Talks

I will discuss the problem of obtaining an asymptotic formula for counting integral solutions to an equation of the form f(x, y, z, w)=N in an expanding box, where N is a non-zero integer and f is an indefinite quadratic form over the integers. For forms of the shape axy-bzw, I will then explain how we can obtain a significantly strong error term by applying deep methods from the spectral theory of automorphic forms. This is a joint work with Rachita Guria.

UTe₂ is an intriguing recently discovered superconductor that exhibits a wide range of exotic properties. Experimental evidence increasingly supports spin-triplet pairing, and under pressure or in magnetic fields, UTe₂ displays multiple superconducting phases, including a remarkable reentrant phase above 40 T. However, conflicting results persist regarding the presence of chiral and time-reversal symmetry breaking. Recent STM measurements have identified a charge density wave in the normal state, which couples with the superconducting state at lower temperatures to form a pair density wave. In this talk, I will provide an overview of the latest developments in understanding the unconventional superconductivity of UTe₂.

Galaxies are the medium through which we study the structure of the universe. However, widely applied statistical models of galaxies are generally over-simplified: even recently proposed models cannot capture the dependencies on environment or formation history. To solve this problem, I will introduce Graph Neural Networks (GNNs), a general and ideal tool for physical modelling. Geometrically constrained GNNs vastly improve our models, and allow us to ask detailed questions about the importance of formation history and environment for cosmological galaxy modeling. I will also prove a surprising equivalence between these two aspects of galaxy formation.

We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over Q defined over at least 10 variables. This is a joint work with Simon L. Rydin Myerson (warwick) and Junxian Li (UC Davis).

n this talk, I will discuss the problem of obtaining lower bounds for the number of rational points of bounded height on cubic hypersufaces. Our main tools will be the circle method, the Ekedahl sieve and the geometry of numbers.

In practice, L-functions appear as generating functions encapsulating information about various objects, such as Galois representations, elliptic curves, arithmetic functions, modular forms, Maass forms, etc. Studying L-functions is therefore of utmost importance in number theory at large. Two of their attached data carry critical information: their zeros, which govern the distributional behavior of underlying objects; and their central values, which are related to invariants such as the class number of a field extension. We discuss a connection between low-lying zeros and central values of L-functions, in particular showing that results about the distribution of low-lying zeros (towards the density conjecture of Katz-Sarnak) implies results about the distribution of the central values (towards the normal distribution conjecture of Keating-Snaith). Even though we discuss this principle in general, we instanciate it in the case of modular forms in the level aspect to give a statement an...

In this talk, I will discuss how M2-M5 intersections in a twisted M-theory background yield the R-matrices of the quantum toroidal algebra of gl(1). These R-matrices are identified with the Miura operators for the q-deformed W- and Y-algebras. Additionally, I will show how the M2-M5 intersection (or equivalently, the Miura operator) generates the qq-characters of the 5d N=1 gauge theory, offering new insight into the algebraic meaning of the latter.

To justify the existence of measurements that can not be performed jointly on quantum systems, Heisenberg put forward a heuristic argument, involving the famous gamma-ray microscope Gedankenexperiment, based on the existence of measurements that irreversibly alter the physical system on which they act. Today, the impossibility of jointly measuring some physical quantities, termed measurement incompatibility, and irreversible disturbance, namely the existence of operations that irreversibly alter the system on which they act, are understood to be distinct but related features of quantum mechanics. In our work, we formally characterized the relationship between these two properties, showing that measurement incompatibility implies irreversible disturbance, though the converse is false. The counterexamples are two toy theories: Minimal Classical Theory and Minimal Strongly Causal Bilocal Classical Theory. These two are distinct as counterexamples because the latter allows for classical conditioning. Our research followed an operational approach exploiting the framework of Operational Probabilistic Theories. In particular, it required the development of two new classes of operational theories: Minimal Operational Probabilistic Theories and Minimal Strongly Causal Operational Probabilistic Theories. These theories are characterized by a restricted set of dynamics, limited to the minimal set consistent with the set of states. In Minimal Strongly Causal Operational Probabilistic Theories, classical conditioning is also allowed.