## Spring 2017 Seminars and Abstracts

Seminars take place on Monday afternoons 14:00 - 15:00. Everyone is welcome.

### Talks

- January 25,
**WEDNESDAY**,**16:00,**(SCI 3.05), Joint Pure and Applied Mathematics Research Seminar,**Razvan Gurau (Ecole****Polytechnique, Palaiseau),**Invitation to Random Tensors - January 27,
**FRIDAY, 15:00,**(EFRY 1.01),**Dag Madsen (Nord University),**Filtrations in Abelian Categories Determined by a Tilting Object. - January 30, (EFRY 1.01),
**Mathew Pugh (Cardiff University),**Jacobi Algebras Associated to Modular Tensor Categories. - February 13, (EFRY 1.01),
**Andrew Brooke-Taylor (University of Leeds),**Products of CW - February 20, (JSC 2.02),
**Vahagn Aslanyan (University of Oxford),**Ax-Schanuel and Existential Closedness for the J-Function. - March 6, (JSC 2.02),
**Philip Welch (University of Bristol),**Games of perfect information and generalized computation. - March 13, (JSC 2.02),
**PhD Student Talks: Jack Keeler,**Forced Solitary Waves with Algebraic Decay at Critical Froude Number**and Gabriele Bocca**, Mutation of Cyclic Quivers and Cutting Sets. - March 20, (JSC 2.02),
**Sebastien Eterovic (University of Oxford),**A Schanuel Property for j - April 24, (JSC 2.03),
**, Hayder Salman (UEA),**From Differential Geometry to the Phase Field in Quantum Mechanics. - May 8, (LT3), Joint Pure and Applied Colloquium Research Seminar,
**Alan Sokal (University College London and New York University),**Total Positivity: A Concept at the Interface Between Algebra, Analysis and Combinatorics.

### Abstracts

**Mathew Pugh: **A modular tensor category $C$ encodes the topological structure underlying a rational 2-dimensional conformal field theory, whilst module categories over $C$ yield boundary conditions. One class of examples is given by the Wess-Zumino-Witten models whose ingredients are a simple finite-dimensional Lie group and a non-negative integer. In this talk I will describe the classification of (finite semisimple) module categories for certain Lie groups of small rank, and the construction of Jacobi (or superpotential) algebras associated to these categories. In the case of $SU(2)$ the module categories are classified by the Dynkin diagrams and the associated Jacobi algebras are the preprojective algebras for Dynkin quivers.

**Razvan Gurau:** Random matrices are ubiquitous in modern theoretical physics and provide insights on a wealth of phenomena, from the spectra of heavy nuclei to the theory of strong interactions or random two dimensional surfaces. The backbone of all the analytical results in matrix models is their 1/N expansion (where N is the size of the matrix). Despite early attempts in the '90, the generalization of this 1/N expansion to higher dimensional random tensor models has proven very challenging. This changed with the discovery of the 1/N expansion (originally for colored and subsequently for arbitrary invariant) tensor models in 2010. I this talk I will present a short introduction to the modern theory of random tensors and its connections to random higher dimensional geometry.

**Dag Madsen (Nord University):** A tilting object of projective dimension one in an abelian category determines a torsion pair and consequently every object has a two-step filtration. In this talk we will discuss a generalization to the case when the tilting object has arbitrary finite projective dimension. In particular we will show that if the projective dimension is two, there is a unique way to define extension-closed subcategories such that every object has a three-step filtration with the right properties. This is joint work with Bernt Tore Jensen and Xiuping Su.

**Andrew Brooke-Taylor: **CW complexes are the topological spaces of choice for algebraic topologists. It has been known for a long time, however, that problems can arise when taking their products - if they have uncountably many cells, the product of two CW complexes need not be a CW complex. In the 1980's some results were obtained under the assumption of various set-theoretic hypotheses giving characterisations of when the product of CW complexes is a CW complex. I will present a complete characterisation, valid in any universe of set theory, of when the product of CW complexes is a CW complex. Moreover, no background knowledge of set theory will be required of the audience, for either the characterisation itself or the proof.

