C3

A central topic of study in analytic number theory is the behaviour of the Riemann zeta function. Many theorems and conjectures in this area are closely connected to concepts from probability theory. In this talk, we will discuss several results on the typical size of the zeta function on the critical line, over different scales. Along the way, we will see the role that is played by some probabilistic phenomena, such as the central limit theorem and multiplicative chaos.

It is known that there exists certain randomness in the values of the Moebius function. It is widely believed that this randomness predicts significant cancellations in the summation of the Moebius function times any 'reasonable' sequence. This rather vague principle is known as an instance of the 'Moebius randomness principle'. Sarnak made this principle precise by identifying the notion 'reasonable' as deterministic. More precisely, Sarnak's Moebius Disjointness Conjecture predicts the disjointness of the Moebius function from any arithmetic functions realized in any topological dynamical systems of zero topological entropy. In this talk, I will firstly introduce some background and progress on this conjecture. Secondly, I will talk about some of my work on this. Thirdly, I will talk some related problems to this conjecture.

We will first introduce the modular method for solving Diophantine Equations, famously used to
prove the Fermat Last Theorem. Then, we will see how to generalize it for a totally real number field $K$ and
a Fermat-type equation $Aa^p+Bb^q=Cc^r$ over $K$. We call the triple of exponents $(p,q,r)$ the 
signature of the equation. We will see various results concerning the solutions to the Fermat equation with
signatures $(r,r,p)$ (fixed $r$). This will involve image of inertia comparison and the study of certain
$S$-unit equations over $K$. If time permits, we will discuss briefly how to attack the very similar family
of signatures $(p,p,2)$ and $(p,p,3)$. 

Let $p$ be a prime, let $1 \le t < d < p$ be integers, and let $S$ be a non-empty subset of $\mathbb{F}_p$ (which may be thought of as being $\{0,1\}$). We will establish that if a polynomial $P:\mathbb{F}_p^n \to \mathbb{F}_p$ with degree $d$ is such that the image $P(S^n)$ does not contain the full image $A(\mathbb{F}_p)$ of any non-constant polynomial $A: \mathbb{F}_p \to \mathbb{F}_p$ with degree at most $t$, then $P$ coincides on $S^n$ with a polynomial $Q$ that in particular has bounded degree-$\lfloor d/(t+1) \rfloor$-rank in the sense of Green and Tao, and has degree at most $d$. Likewise, we will prove that if the assumption holds even for $t=d$ then $P$ coincides on $S^n$ with a polynomial determined by a bounded number of coordinates and with degree at most $d$.

In 1993, Erdős, Sárközy and Sós posed the question of whether there exists a set $S$ of positive integers that is both a Sidon set and an asymptotic basis of order $3$. This means that the sums of two elements of $S$ are all distinct, while the sums of three elements of $S$ cover all sufficiently large integers. In this talk, I will present a construction of such a set, building on ideas of Ruzsa and Cilleruelo. The proof uses a powerful number-theoretic result of Sawin, which is established using cutting-edge algebraic geometry techniques.

Serre's conjecture (now a theorem) predicts that an irreducible 2-dimensional odd
Galois representation of $\mathbb Q$ with coefficients in $\bar{\mathbb F}_p$ comes from the mod p reduction of
a modular form. A key feature is that two modular forms of different weights can have the same
mod p reduction. Fixing a modular form $f$, the weight part of Serre's conjecture seeks to find all
the possible weights where one can find a modular form congruent to $f$ mod $p$. The recipe for these
weights was conjectured by Serre, and it depends only on the local Galois representation at $p$. I
will explain the ideas involved in Edixhoven's proof of the weight part, and if time allows, I
will briefly say something about what the generalizations beyond $\operatorname{GL}_2/\mathbb Q$ might look like. 

Hydrocephalus is a serious medical condition which causes an excess of cerebrospinal fluid (CSF) to build up within the brain. A common treatment for congenital hydrocephalus is to implant a permanent drainage shunt, removing excess CSF to the stomach where it can be safely cleared. However, this treatment carries the risk of vascular brain tissues such as the Choroid Plexus (CP) being dragged into the shunt during drainage, causing it to block, and also preventing the shunt from being easily replaced. In this talk I present results from our fluid-structure interaction model which simulates the deflection of the CP during the operation of the hydrocephalus shunt. We seek to improve the shunt component by optimising the geometry with respect to CP deflection.

A C*-algebra has the lifting property (LP) if any unital completely positive map into a quotient C*-algebra admits a completely positive lift. The local lifting property (LLP), introduced by Kirchberg in the early 1990s, is a weaker, local version of the LP.  I will present a method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C*-algebras of countable groups with (relative) property (T). This allows us to derive that the full C*-algebras of the groups $Z^2\rtimes SL_2(Z)$ and $SL_n(Z)$, for n>2, do not have the LLP. The same method allows us to prove that the full C*-algebras of a large class of groups with property (T), including those admitting a probability measure preserving action with non-vanishing second real-valued cohomology, do not have the LP.  In a different direction, we prove that the full C*-algebras of any non-finitely presented groups with property (T) do not have the LP. Time permitting, I will also discuss a connection with the notion of Hilbert-Schmidt stability for countable groups. This is based on a joint work with Pieter Spaas and Matthew Wiersma.

Lie groupoids are closely connected to pseudo-differential calculus. On a vector bundle considered as a `commutative Lie groupoid' (i.e. as a family of commutative Lie groups), they can be treated using the Fourier transform. In this talk, we explore the extension of this idea to the noncommutative space by employing the tubular neighborhood construction and subsequently adopting a global approach through the introduction of deformation to the normal cone (groupoid). By utilizing this groupoid, we can construct the analytic index of pseudo-differential operators without relying on pseudo-differential calculus.

Furthermore, through the canonical construction of the space of functions with Schwartz decay, pseudo-differential operators on a manifold can be represented as an integral associated with smooth functions on the deformation to the normal cone. This perspective provides a geometric characterization that allows for the direct proof of fundamental properties of pseudo-differential operators.