Past

Invariance under selected transformations has recently proven to be a powerful inductive bias in several machine learning models. One class of such models are tensor train networks. In this talk, we impose invariance relations on tensor train networks. We introduce a new numerical algorithm to construct a basis of tensors that are invariant under the action of normal matrix representations of an arbitrary discrete group. This method can be up to several orders of magnitude faster than previous approaches. The group-invariant tensors are then combined into a group-invariant tensor train network, which can be used as a supervised machine learning model. We applied this model to a protein binding classification problem, taking into account problem-specific invariances, and obtained prediction accuracy in line with state-of-the-art invariant deep learning approaches. This is joint work with Brent Sprangers.

We argue that the existence of solitons in theories in which local symmetries are spontaneously broken requires spacetime to be enlarged by additional coordinates that are associated with large local transformations. In the context of gravity theories the usual coordinates of spacetime can be thought of arising in this way. E theory automatically contains such an enlarged spacetime. We propose that spacetime appears in an underlying theory when the local symmetries are spontaneously broken.

North Wing talk: Dr Thomas Karam
Title: Ranges control degree ranks of multivariate polynomials on finite prime fields.

Abstract: Let $p$ be a prime. It has been known since work of Green and Tao (2007) that if a polynomial $P:\mathbb{F}_p^n \mapsto \mathbb{F}_p$ with degree $2 \le d \le p-1$ is not approximately equidistributed, then it can be expressed as a function of a bounded number of polynomials each with degree at most $d-1$. Since then, this result has been refined in several directions. We will explain how this kind of statement may be used to deduce an analogue where both the assumption and the conclusion are strengthened: if for some $1 \le t < d$ the image $P(\mathbb{F}_p^n)$ does not contain the image of a non-constant one-variable polynomial with degree at most $t$, then we can obtain a decomposition of $P$ in terms of a bounded number of polynomials each with degree at most $\lfloor d/(t+1) \rfloor$. We will also discuss the case where we replace the image $P(\mathbb{F}_p^n)$ by for instance $P(\{0,1\}^n)$ in the assumption.
 

South Wing talk: Dr Hamid Rahkooy
Title: Toric Varieties in Biochemical Reaction Networks

Abstract: Toric varieties are interesting objects for algebraic geometers as they have many properties. On the other hand, toric varieties appear in many applications. In particular, dynamics of many biochemical reactions lead to toric varieties. In this talk we discuss how to test toricity algorithmically, using computational algebra methods, e.g., Gröbner bases and quantifier elimination. We show experiments on real world models of reaction networks and observe that many biochemical reactions have toric steady states. We discuss complexity bounds and how to improve computations in certain cases.

When a biological system is modelled using a mathematical process, the next step is often to estimate the system parameters. Although computational and statistical techniques have been developed to estimate parameters for complex systems, this can be a difficult task. As a result, researchers have focused on revealing parameter-independent dynamical properties of a system. In this talk, we will discuss the study of qualitative behaviors of stochastic biochemical systems using reaction networks, which are graphical configurations of biochemical systems. The goal of this talk is to (1) introduce the basic modelling aspects of stochastically-modelled reaction networks and (2) discuss important results in this literature, including the random time representation, relationships between stochastic and deterministic models, and derivation of stability via network structures.

A complex irreducible admissible representation of a reductive p-adic group is typically infinite-dimensional. To quantify the "size" of such representations, we introduce the concept of canonical dimension. To do so we have to discuss the Moy-Prasad filtrations. These are natural filtrations of the parahoric subgroups. Next, we relate the canonical dimension to the Harish-Chandra local character expansion, which expresses the distribution character of an irreducible representation in terms of nilpotent orbital integrals. Using this, we consider the wavefront set of a representation. This is an invariant the naturally arises from the local character expansion. We conclude by explaining why the canonical dimension might be considered a weaker but more computable alternative to the wavefront set.

