Past

Gotzmann's regularity and persistence theorems provide tools which allow us to find explicit equations for the Hilbert scheme Hilb_P(P^n). A natural next step is to generalise these results to the multigraded Hilbert scheme Hilb_P(X) of a smooth projective toric variety X. In 2003 Maclagan and Smith generalise Gotzmann's regularity theorem to this case. We present new persistence type results for the product of two projective spaces, and time permitting discuss how these may be applied to a more general smooth projective toric variety.

One can get fairly good estimates for primes in short
intervals under the assumption of the Riemann Hypothesis. Weaker
estimates can be shown unconditionally by using a 'zero density
estimate' in place of the Riemann Hypothesis. These zero density
estimates are typically proven by bounding how often a Dirichlet
polynomial can take large values, but have been limited by our
understanding of the number of zeros with real part 3/4. We introduce a
new method to prove large value estimates for Dirichlet polynomials,
which improves on previous estimates near the 3/4 line.

This is joint work (still in progress) with Larry Guth.

Joint work Marta Betcke and Bolin Pan

In photoacoustic tomography (PAT) with flat sensor, we routinely encounter two types of limited data. The first is due to using a finite sensor and is especially perceptible if the region of interest is large relatively to the sensor or located farther away from the sensor. In this talk we focus on the second type caused by a varying sensitivity of the sensor to the incoming wavefront direction which can be modelled as binary i.e. by a cone of sensitivity. Such visibility conditions result, in Fourier domain, in a restriction of the data to a bowtie, akin to the one corresponding to the range of the forward operator but further narrowed according to the angle of sensitivity. 

We show how we can separate the visible and invisible wavefront directions in PAT image and data using a directional frame like Curvelets, and how such decomposition allows for decoupling of the reconstruction involving application of expensive forward/adjoint solvers from the training problem. We present fast and stable approximate Fourier domain forward and adjoint operators for reconstruction of the visible coefficients for such limited angle problem and a tailored UNet matching both the multi-scale Curvelet decomposition and the partition into the visible/invisible directions for learning the invisible coefficients from a training set of similar data.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Liquid crystal elastomers are rubbery solids containing molecular LC rods that align along a common director. On heating, the alignment is disrupted, leading to a substantial (~50%) contraction along the director.  In recent years, there has been a great deal of interest in fabrication LCE sheets with a bespoke alignment pattern. On heating, these patterns generate  corresponding patterns of contraction that can morph a sheet into a bespoke curved surface such as a cone or face. Moreover, LCEs can also be activated by light, either photothermally or photochemically, leading to similarly large contractions. Stimulation by light also introduces an important new possibility: using spatio-temporal patterns of illumination to morph a single LCE sample into a range of different surfaces. Such stimulation can enable non-reciprocal actuation for viscous swimming or pumping, and control over the whole path taken by the sheet through shape-space rather than just the final destination. In this talk, I will start by with an experimental example of a spatio-temporal pattern of illumination being used to actuate an LCE peristaltic pump. I will then introduce a second set of experiments, in which a monodomain sheet morphs first into a cone, an anti-cone and then an array of cones upon exposure to different patterns of illumination. Finally, I will then discuss the general problem of how to choose a pattern of illumination to morph a director-patterned sheet into an arbitrary surface, first analytically for axisymmetric cases, then numerically for low symmetry cases. This last study exceeds our current experimental capacity, but highlights how, with full spatio-temporal control over the stimulation magnitude, one can choreograph an LCE sheet to undergo almost any pattern of morphing.

Consider definable sets over the family of finite fields $\mathbb{F}_q$. Ax proved a quantifier-elimination result for this theory, in a reasonable geometric language. Chatzidakis, Van den Dries and Macintyre showed that to a first-order approximation, the cardinality of a definable set $X$ is definable in a very mild expansion of Ax's theory.  Can such a statement be true of the next higher order approximation, i.e. can we write $|X(\mathbb{F}_q)| = aq^{d} + bq^{d-1/2} + o(q^{d-1/2})$, with $d,a,b$ varying definably with $X$ in a tame theory?    Here $b$ must be viewed as real-valued so continuous logic is needed. I will report on joint work in progress with Will Johnson.

This talk will introduce Khovanov and Knot Floer Homology as tools for studying knots. I will then cover some applications to problems in knot theory including distinguishing embedded surfaces and how they can be used in the context of ribbon concordances. No prior knowledge of either will be necessary and lots of pictures are included.

A group is called W*-superrigid if its group von Neumann algebra completely remembers the original group. In this talk, I will present a recent joint work with Stefaan Vaes in which we generalize W*-superrigidity for groups in two directions. Firstly, we find a class of groups for which W*-superrigidity holds in the presence of a twist by an arbitrary 2-cocycle: the twisted group von Neumann algebra completely remembers both the original group and the 2-cocycle. Secondly, for the same class of groups, the superrigidity also holds up to virtual isomorphism.