**Vahagn Aslanyan:** I will discuss the Ax-Schanuel inequality for the j-function proved recently by Pila and Tsimerman. I will also formulate a matching ``existential closedness'' (EC) statement which tells us which differential equations (in terms of the equation of j) do have a solution. Further, I will make a conjecture stating that this EC property holds in differential fields, and discuss the difficulties that I encountered trying to prove it. The work is motivated by Schanuel's conjecture and Zilber's approach to it, and Kirby's work on exponential differential equation and the Ax-Schanuel theorem in that setting.

**Philip Welch: **Two person perfect information games played into a set A in Baire or Cantor space have been known for a long while to be determined (that is one of the players has a winning strategy for forcing the infinite sequence of plays to lie in A, or to lie out of A) for simple A using only analytical methods (that is for A open, or G_delta, or ...). Work of Kleene, Moschovakis and others showed that certain kinds of generalised Turing computation leads to computations for strategies for open games. This talk indicates how this work can be raised to the G_{\delta,\sigma} level using a Kleenean approach to higher type recursion which employs the so-called `infinite time Turing machines'. In the latter kind of 'machine', computations are allowed to take transfinitely many steps. No prior knowledge of such machines, or very technical results from determinacy will be required.

**Jack Keeler:** Flow over bottom topography at critical Froude number (when $F=1$) is examined with an emphasis on steady, forced solitary wave solutions with algebraic decay in the far-field. We focus on the case of a negative Gaussian topography representing a smooth bottom dip (steady solutions for critical flow are not possible for positive-definite topographies). We study the flow in the weakly-nonlinear limit by way of the forced KdV equation and use a combination of numerical and asymptotic methods to probe the steady solution space, and to conduct time-dependent simulations.

For large amplitude negative Gaussian forcing, we construct an asymptotic solution using boundary layer theory; one point of interest here is an internal layer away from the origin which mediates a change from exponential decay away from the central dip to algebraic decay in the far-field. Intriguingly solutions with different numbers of waves trapped around the central dip are also found for large amplitude topography but these cannot be captured by the boundary-layer analysis. In fact a seemingly infinite sequence of solution branches is uncovered using numerical methods, and in general the solution for any given topography amplitude is non-unique.

**Gabriele Bocca:** Path algebras of quivers with relations are very important in Representation Theory since they provide an easy description of a wide class of finite dimensional algebras. For some particular classes of path algebras it is possible to define rules to "mutate" a quiver into one other such that the corresponding algebras share some useful algebraic properties. In my talk I will describe this procedure for a particular class of algebras that can be represented by path algebras over a cyclic quiver with relations.

**Sebastien Eterovic:** I will explain how some ideas from model theory and differential algebra can be used to obtain transcendence properties for the j-function coming from elliptic curves. The method I will present is adapted from work of M. Bays, J. Kirby and A.J. Wilkie for the exponential function.

**Hayder** **Salman:** In this talk, I will discuss three disparate and seemingly unrelated areas to help understand the properties of the phase of a quantum wave-function; namely differential geometry, fluid mechanics, and quantum mechanics. In quantum mechanics, the state of a system is described by a complex scalar field $\psi:\mathbb{R}^3\times \mathbb{R} \rightarrow \mathbb{C}$, the so-called wave-function of a system. It follows that the phase of this complex scalar field determines the subsequent time evolution of the system even though, in practice, it can not be measured directly. It is now well established that although the time development of this complex scalar field is governed by a Schr\"{o}dinger equation, an alternative fluid-mechanical interpretation exits in which the modulus and argument of $\psi$ are related to the density and velocity potential of the fluid. In classical fluids, under suitable assumptions, the velocity potential is considered to be a useful construct from which the physically more relevant velocity field can be derived. However, its interpretation as the quantum-mechanical phase elevates it to a more fundamental status. The correspondence between the quantum-mechanical and the fluid-mechanical views allows phase defects to be identified with vortex line filaments embedded in 3D space. The possibility to knot these line filaments in both optical and superfluid systems, allowing knotted optical fields as well knotted superfluid vortices to be realized, have generated further interest in understanding properties of the quantum phase.

