a natural field theory description of the self-dual transition. Moreover, it can be generalized to an entire series of theories labeled by an integer k. For each k>2, the theory describes an analogous transition involving different non-Abelian topological orders, such as the double Fibonacci order (k=3) and the S_3 quantum double (k=4). For k=1, we conjecture that the corresponding CSH transition is in fact infrared-dual to the 3d Ising transition, in close analogy with the particle-vortex duality of a complex scalar.

A numerical model is built, simulating the principles of kinetic gas theory, to predict pressures of molecules in a spherical pressure vessel; the model tracks a single particle and multiplies the force on the spherical walls by a mole of molecules to predict the net pressure. An intermolecular attractive force is added for high-density simulations to replicate a real fluid. The force is chosen to ensure the fluid matches the Peng-Robinson equation of state as it is compressed to a near supercritical density. The standard deviations of the molecule velocity with respect to temperature and density is studied to define the entropy. A parametric study of a Stirling cycle heat engine utilizing near-supercritical densities is modeled to study how the temperature dependence of the attractive intermolecular Van der Waal forces can affect the net total entropy change to the surrounding environment. A practical, macroscopic-scale piston-cylinder engine was then built and demonstrated, utilizing a novel thermodynamic cycle that closely resembles the Carnot heat engine cycle. This engine utilized non-ideal carbon dioxide as the working fluid to take advantage of the entropy effects demonstrated in the Argon simulations to boost thermodynamic efficiency. This engine offers significant opportunities for practical energy generation. <a href="https://arxiv.org/abs/2004.03584">https://arxiv.org/abs/2004.03584</a> USPTO Publication US-20260078714-A1.

Elliott Lieb’s Simplified Approach, first formulated in 1963, offers a novel mathematical framework to analyze an interacting Bose gas. We assume an integrable (soft), non-negative pair potential with finite scattering length. In this framework, a system of many particles is described by a single nonlinear integro-differential equation on \R^d. In recent joint work with Ian Jauslin, we computed the low-density expansion of the Simple equation’s prediction of the ground state energy and condensate fraction. We found that the Simple equation reproduces the correct leading-order and first correction terms for the energy, agreeing with previous predictions and recent rigorous results. For the condensate fraction, we found agreement with Bogoliubov theory and Schick’s 1971 prediction.