Quantum Matter, Information, Dynamics, and Computation

Fall 2027: Topology, Spectra, Disorder, and Emergent Structure

Quantum materials create a sharp test of how far mathematical structure can guide physical understanding when periodicity, weak interaction assumptions, or simple band descriptions fail. Quasiperiodic and disordered systems can support localization, critical behavior, and unusual transport; twisted multilayers introduce long moiré scales and effective Hamiltonians whose approximation regimes must be understood; boundaries and defects can host states whose stability depends on spectral and topological structure. The semester will place these phenomena on common mathematical ground by treating models, spectra, topology, and observables as parts of one problem. Three clusters provide concrete entry points:

  • The first concerns aperiodicity, disorder, and spectral type: when does departure from periodicity localize states, and when does it support extended or critical behavior?
  • The second concerns interfaces, defects, and embedded states, including algebraically induced surface states connected with reducible Fermi surfaces and bound states in the continuum.
  • The third concerns geometry and emergent order in layered quantum systems: effective descriptions of twisted multilayers, propagation along moiré structures, wave-function geometry, superconductivity, and correlated or fractionalized phases. These questions require spectral theory, PDE, topology, mathematical physics, numerical computation, and condensed-matter insight to meet on the same examples.

At TIAMS, participants will work with explicit Hamiltonians, spectral varieties, interface models, and numerical data. A graduate research school will establish common tools in self-adjoint spectral theory, Fourier analysis, effective Schrödinger models, compact/Fredholm operators, and index ideas. Working groups will compare controlled limits with finite computations, identify perturbations that preserve or destroy proposed states, and record which conclusions depend on symmetry, topology, or algebraic structure. Progress may take the form of a theorem, a sharper effective model, a computable invariant, a falsified mechanism, or a clean statement of the regime in which a physical prediction is trustworthy. A central discipline of the semester will be to distinguish effects that persist under controlled limits from those created by truncation, finite size, symmetry assumptions, or a particular numerical representation.

  • How do quasiperiodicity and disorder create, destroy, or reorganize electronic criticality and localization?
  • What are the rigorous spectral types of twisted and incommensurate multilayer models in physically relevant regimes?
  • Which algebraic or topological indices govern localized surface or interface states embedded in a continuum?
  • How do wave-function geometry and spatial structure enter superconductivity and other collective phases?
  • How can topology be defined and detected when translation symmetry is absent, interactions matter, or the relevant states are not isolated bands?
  • Which model reductions preserve the physical mechanisms of interest, and how can their domains of validity be quantified?

  • Spectral theory of self-adjoint operators
  • Quasiperiodic and disordered systems
  • Twisted multilayers and moiré models
  • Topological phases without translation symmetry
  • Bound states in the continuum
  • Interfaces, defects, and edge propagation
  • Wave-function geometry and superconductivity

  • Justin H. Wilson — TIAMS scientific lead; LSU
  • Stephen P. Shipman — TIAMS mathematical/scientific lead; LSU
  • Michael I. Weinstein — external scientific leader; Columbia University
  • Allan H. MacDonald  — intended external scientific leader; The University of Texas at Austin
  • Piers Coleman (pending) — intended scientific adviser / international-network partner; Rutgers University

The semester centers on overlapping research residence and active working groups. Opening instruction should bring mathematicians and physicists onto the same models; a focused workshop should concentrate the larger community around the core spectral/topological questions; October residence should be protected for sustained small-group work; a later intensive can target a problem that has sharpened during the semester; and a closing synthesis should identify results, failures, and problems ready for continued collaboration.

Preliminary Workshops, Schools, or Focused Meetings

Mathematical Foundations of Quantum Materials — graduate research school

Dates TBA

Focused workshop on spectra, topology, disorder, and emergent structure

Dates TBA

Working-group intensive on layered and interface models

Dates TBA

Semester synthesis meeting

Dates TBA

 

Seminars and Working Groups

  • Aperiodic spectra and localization: spectral type, criticality, transport, and controlled approximations.
  • Fermi-surface geometry and embedded states: reducibility, topological indices, interface/edge states, and perturbative stability.
  • Moiré and multilayer models: effective Hamiltonians, edge propagation, approximation regimes, and numerical comparison.
  • Wave-function geometry and collective order: geometric information in quantum states and consequences for superconducting or correlated phases.

The semester is intended for spectral theorists, analysts, PDE researchers, condensed-matter theorists, mathematical physicists, topologists interested in quantum systems, and computational researchers working on electronic structure or many-body models. Faculty at all career stages, postdocs, and graduate students are invited. Researchers in operator algebras, probability, numerical analysis, materials science, and quantum engineering may find natural entry points when their methods address localization, effective Hamiltonians, interfaces, topological indices, or layered systems. Participants should be ready to work on shared examples and to explain what their preferred models assume. Experimental or computational materials researchers are also welcome when they can bring data, model constraints, or concrete observables that help distinguish competing mathematical mechanisms.

  • Semester or extended residence as a scientific anchor.
  • One- to three-month visits overlapping with a working group.
  • Short targeted visits around a defined model or calculation.
  • Participation in the graduate research school or focused workshop.
  • Weekly seminars, research discussions, and working-group meetings.
  • Expression of interest through the TIAMS participation form.

The research school is designed as an entry ramp into the semester's active problems. Students and postdocs should learn the analytic tools on examples that visitors are actually studying, then join working groups where those tools are used. They will be encouraged to present computations and partial arguments, ask specialists to make hidden assumptions explicit, and remain with a problem long enough to see how numerical evidence, model reduction, and proof constrain one another. Informal progress sessions should give them a low-friction way to test ideas before presenting polished results.

Possible outcomes include spectral and localization results, rigorous statements for effective multilayer models, topological or algebraic criteria for interface states, model-to-model comparison tables, computations linked to explicit error regimes, open-problem documents, reproducible code, and collaborations between mathematicians and materials theorists.

Use the TIAMS Program Participation form to describe your research interests, preferred working group, possible contribution, requested visit dates, and any support needs.

Applicants may request a long visit, a focused shorter stay, or participation tied to the school or workshop.