I will show how the phase can be understood using important results from differential geometry. Starting with properties of a planar curve, I will subsequently present generalisations of these results for a curve on a manifold that relates its local geometric properties to the global topological properties of the manifold. These results culminate in a generalized form of the Gauss-Bonnet theorem. An important ingredient in the formulation we present is an application of Elie Cartan's spinors for representing the group of SO(3) rotations of a bi-vector. By combining results from these different areas, we will show how a mathematically elegant expression can be obtained for the quantum mechanical phase/velocity potential that is analogous to the Biot-Savart law that arises in vortex dynamics and electromagnetism. The formulae we will present generalises previous results cited in classic texts on the subject.

**Alan Sokal: **A matrix $M$ of real numbers is called {\em totally positive} if every minor of $M$ is nonnegative. This somewhat bizarre concept from linear algebra has surprising connections with analysis --- notably polynomials and entire functions with real zeros, and the classical moment problem and continued fractions --- as well as combinatorics. I will explain briefly some of these connections, and then introduce a generalization: a matrix $M$ of polynomials (in some set of indeterminates) will be called {\em coefficientwise totally positive} if every minor of $M$ is a polynomial with nonnegative coefficients. Also, a sequence $(a_n)_{n \ge 0}$ of real numbers (or polynomials) will be called {\em (coefficientwise) Hankel-totally positive}\/ if the Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$ associated to $(a_n)$ is (coefficientwise) totally positive. It turns out that many sequences of polynomials arising in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive; in some cases this can be proven using continued fractions, while in other cases it remains a conjecture.

Total Positivity Seminar Slide (PDF, 203 KB)

## Autumn 2016 Seminars and Abstracts

Seminars take place on Monday afternoons 14:00 - 15:00. Everyone is welcome.

### Talks

- October 10, (SCI 3.05),
**Chris Le Sueur (UEA),**Infinite Games: Infinite Fun! - October 17, (SCI 3.05),
**James East (University of Western Sydney),**Idempotents in Planar Diagram Monoids. - October 24, (SCI 0.31),
**Dr Lorna Gregory (Manchester),**Interpretation Functors, Representation Type and Decidability. - November 7, (C.HALL 01.19)
**Wolfram Bentz (Hull),**The Rank of Partition-Stabilising Monoids of Transformations. - November 14, (ARTS 2.02),
**Alex Zalesski (UEA),**Invariants of Maximal Tori and Individual Elements of Chevalley Groups in Their Representations. - November 21, (SCI 3.05),
**Shaun Stevens (UEA),**Propagating results in complex representation theory via the modular theory. - November 28, (LT4),
**Beth Romano (Cambridge),**Representations of p-adic groups via geometric invariant theory. - December 5, (LT4),
**Johannes Siemons (UEA),**On the shape of permutations in a finite primitive permutation group. - December 12,
**Starting at 15:00,**(SCI 3.05),**Professor Shahn Majid (QMUL),**Hodge operator on quantum groups and braided Fourier transform.

### Abstracts

**Chris Le Sueur: **Infinite games arose out of the study of game theory in the 50s and grew into the field of Determinacy in Set Theory, where it is now one of the most important topics. I will give a "suitable for non-set-theorists" introduction to the area, culminating in a taste of the connections with large cardinals. I'll try to assume as little prior knowledge as possible, though if you can remember what ordinals and cardinals are, that will help!

**James East: **I will discuss recent joint projects with Igor Dolinka, Des FitzGerald, Robert Gray, James Mitchell, and others, on planar diagram monoids. These include families such as Temperley-Lieb monoids, Kauffman monoids and Motzkin monoids. Questions considered include: What are the idempotents of these monoids? How many idempotents are there? What is the idempotent-generated submonoid? What it the minimal number of idempotents required to generate these submonoids?

**Lorna Gregory:** In this talk I will give an overview of recent developments in the model theory of representations of finite-dimensional k-algebras.

These results will connect the representation type of a finite-dimensional algebra with the decidability of its theory of modules.

The theory of modules of a ring R is said to be decidable if there exists an algorithm which given a first order sentence in the language R-modules answers whether this sentence is true of all R-modules or not.

In order investigate the relationship between the representation type of a finite-dimensional algebra with the decidability of its theories of modules, we use interpretation functors which are an additive version of the model theoretic notion of interpretation. Algebraically, they are characterised as additive functors between module categories which commute with direct limits and products.