If a group quasiisometrically embeds into a finite product of infinite valence trees then a number of things are implied; for example, the group will have finite Assouad-Nagata dimension and finite asymptotic dimension. An even stronger statement is that the group quasiisometrically embeds into a finite product of uniformly bounded valence trees. The research on which groups quasiisometrically embed into finite products of uniformly bounded valence trees is limited, however a notable result of Buyalo, Dranishnikov and Schroeder from 2007 proves that all hyperbolic groups do admit these quasiisometric embeddings. In a recently released preprint, I extend their result to cover groups which are relatively hyperbolic with respect to virtually abelian peripheral subgroups. 

This talk will focus on the ideas at the core of Buyalo, Dranishnikov and Schroeder’s result and the extension that I proved, and in particular I will attempt to provide a general framework for upgrading quasiisometric embeddings into infinite valence trees so that they are now quasiisometric embeddings into uniformly bounded valence trees. The central concept is called a diary which I will define. 

Let $G$ be a $d$-regular graph of growing degree on $n$ vertices. Form a random subgraph $G_p$ of $G$ by retaining edge of $G$ independently with probability $p=p(d)$. Which conditions on $G$ suffice to observe a phase transition at $p=1/d$, similar to that in the binomial random graph $G(n,p)$, or, say, in a random subgraph of the binary hypercube $Q^d$?

We argue that in the supercritical regime $p=(1+\epsilon)/d$, $\epsilon>0$ a small constant, postulating that every vertex subset $S$ of $G$ of at most $n/2$ vertices has its edge boundary at least $C|S|$, for some large enough constant $C=C(\epsilon)>0$, suffices to guarantee likely appearance of the giant component in $G_p$. Moreover, its asymptotic order is equal to that in the random graph $G(n,(1+\epsilon)/n)$, and all other components are typically much smaller.

We also give examples demonstrating tightness of our main result in several key senses.

A joint work with Sahar Diskin, Joshua Erde and Mihyun Kang.

In this talk I will describe the systematic construction of strongly interacting RG fixed points with a finite disorder strength.  Such random-field disorder is quite common in condensed matter experiment, necessitating an understanding of the effects of this disorder on the properties of such fixed points. In the past, such disordered fixed points were accessed using e.g. epsilon expansions in perturbative quantum field theory, using the replica method to treat disorder.  I will show that holography gives an alternative picture for RG flows towards disordered fixed points.  In holography, spatially inhomogeneous disorder corresponds to inhomogeneous boundary conditions for an asymptotically-AdS spacetime, and the RG flow of the disorder strength is captured by the solution to the Einstein-matter equations. Using this construction, we have found analytically-controlled RG fixed points with a finite disorder strength.  Our construction accounts for, and explains, subtle non-perturbative geometric effects that had previously been missed.  Our predictions are consistent with conformal perturbation theory when studying disordered holographic CFTs, but the method generalizes and gives new models of disordered metallic quantum criticality.

We consider a quantum field model with exponential interactions on the two-dimensional torus,  which is called the ${¥rm{exp}(¥Phi)_{2}$-quantum field model or Hoegh-Krohn’s model. In this talk, we discuss the stochastic quantization of this model. Combining key properties of Gaussian multiplicative chaos with a method for singular SPDEs, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full $L_{1}$-regime $¥vert ¥alpha ¥vert<{¥sqrt{8¥pi}}$ of the charge parameter $¥alpha$. We also identify the solution with an infinite dimensional diffusion process constructed by the Dirichlet form approach. 

The main part of this talk is based on joint work with Masato Hoshino (Osaka University) and  Seiichiro Kusuoka (Kyoto University), and the full paper can be found on https://link.springer.com/article/10.1007/s00440-022-01126-z

We present a general concept that is suitable for studying the stability of equilibria for open systems in continuum thermodynamics. We apply such concept to a generalized Newtonian incompressible heat conducting fluid with prescribed nonuniform temperature on the boundary and with the no-slip boundary conditions for the velocity in three dimensional domain. For large class of constitutive relation for the Cauchy stress, we identify a class of proper solutions converging to the equilibria exponentially in a suitable metric and independently of the distance to equilibria at the initial time. Consequently, the equilibrium is nonlinearly stable and attracts all weak solutions from that class. The proper solutions exist and satisfy entropy (in)equality.

In 1735, Euler observed that $ζ(2) = 1 + \frac{1}{2²} + \frac{1}{3²} + ⋯ = \frac{π²}{6}$. This is related to the famous identity $ζ(−1) "=" 1 + 2 + 3 + ⋯ "=" \frac{−1}{12}$. In general, values of the Riemann zeta function at positive even integers are equal to rational numbers multiplied by a power of $π$. The values at positive odd integers are much more mysterious; for example, Apéry proved that $ζ(3) = 1 + \frac{1}{2³} + \frac{1}{3³} + ⋯$ is irrational, but we still don't know if $ζ(5) = 1 + \frac{1}{2⁵} + \frac{1}{3⁵} + ⋯$ is rational or not! In this talk, we will explain the arithmetic significance of these values, their generalizations to Dirichlet/Dedekind L−functions, and to L−functions of elliptic curves. We will also present a new formula for $ζ(3) = 1 + \frac{1}{2³} + \frac{1}{3³} + ...$ in terms of higher algebraic cycles which came out of an ongoing project with Lambert A'Campo.

