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Yunqing Tang, Berkeley

The arithmetic of power series and applications to irrationality

We will discuss a new approach to prove irrationality of certain periods, including the value at 2 of the Dirichlet L-function associated to the primitive quadratic character with conductor -3. Our method uses rational approximations from the literature and we develop a new framework to make use of these approximations. The key ingredient is an arithmetic holonomy theorem built upon earlier work by André, Bost, Charles (and others) on arithmetic algebraization theorems via Arakelov theory. This is joint work with Frank Calegari and Vesselin Dimitrov.

MC 5417

Stanley Xiao, University of Northern British Columbia

Elliptic curves admitting a rational isogeny of prime degree, ordered by conductor

We consider explicit parametrizations of rational points on the modular curves X_0(p) for p in {2,3,5,7}, which corresponds to elliptic curves E/Q$ admitting a rational isogeny of degree p, and consider conductor polynomials of such curves. Conductor polynomials are polynomial divisors of the discriminant which more closely approximate the conductors of elliptic curves. By using results on almost-prime values of polynomials, including recent breakthrough work of Ben Green and Mehtaab Sawhney, we count such curves whose conductors have the least number of distinct prime factors, ordered by conductor. This is joint work with Alia Hamieh and Fatma Cicek. 

MC 5417

Brent Nelson, Michigan State University

Closable derivations are anticoarse, of course

The anticoarse space of an inclusion $N\subset M$ of tracial von Neumann algebras is an $N$-subbimodule of $L^2(M)$ whose size is sensitive to several structural properties of the inclusion. It has become a staple of so-called microstates techniques in free probability, where it allows one to parlay finite dimensional approximations into algebraic properties. On the other hand, non-microstates techniques, which exploit the regularity of certain derivations on a von Neumann algebra, have not made use of the anticoarse space, until now. In this talk, I will discuss how deformations given by closable derivations provide a natural connection to anticoarse spaces and consequently yield new applications of free probability. This is based on joint work with Yoonkyeong Lee.

QNC 1507 or Join on Zoom

Yunqing Tang, Berkeley

Irrationality of periods

Periods are interesting numbers arising from algebraic geometry. Grothendieck’s period conjecture provides predictions on irrationality and transcendence of periods. There have been some systematic studies on certain periods, such as Baker’s theory on linear forms of logarithms of algebraic numbers. However, beyond special cases, we do not know the irrationality of simple-looking periods such as the product of two logs. In this talk, I will discuss the joint work with Calegari and Dimitrov on an irrationality result of certain product of two logs and some other periods. A classical prototype of the method was first used by Apéry to prove the irrationality of zeta(3). The key ingredient is an arithmetic holonomy theorem built upon earlier work by André, Bost, Charles (and others) on arithmetic algebraization theorems via Arakelov theory.

MC 5501

Josse van Dobben de Bruyn, Charles University

Asymmetric graphs with quantum symmetry

Quantum isomorphisms of graphs form a bridge between noncommutative geometry (NCG) and quantum information theory (QIT), as they connect quantum automorphism groups of graphs with nonlocal games. This makes it possible to use techniques from QIT in NCG and vice versa. In this talk, I will present a striking application of this connection, where we use ideas from QIT to prove a surprising result in NCG. Using a construction similar to the Mermin–Peres magic square from QIT, we construct graphs with trivial automorphism group and non-trivial quantum automorphism group, which shows that even graphs with no symmetry at all can have hidden quantum symmetries. These are the first known examples of any kind of commutative spaces in NCG with this property.

This talk is based on joint work with David E. Roberson (Technical University of Denmark) and Simon Schmidt (Ruhr University Bochum).

QNC 1507 or Join on Zoom

Java Villano, University of Toronto

Relativizing computable categoricity

A computable structure A is said to be computably categorical if for all computable copies B of A, there exists a computable isomorphism between A and B. We can relativize this notion to any Turing degree d by asking that for any d-computable copy B of A, there is a d-computable isomorphism between A and B. In this talk, we will discuss results about this relativization. In particular, we will discuss how for directed graphs, categoricity relative to a degree need not be monotonic in the c.e.~ degrees, and how for other structures besides directed graphs, categorical behavior relative to a degree stabilizes on certain Turing cones.

MC 5403

Gian Cordana Sanjaya, University of Waterloo

Density of Special Classes of Polynomials with Squarefree Discriminant

In this talk, we compute the density of monic polynomials of fixed degree over Z_p, which has squarefree discriminant, provided some restriction on the coefficients of the polynomial. This is a natural extension to a previous result by Yamamura, who solved the case where the coefficients have no restrictions at all.

MC 5479

Amanda Petcu, University of Waterloo

Stability and Lyapunov Functions

When working with a nonlinear system of differential equations, finding explicit, closed-form solutions can be difficult. A tool in such situations is to determine the stability of the equilibrium points of the system. This analysis allows us to predict the long-term behavior of the system by examining its trajectories and how they behave near an equilibrium point: specifically, do they remain bounded in some compact set, converge to the point, or escape to infinity? In this talk, we will discuss Lyapunov's Direct Method, a technique that allows us to determine the stability of an equilibrium point without explicitly solving the differential equations.

MC 5403

Elan Roth, University of Waterloo

A Continuation of Random Binary Sequences

We will build on the notions of ML-random and 1-random by defining a third interpretation of unpredictability using martingales. We will then prove the equivalence of these definitions and discuss some of their nice properties. Then, we will see how randomness is spread among the Turing degrees.

MC 5403

Joel Kamnitzer, McGill University

The top-heavy conjecture and the topology of (real) matroid Schubert varieties

Suppose we are giving a spanning set S in a vector space V and we consider all subspaces of V spanned by subsets of S. The top-heavy conjecture states that the number of dimension k subspaces is less than or equal to the number for codimension k subspaces. This elementary statement was first conjectured by Dowling and Wilson in 1975 and resisted any proof for 40 years. Finally though, it was resolved by Huh and Wang in 2017, and partially led to Huh's 2022 Fields Medal. I will outline the details of the proof, which relies on the study of the topology of a beautiful space called a matroid Schubert variety. Finally, I will discuss our own contribution to this subject, which is the study of the topology of the real locus of this space (which unfortunately does not lead to the proof of any famous conjecture).

MC 5501