The representation type of a finite-dimensional k-algebra is a measure of how hard it is to classify its finite-dimensional indecomposable modules. Roughly, a finite-dimensional k-algebra is of wild representation type if classifying its finite-dimensional indecomposable modules is as hard as classifying those of the polynomial ring in two non-commuting variables. On the other hand, a finite-dimensional algebra is tame if for every dimension d, all but finitely many of the finite-dimensional indecomposable modules of dimension d are in finitely many 1-parameter families. According to Drozd, when k is algebraically closed, a finite-dimensional k-algebra is either tame or wild.

All terminology coming from model theory will be explained, as will the definitions of tame and wild representation type.

**Wolfram Bentz:** Let $\mathcal{P}$ be a partition of a finite set $X$. We say that a transformation $f:X\to X$ stabilises the partition $\mathcal{P}$ if for all $P\in \mathcal{P}$ there exists $Q\in \mathcal{P}$ such that $Pf\subseteq Q$. Let $T(X,\mathcal{P})$ denote the monoid of all full transformations of $X$ that preserve the partition $\mathcal{P}$.

In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, he conjectured that his bound was exact. In 2009, Ara\’ujo and Schneider used representation theory to confirm this conjecture.

A more general task is to find the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is an arbitrary partition. In this talk we presents the solution of this problem and discuss some of the proof techniques, which range from representation theory to combinatorial arguments

**Alex Zalesski**: I shall concentrate on the group $SLn(q)$. Let $T$ be a maximal abelian irreducible subgroup of $G$. This is known to be cyclic, so let $T = \langle t \rangle$. One of problems of interest is to determine the eigenvalue 1 multiplicities of t in irreducible representations of $G$, or at least to decide when this is non-zero. The problem nature depends on the characteristic $p$ of the representation field. I shall report some new results for $p = 0$ and for $p$ dividing $q$.

**Shaun Stevens: **The Jacquet—Langlands correspondence is a bijection between so-called “discrete series” irreducible complex representations of two p-adic groups. Among these discrete series representations, the basic ones, which are building blocks, are called “cuspidal”. To a certain extent, the Jacquet—Langlands can be described explicitly when two cuspidal representations correspond. I will try to explain what these words mean and how techniques from modular representation theory can be used in a surprising way to extend these results to arbitrary discrete series representations, even though the representations we are interested in are all complex! This is joint work with Vincent Sécherre.

**Beth Romano: **The structure of reductive p-adic groups arises from the interaction of Euclidean geometry and the arithmetic of p-adic fields. In recent work Reeder and Yu draw upon this interaction to construct “epipelagic” supercuspidal representations. A p-adic group G is associated to a tiling of a certain affine space A. Each point x in A determines a group G_x and an F_p-representation V_x of G_x. Given a stable linear functional on V_x, the recipe of Reeder–Yu produces a finite set of supercuspidal representations. For small p, these would be new representations of G which are especially interesting for the local Langlands correspondence. In joint work, Jessica Fintzen and I have classified those points x in A such that V_x has stable linear functionals; as a corollary the construction of Reeder–Yu gives new representations when p is small. In my talk, I will give an overview of this work, as well as explicit examples for the case when G = G_2, the automorphism group of the octonions. For these examples, I will explicitly describe the locus of all stable vectors, as well as the Langlands parameters which correspond under the local Langlands correspondence to the representations of G.

**Johannes Siemons:** Let G be a permutation group on a finite set X, and let g be an element in G. Now write g as a product of distinct cycles and denote the length of these cycles by n_1, n_2,..,n_t. We may think of this information as the shape of g. It contains essential information about g, and even about G itself. For instance, if g has order k, one may conclude - naively - that one of the cycles of g has length k. Of course, this is not true, the permutation (1,2,3)(4,5) has order 6.

Nevertheless, and this should surprise you, under the right additional assumption the naive conclusion does hold true: For all but two types of finite primitive permutation groups G each element g in G has a cycle whose length is the order of g. This is work of P Spiga, C Praeger, M Giudici, L Emmett, A Zalesskii and myself, some of it going back to 2002 but with renewed interest and new results by the first three authors.

A large part of the talk will be elementary. I will explain why elements with long cycles are interesting, what primitive means and what types of primitive groups there are. There are still open conjectures, and we give some ideas about proofs. Finally we mention consequences for recognition problems for finite simple groups.