(joint with Chernikov) 1-dependent theories, better known as NIP theories, are the first class of the strict hierarchy of n-dependent theories. The random n-hypergraph is the canonical object which is n-dependent but not (n−1)-dependent. We proved the existence of strictly n-dependent groups for all natural numbers n. On the other hand, there are no known examples of strictly n-dependent fields and we conjecture that there aren’t any. 

We were interested which properties of groups and fields for NIP theories remain true in or can be generalized to the n-dependent context. A crucial fact about (type-)definable groups in NIP theories is the absoluteness of their connected components. Our first aim is to give examples of n-dependent groups and discuss a adapted version of absoluteness of the connected component. Secondly, we will review the known properties of NIP fields and see how they can be generalized.

The elliptic Gamma function — a generalization of the q-Gamma function, which is itself the q-analog of the ordinary Gamma function — is a meromorphic special function in several variables that mathematical physicists have shown to satisfy modular functional equations under SL(3,Z). In this talk I will present evidence (numerical and theoretical) that this function often takes algebraic values that satisfy explicit reciprocity laws and that are related to derivatives of Hecke L-functions at s=0. Thus this function conjecturally allows to extend the theory of complex multiplication to complex cubic fields as envisioned by Hilbert's 12^th problem. This is joint work with Nicolas Bergeron and Pierre Charollois.

Tensor decomposition express a tensor as a linear combination of elementary tensors. They have applications in chemometrics, computer science, machine learning, psychometrics, and signal processing. Their uniqueness properties render them suitable for data analysis tasks in which the elementary tensors are the quantities of interest. However, in applications, the idealized mathematical model is corrupted by measurement errors. For a robust interpretation of the data, it is therefore imperative to quantify how sensitive these elementary tensors are to perturbations of the whole tensor. I will give an overview of recent results on the condition number of tensor decompositions, established with my collaborators C. Beltran, P. Breiding, and N. Dewaele.

Metal forming involves permanently deforming metal into a required shape.  Many forms of metal forming are used in industry: rolling, stamping, pressing, drawing, etc; for example, 99% of steel produced globally is first rolled before any subsequent processing.  Most theoretical studies of metal forming use Finite Elements, which is not fast enough for real-time control of metal forming processes, and gives little extra insight.  As an example of how little is known, it is currently unknown whether a sheet of metal that is squashed between a large and a small roller should curve towards the larger roller, or towards the smaller roller.  In this talk, I will give a brief overview of metal forming, and then some progress my group have been making on some very simplified models of cold sheet rolling in particular.  The mathematics involved will include some modelling and asymptotics, multiple scales, and possibly a matrix Wiener-Hopf problem if time permits.

PhD_course_Roux_1.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

We will be joined by Philip Maini, Professor of Mathematical Biology and Ethnic Minorities Fellow at St John's College, to discuss his mathematical journey and experiences.

In the critical beta-splitting model of a random $n$-leaf rooted tree, clades (subtrees) are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$. This model turns out to have interesting properties. There is a canonical embedding into a continuous-time model ($\operatorname{CTCS}(n)$). There is an inductive construction of $\operatorname{CTCS}(n)$ as $n$ increases, analogous to the stick-breaking constructions of the uniform random tree and its limit continuum random tree. We study the heights of leaves and the limit fringe distribution relative to a random leaf. In addition to familiar probabilistic methods, there are analytic methods (developed by co-author Boris Pittel), based on explicit recurrences, which often give more precise results. So this model provides an interesting concrete setting in which to compare and contrast these methods. Many open problems remain.
Preprints at https://arxiv.org/abs/2302.05066 and https://arxiv.org/abs/2303.02529

More than 30 year ago Jerrum studied the planted clique problem and proved that under worst-case initialization Metropolis fails to recover planted cliques of size $\ll n^{1/2}$ in the Erdős-Rényi graph $G(n,1/2)$. This result is classically cited in the literature of the problem, as the "first evidence" that finding planted cliques of size much smaller than square root $n$ is "algorithmically hard". Cliques of size $\gg n^{1/2}$ are easy to find using simple algorithms. In a recent work we show that the Metropolis process actually fails to find planted cliques under worst-case initialization for cliques up to size almost linear in $n$. Thus the algorithm fails well beyond the $\sqrt{n}$ "conjectured algorithmic threshold". We also prove that, for a large parameter regime, that the Metropolis process fails also under "natural initialization". Our results resolve some open questions posed by Jerrum in 1992. Based on joint work with Zongchen Chen and Iias Zadik.

