Degree bounds have a long history in invariant theory. The Noether bound on the degrees of algebra generators for a ring of invariants is over a century old, and there is a vast literature sharpening and generalizing it. In the last two decades, there has also been an active program on degree bounds for invariants which are able to distinguish orbits as well as algebra generators can (known as separating invariants).

In this talk I make the case that generators for the field of rational invariants represent an exciting avenue for research on degree bounds as well. I present new lower and upper bounds. It will transpire that even the case of G=Z/pZ, uninteresting from the point of view of generating and separating invariants, has a story to tell for rational invariants. The methods involve the classical Minkowski “geometry of numbers”. I argue that this domain of inquiry is both interesting in itself and well-motivated by applications.

This talk is based on joint work with Thays Garcia, Rawin Hidalgo, Consuelo Rodriguez, Alexander Kirillov Jr., Sylvan Crane, Karla Guzman, Alexis Menenses, Maxine Song-Hurewitz, Afonso Bandeira, Joe Kileel, Jonathan Niles-Weed, Amelia Perry, and Alexander Wein.

We discuss a kind of weak-type inequality for the Hardy-Littlewood maximal operator that was first studied by Muckenhoupt and Wheeden. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior; in particular, as shown by Muckenhoupt and Wheeden, the class of weights characterizing such inequalities is strictly larger than $A_p$. In this talk, I will present recent work on the full characterization of the weights for which these inequalities hold for the Hardy-Littlewood maximal operator. Connections to weak-type estimates for the Bergman projection on bounded simply connected planar domains will also be discussed.

The general motto of noncommutative algebraic geometry is that any category sufficiently close to the derived category of a variety should be regarded as a noncommutative variety. Using this principle, pioneering work of Artin-Tate-Van den Bergh-Zhang extends important aspects of projective geometry to the noncommutative setting. I’ll talk about extensions of this theory to noncommutative spaces associated to dg-algebras with a focus on how it feeds back into the commutative setting. In particular, I’ll discuss a generalization of a celebrated theorem of Orlov concerning the derived category of a projective complete intersection. Joint work with Michael K. Brown.

Signed persistence diagrams provide a concise representation of the rank invariant (persistent Betti numbers) or birth-death invariant for filtrations of topological spaces. When these filtrations are indexed by a linear poset, the signed persistence diagrams consist of nonnegative values, allowing for straightforward interpretation as the multiplicity of the bars in the barcode of the filtration. However, for filtrations indexed by non-linear posets, signed persistence diagrams can include negative values, which complicates their interpretability. In this talk, we introduce Grassmannian persistence diagrams, a notion that extends the capabilities of signed persistence diagrams while preserving interpretability. Moreover, when the indexing poset is linear, Grassmannian persistence diagrams also reveal canonical cycle spaces for each bar in the barcode associated with the filtration.

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.