I will talk about link and three-manifold invariants defined in terms of a non-semisimple finite ribbon category C together with a choice of tensor ideal and a trace on it. If the ideal is all of C, these invariants agree with those defined by Lyubashenko in the 90’s, and as we show, they only depend on the Grothendieck class of the objects labelling the link. These invariants are therefore not able to determine non-split extensions, or they are algebraically weak. However, we observed an interesting phenomenon: if one chooses an intermediate proper ideal between C and the minimal ideal of projective objects, the invariants become algebraically much stronger because they do distinguish non-trivial extensions. This is demonstrated in the case of C being the super-modular category of an exterior algebra. That is why these invariants deserve to be called “non-semisimple”. This is a joint work with J. Berger and I. Runkel.

The relationship between the topological or homotopy-invariant properties of a symplectic manifold X and the set of possible immersed or embedded Lagrangian submanifolds of X is rich and mostly mysterious.  In 2020, D. Alvarez-Gavela, Y. Eliashberg, and D. Nadler proved that any Weinstein manifold (e.g. an affine variety) admitting a Lagrangian plane field retracts onto a Lagrangian submanifold with arboreal singularities (a certain class of singularities which can be described combinatorially). I will discuss work in progress with D. Alvarez-Gavela and T. Large investigating the other direction, in which we prove a partial converse to the AGEN result and show that most Weinstein manifolds do not admit such skeleta. This suggests that the Floer-theoretic invariants of some well-known open symplectic manifolds may be more complicated than expected.

There will be free pizza provided for all attendees directly after the event just outside L1, so please do come along!

North Wing
Speaker: Alexandru Pascadi 
Title: Points on modular hyperbolas and sums of Kloosterman sums
Abstract: Given a positive integer c, how many integer points (x, y) with xy = 1 (mod c) can we find in a small box? The dual of this problem concerns bounding certain exponential sums, which show up in methods from the spectral theory of automorphic forms. We'll explore how a simple combinatorial trick of Cilleruelo-Garaev leads to good bounds for these sums; following recent work of the speaker, this ultimately has consequences about multiple problems in analytic number theory (such as counting primes in arithmetic progressions to large moduli, and studying the greatest prime factors of quadratic polynomials).

South Wing
Speaker: Tim LaRock
Title: Encapsulation Structure and Dynamics in Hypergraphs
Abstract: Within the field of Network Science, hypergraphs are a powerful modelling framework used to represent systems where interactions may involve an arbitrary number of agents, rather than exactly two agents at a time as in traditional network models. As part of a recent push to understand the structure of these group interactions, in this talk we will explore the extent to which smaller hyperedges are subsets of larger hyperedges in real-world and synthetic hypergraphs, a property that we call encapsulation. Building on the concept of line graphs, we develop measures to quantify the relations existing between hyperedges of different sizes and, as a byproduct, the compatibility of the data with a simplicial complex representation–whose encapsulation would be maximum. Finally, we will turn to the impact of the observed structural patterns on diffusive dynamics, focusing on a variant of threshold models, called encapsulation dynamics, and demonstrate that non-random patterns can accelerate spreading through the system.

The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) on singular data sets. However, unlike in the non-singular scenario, the homotopy type (and consequently homology) are rather course invariants of singular spaces, even in low dimension. This suggests the use of finer invariants of singular spaces for TDA, making use of stratified homotopy theory instead of classical homotopy theory.
After an introduction to stratified homotopy theory, I will describe the construction of a persistent stratified homotopy type obtained from a sample with two strata. This construction behaves much like its non-stratified counterpart (the Cech complex) and exhibits many properties (such as stability, and inference results) necessary for an application in TDA.
Since the persistent stratified homotopy type relies on an already stratified point-cloud, I will also discuss the question of stratification learning and present a convergence result which allows one to approximately recover the stratifications of a larger class of two-strata stratified spaces from sufficiently close non-stratified samples. In total, these results combine to a sampling theorem guaranteeing the (approximate) inference of (persistent) stratified homotopy types from non-stratified samples for many examples of stratified spaces arising from geometrical scenarios.

Kashiwara’s theory of crystal bases provides a powerful tool for studying representations of quantum groups. Crystal bases retain much of the structural information of their corresponding representations, whilst being far more straightforward and ‘stripped-back’ objects (coloured digraphs). Their combinatorial description often enables us to obtain concrete realizations which shed light on the representations, and moreover turn challenging questions in representation theory into far more tractable problems.

After reviewing the construction and basic theory regarding quantum groups, I will introduce and motivate crystal bases as ‘nice q=0 bases’ for their representations. I shall then present (in both finite and affine types) the construction of Young wall models in the important case of highest weight representations. Time permitting, I will finish by discussing some applications across algebra and geometry.