Schedule and Abstracts

When he built his tower, Gustave Eiffel engraved the names of 72 French scientists, all men. Recently, it has been decided to engrave the names of 72 French women scientists on the tower. Eleven mathematicians were selected, including Marie-Hélène Schwartz. One of her major works, in 1965, was the construction of characteristic classes for singular complex analytic varieties. This result was subsequently part of a conjecture by Grothendieck and Deligne, proved by Robert MacPherson. The classes are now called "Schwartz-MacPherson classes. The elementary presentation will focus as much on the life of Marie-Hélène Schwartz as on her proof of the Poincaré-Hopf Theorem for singular varieties. 

That is a join work with Thuy Nguyen (UNESP, Rio Preto, Brazil) and Tadeusz Mostowski (Warsaw, Poland).

Link to PDF abstract

I’ll start by giving an introduction to index theory in the presence of symmetries and explain how this fits into the framework of noncommutative geometry.  After that I will focus on the case of proper actions of groups and explain a recent ``higher fixed point formula” obtained in joint work with Paolo Piazza, Yanli Song and Xiang Tang.

Symmetries and anomalies provide powerful constraints on the dynamics of quantum many-body systems. In this talk, I will revisit a class of models known as quantum dimer models, and argue for the presence of a mixed anomaly involving lattice rotations and a higher-form symmetry. This anomaly forbids the existence of any symmetry preserving gapped phase, consistent with the known phase diagrams of these models. This talk is based on work in progress with Wilbur Shirley and Clay Córdova.

We consider a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We focus on a quantum dimer model on the triangular lattice, with doubly-periodic edge weights. For simplicity we consider a 2x1 periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled α. We analytically show that the model exhibits a continuous quantum phase transition at α = 3, changing from a topological Z2 quantum spin liquid (α < 3) to a columnar ordered state (α > 3). The dimer-dimer correlator decays exponentially on both sides of the transition, while the vison correlator exhibits an exponential decay in the spin liquid phase but becomes a constant in the ordered phase. We analytically show that the topological Rényi entropy of order ∞ (topological min-entropy) changes from log(2) for the quantum spin liquid phase α < 3, to 0 for the ordered phase α > 3, thereby analytically confirming the topological nature of the phase transition. Joint work with Jeet Shah, Matthew Lerner-Brecher, Amol Aggarwal, Alexei Borodin, and Victor Galitski (https://arxiv.org/abs/2601.15377).

It is expected that a generic closed many-body system prepared in a well-behaved initial state will eventually thermalize, i.e. approach a Gibbs state. This property, while compatible with and even demanded by the physical intuition, is much stronger than ergodicity or mixing and is difficult to justify mathematically. Similarly, a generic many-body system subject to a periodic drive (a Floquet system) is expected to heat up and approach the state of maximal entropy. In this talk I will  describe an infinite set of many-body Floquet systems of algebraic origin, both classical and quantum, for which thermalization of very general initial states can be studied analytically as well as numerically.

The Born-Oppenheimer or adiabatic approximation separates fast electronic dynamics from slow nuclear dynamics.  In this approximation, the electronic Schrödinger equation is solved for fixed nuclei and then each electronic energy is used as a potential energy surface for solving the vibrational Schrödinger equation. Von Neuman and Wigner put forth a dimensional argument that the potential curves should avoid each other in diatomic molecules but that potential surfaces should have intersections in polyatomic molecules. With the local shape of a double-cone in 2D, these conical intersections have been identified by their associated topological phase (Berry phase) and are now known to occur in many photochemical reactions. Nevertheless, they are features of an approximation. Several years ago, we were exploring a non-adiabatic model for energy transfer in photosynthesis and found that the exact non-adiabatic eigenfunctions have a weird nodal structure that parallels the dimensional arguments of Von Neuman and Wigner. Instead of the usual nodal points in 1D, nodal curves in 2D, etc. the nonadiabatic eigenfunctions have avoided nodes in 1D, point nodes in 2D, etc. In the neighborhood of a point node in 2D, the nonadiabatic eigenfunctions have the local shape of a cone. High symmetry can generate higher order nodes. The nodes have a signed integer electronic index that measures electronic state vector rotation around a closed vibrational path and is locally stable to perturbations. I will discuss examples of conical nodes that occur by accident and near symmetry-required conical intersections.