Generating n-tuples of a group G, or in other words epimorphisms Fₙ→G are usually studied up to the natural right action of Aut(Fₙ) on Epi(Fₙ,G); here Fₙ is the free group of n generators. The orbits are then called Nielsen classes. It is a classic result of Nielsen that for any n ≥ k there is exactly one Nielsen class of generating n-tuples of Fₖ. This result was generalized to surface groups by Louder.

In this talk the case of Fuchsian groups is discussed. It turns out that the situation is much more involved and interesting. While uniqueness does not hold in general one can show that each class is represented by some unique geometric object called an "almost orbifold covers". This can be thought of as a classification of Nielsen classes. This is joint work with Ederson Dutra.

PhD_course_Roux_0.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

For a reductive group defined over a p-adic field, the wavefront set is an invariant of an admissible representations which roughly speaking measures the direction of the singularities of the character near the identity. Studied first by Roger Howe in the 70s, the wavefront set has important connections to Arthur packets, and has been the subject of thorough investigation in the intervening years. One of main lines of inquiry is to determine the relation between the wavefront set and the L-parameter of a representation. In this talk we present new results answering this question for unipotent representations with real infinitesimal character. The results are joint with Dan Ciubotaru and Lucas Mason-Brown.

A self-similar group is a group $G$ acting on a regular, infinite rooted tree by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups, like Grigorchuk's 2-group of intermediate growth, are of this form. Nekrashevych associated $C^*$-algebras and algebras with coefficients in a field to self-similar groups. In the case $G$ is trivial, the algebra is the classical Leavitt algebra, a famous finitely presented simple algebra. Nekrashevych showed that the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims, and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example). The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contacting groups are finitely presented.

In this talk, we discuss a result of the speaker and Benjamin Steinberg characterizing simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm for deciding simplicity given an automaton generating the group. We apply our results to several families of contracting groups like GGS groups and Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.

TBA

We talk about the Hausdorff measure of level sets of the fields, say length of level lines of a planar field. Given two coupled stationary fields  $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F=f_1-f_2$. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using the Kac-Rice type formula as the main tool in the analysis. 

General Relativity and Quantum Theory are the two main achievements of physics in the 20th century. Even though they have greatly enlarged the physical understanding of our universe, there are still situations which are completely inaccessible to us, most notably the Big Bang and the inside of black holes: These circumstances require a theory of Quantum Gravity — the unification of General Relativity with Quantum Theory. The most natural approach for that would be the application of the astonishingly successful methods of perturbative Quantum Field Theory to the graviton field, defined as the deviation of the metric with respect to a fixed background metric. Unfortunately, this approach seemed impossible due to the non-renormalizable nature of General Relativity. In this talk, I aim to give a pedagogical introduction to this topic, in particular to the Lagrange density, the Feynman graph expansion and the renormalization problem of their associated Feynman integrals. Finally, I will explain how this renormalization problem could be overcome by an infinite tower of gravitational Ward identities, as was established in my dissertation and the articles it is based upon, cf. arXiv:2210.17510 [hep-th].

Reaction–diffusion equations are widely used to model spatial propagation, and constant-speed "traveling waves" play a central role in their dynamics. These waves are well understood in "essentially 1D" domains like cylinders, but much less is known about waves with noncompact transverse structure. In this direction, we will consider traveling waves of the KPP reaction–diffusion equation in the Dirichlet half-space. We will see that minimal-speed waves are unique (unlike faster waves) and exhibit curious asymptotics. The arguments rest on potential theory, the maximum principle, and a powerful connection with the probabilistic system known as branching Brownian motion.

This is joint work with Julien Berestycki, Yujin H. Kim, and Bastien Mallein.

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.