The goal of this talk is to discuss spectral theory for quantum Hamiltonians describing  a system of two  identical spinless bosons on the two-dimensional lattice particles.

We assume that the particles are interact via on-site and first and second nearest-neighboring site interactions between the bosons are only nontrivial and that these interactions are of magnitudes \(\gamma\), \(\lambda\), and \(\mu\), respectively.

Introducing the quasi-momentum \(k\in (-\pi,\pi]^2\)  for the system of two identical bosons,  and decomposing the  Hamiltonians into the direct von Neumann integral reduces the problem to the study  two-particle fiber lattice Schrödinger operators that depend on the two-particle quasi-momentum \(k\).

In this model, we determine both the exact number and location of the eigenvalues for the lattice Schrödinger operator \(H_{\gamma\lambda\mu}(0)\), depending on the interaction parameter \(\gamma\), \(\lambda\) and \(\mu\).

We find a partition of the \((\gamma,\lambda,\mu)\)-space into connected components such that, in each connected component, the two-boson Schrödinger operator corresponding to the zero quasi-momentum of the center of mass has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential (continuous) spectrum and above its top. Moreover,  for each connected component, a sharp lower bound is established on the number of isolated eigenvalues for the two-boson Schrödinger operator corresponding to any admissible nonzero value of the center-of-mass quasimomentum.

We will reveal the mechanisms of emergence and absorption of the eigenvalues at the thresholds of the essential (continuous) spectrum of \(H_{\gamma\lambda\mu}(0)\), as the parameters \(\gamma\), \(\lambda\) and \(\mu\) vary.

In addition, we have identified the sufficient conditions for the exact number of eigenvalues of the lattice Schrödinger operator for all values of the interaction parameters. 

We extend the asymptotic Samuel function of an ideal to a filtration and study some properties of the function. We look at the relation between integral closure of filtrations and the asymptotic Samuel function. We further study the notion of projective equivalence of filtrations. The talk follows the pre-print ‘The Asymptotic Samuel Function of a Filtration’ by Dale Cutkosky and Smita Praharaj.

Uniform expressivity guarantees that a Graph Neural Network (GNN) can express a query without the parameters depending on the size of the input graphs. This property is desirable in applications in order to have number of trainable parameters that is independent of the size of the input graphs. Uniform expressivity of the two variable guarded fragment (GC2) of first order logic is a well-celebrated result for Rectified Linear Unit (ReLU) GNNs [Barcelo & al., 2020]. In this talk, we prove that uniform expressivity of GC2 queries is not possible for GNNs with a wide class of Pfaffian activation functions (including the sigmoid and tanh), answering a question formulated by [Grohe, 2021]. We also show that despite these limitations, many of those GNNs can still efficiently express GC2 queries in a way that the number of parameters remains logarithmic on the maximal degree of the input graphs. Furthermore, we demonstrate that a log-log dependency on the degree is achievable for a certain choice of activation function. This shows that uniform expressivity can be successfully relaxed by covering large graphs appearing in practical applications. Our experiments illustrates that our theoretical estimates hold in practice.

Paraproducts are one of the essential tools of harmonic analysis, used to decompose a product of two functions. Motivated by a similar question in complex analysis, Pott, Reguera, Sawyer and Wick studied the composition of "paraproduct-type operators", to which classical paraproducts belong. Their goal was to find joint conditions on the symbol functions for the composition to be bounded. They classified many paraproduct-type compositions. One of the operators to remain unclassified was the composition of two classical dyadic paraproducts. In this talk, we will discuss the joint conditions for the boundedness of two paraproducts as well as certain weighted inequalities.

In this talk we discuss the existence and the importance of asymptotic colengths for families of \(m\)-primary ideals in a Noetherian local ring \((R,m)\).  We explore various families such as weakly graded families, weakly \(p\)-families and weakly inverse \(p\)-families and discuss a new analytic method to prove the existence of limits. Additionally, if time permits we will talk about Minkowski type inequalities, positivity results, and volume = multiplicity formulas for these families of ideals. This talk is based on a joint work with Cheng Meng.

Parseval frames provide redundant encodings, with equal-norm Parseval frames offering optimal robustness against one erasure. However, constructing such frames can be challenging. The Paulsen Problem asks to determine how far an ε-nearly equal-norm Parseval frame is from the set of all equal-norm Parseval frames. In this talk, I will present an approach to the Paulsen’s problem based on quiver invariant theory

Let \(R\) be a commutative Noetherian domain with identity and finite Krull dimension \(d\). If \(R\) is non-singular, then for every ideal \(I \subseteq R\) and \(n \in \mathbb{N}\), the symbolic power \(I^{(dn)}\) is contained in the ordinary power \(I^n\). This property is known as the Uniform Symbolic Topology Property, reflecting a uniform comparison between symbolic and ordinary powers of ideals in \(R\).

Historically, this property was first established for rings over complex numbers by Ein, Lazarsfeld, and Smith, then extended to non-singular rings containing a field by Hochster and Huneke. Later, Ma and Schwede proved it for reduced ideals in excellent non-singular rings of mixed characteristic, and Murayama extended it to all regular rings, even non-excellent ones. Their proofs have profound connections with the subjects of multiplier/test ideal theory, closure operations, and constructions of big Cohen-Macaulay algebras.

The study of symbolic powers becomes significantly more challenging in the presence of singularities. In this talk, we focus on prime characteristic strongly \(F\)-regular singularities that arose from Hochster and Huneke's tight closure theory. We show that an \(F\)-finite strongly \(F\)-regular domain satisfies the Uniform Symbolic Topology Property, meaning there exists a constant \(C\) such that \(I^{(Cn)} \subseteq I^n\) for all ideals \(I \subseteq R\) and \(n \in \mathbb{N}\). We will explore the role of splitting ideals and Cartier linear maps in establishing this result.

This talk considers provable methods for cryo-electron microscopy, which is an increasingly popular imaging technique for reconstructing 3-D biological macromolecules from a collection of noisy and randomly oriented projection images, with applications in e.g., drug design. The talk will present two uniqueness guarantees for recovering these structures from the second moment of the projection images, as well as two associated numerical algorithms. Mathematically, the results boil down to ensuring unique solutions to highly structured non-linear equations.