M. Born, Gott. Nachr. math. phys. Kl. 6, 1 (1951).

J. von Neumann and E. Wigner, Phys. Z. 30, 467 (1929).

M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).

M. Klessinger and J. Michl, Excited States and Photochemistry of Organic Molecules. (VCH Publishers, New York, 1995).

V. Tiwari, W. K. Peters and D. M. Jonas, Proc. Natl. Acad. Sci. USA 110, 1203 (2013).

P. W. Foster, W. K. Peters and D. M. Jonas, Chem. Phys. Lett. 683, 268 (2017).

P. W. Foster and D. M. Jonas, J. Phys. Chem. A 121, 7401 (2017); 123, 1273 (2019).

P. W. Foster and D. M. Jonas, Chem. Phys. 520, 108 (2019).

Matrix product states form an especially tractable and well-understood class of states of quantum spin chains. In particular, when working with matrix product states, one obtains elegant derivations of various topological indices that classify phases of both symmetry protected states as well as continuously parametrized families of states. I will begin this talk with a brief introduction to matrix product states on infinite lattices, move on to discuss joint work with my CU Boulder colleagues on the homotopy type of the space of matrix product states, and finally I will mention work in progress on the classification of matrix product states that are both symmetry-protected and parametrized.

Let G be a connected linear real reductive group with a maximal compact subgroup K. In this talk, we will discuss an approach to studying the large-time behavior of the heat kernel on the corresponding homogeneous space G/K using  Bismut's formula. We will try to explain how Bismut's formula provides a natural link between the index theory and representation theory.  In particular, Vogan's lambda-map in the representation theory of G plays a central role in the large time asymptotic analysis about the trace of the heat kernel. This talk is based on joint works with Shu Shen and Yanli Song. 

I will describe how to deal with single-particle dynamics over complex aperiodic discrete geometries using C*-algebras. Using numerical examples, I will first convey that any local Hamiltonian over such a geometry falls into one and same algebra, completely determined by the underlying discrete geometry. I will then describe this algebra based on the notion of phason spaces, and compute it for particular cases (e.g. twisted bilayers). K-theory of this algebra tells us what topological invariants are available for a given discrete geometric structure. For example, a twisted bilayer can host up to second-Chern numbers but nothing above that. The punch line of the talk would be a C*-formulation of ordinary and higher order bulk-defect correspondences. The claim is that, given a geometry with defects (surfaces, hinges, corners, dislocations, disclinations, etc), the C*-algebraic framework supplies a device to enumerate all possible non-trivial defect-boundary correspondences, without missing one. I will include numerical demonstrations of these principles.

References:

• P. D. Ojito, E. Prodan, T. Stoiber, C*-framework for higher-order bulk-boundary correspondences, Comm. Math. Phys 406, 233 (2025).

• B. Mesland, E. Prodan, Classifying the dynamics of architected materials by groupoid methods, Journal of Geometry and Physics 196, 105059 (2024).

My talk centers on an “ordinary-looking” ordinary differential equation—the prolate spheroidal wave operator—which has played a surprisingly rich and unexpected role across several fields.  It first arose as a “lucky accident” in the 1960s solution by Slepian, Landau, and Pollak of the problem of time-and band-limiting of signals, raised by Claude Shannon.

The same operator reappeared in the late 1990s through a cutoff mechanism in Alain Connes’ trace formula reformulation of the Weil explicit formula in number theory. More recently, in joint work with Connes (2022), we found that when this operator is extended from a finite interval to the whole line, its spectrum reproduces—with striking accuracy—the squares of the imaginary parts of the zeros of the Riemann zeta function.

I will conclude with a refinement of this phenomenon based on analyzing the operator in the complex domain.

Gapped phases of two-dimensional free-fermionic systems without any symmetry are classified by an integer-valued topological invariant called the Chern number, which counts the net chirality of boundary modes. It has long been conjectured that systems with distinct Chern numbers cannot be deformed into each other without a phase transition, even in the presence of arbitrarily strong interactions. In this talk, I will present a construction of a many-body index that generalizes the Chern number to the interacting setting and allows us to prove this conjecture. When the edge modes can be described by a conformal field theory, the index provides a microscopic characterization of the chiral central charge.

There are now many examples of gapped fracton models, which are defined by the presence of restricted-mobility excitations above the quantum ground state. This complex landscape of examples is far from being mapped out. In this talk, I will describe recent progress on characterization and classification of fracton orders, and on related problems for subsystem symmetries.