## Spring 2015 Spring 2015

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

### Talks

• January 26, (ARTS 01.01), Mark Blyth (UEA), Axisymmetric Solitary Waves on a Ferrofluid Jet.
• February 2 (JSC 0.01), Joseph Grant (UEA),  Braid Groups and Quiver Mutation
• February 16, (ARTS 2.02), Jacob Hilton (Leeds), The Topological Pigeonhole Principle for Ordinals.
• February 23, (ITCS 01.26), Hans Jakob Rivertz (Visiting Professor UEA and Sørtrøndelag University College), Self-Similarity of Dihedral Tilings.
• March 2, (ARTS 2.01), Kaisa Kangas (Helsinki), From Strongly Minimal Sets to Zariski-like Structures.
• March 9, (ARTS 2.01), Alexei Vernitski (Essex), Knot Semigroups: Semigroup-Theory Problems and Knot-Theory Problems
• March 16, 15:00, (EFRY 01.02), Joint PhD Research Seminars
• April 20 (LT1), Teresa Conde (Oxford), The Quashihereditary Structure of a Schur-Like Algebra.
• April 27, (SCI 3.05), András Pongrácz (Middlesex University), The Topological Monoid Structure of End(F).
• May 11, (SCI 3.05), Philip Welch (Bristol), Turing Abound: Black Holes, Neural Nets, Bangs, Shrieks, Transfinite Ordinals and the Kitchen Sink.
• May 18, (SCI 3.05), Stefanie Zegowitz (Exeter), Closed Orbits in Quotient Systems.
• June 1, (SCI 3.05), Prof Alessandro Beraducci (universita'di Pisa),
• June 8, (SCI 3.05), Aaron Chan (Uppsala University), Tilting Theory of Brauer Graph Algebras and Curves of Riemann Surfaces.
• June 8, (SCI 3.05), Jeremie Guilhot (Tours), Kazhdan-Lusztig Cells in (Affine) Weyl Group.

### Abstracts

Joseph Grant: The possible ways to braid a given number of strings give a group, known as the braid group.  Artin gave generators and relations for these groups, which can be encoded in a graph of Dynkin type A.  This presentation can be generalized to other graphs.

A quiver is a directed graph.  Mutation of quivers was introduced by Fomin and Zelevinsky as part of their theory of cluster algebras.  I will describe joint work with Robert Marsh which links presentations of braid groups with mutation of quivers, and explain how this is related to triangulations of polygons.

The main result of this talk involves no complicated tools. I will assume the audience knows what a group is but, until the last ten minutes or so, I will explain any other concepts I use.  However, the results are connected to some of the most important objects in modern mathematics and mathematical physics, as I will outline towards the end of the talk.

Jacob Hilton: The ordinary pigeonhole principle states that if you put more than n items into n containers, then one container must contain 2 or more items. The infinite pigeonhole principle states that if you put infinitely many items into finitely many containers, then one container must contain infinitely many items. What happens if you divide up a countable ordinal into finitely many containers? Must there be some container that contains a set of order type omega+1? We will study this question and see how the answer changes when we topologize the ordinals. We will also take a glance at joint work with Andrés Caicedo on Ramsey-style analogues of this question.

Hans Jakob Rivertz: Aperiodic tilings have been studied since the seventies. In 1972, Roger Penrose discovered a tessellation of the plane with pentagonal symmetry. This tessellation has two kinds of rhombic tiles. One thin tile with angle 36 degree and one fat tile with angle 72 degree. Although, this tessellation has no translational symmetries it has almost translational symmetries, i.e. translations that preserves all tiles but an arbitrary sparse set of tiles. There is also a subdivision of the tiles which give a new tessellation of the same type. A remarkable property is that the vertex set of the tessellation has a subset which is a vertex set of a tessellation with bigger tiles but of the same type as the original.

In a joint paper with J.-H. Eschenburg, we show that for any prime number $n = 2r+1\geq 5$ there exist $r$ planar tilings with self-similar vertex set and the symmetry of a regular n-gon ($D_n$-symmetry). The tiles are the rhombi with angle $\pi k/n$ for $k = 1,...,r.$

Kaisa Kangas: In the context of first order logic, strongly minimal theories provide a setting where mathematical structures can be classified based on dimension-like invariants. Assuming the existence of a Zariski-like topology, Zilber's trichotomy principle holds in this setting: a strongly minimal set is either trivial, interprets a vector space over some division ring or interprets an algebraically closed field. Finally, we look at generalizations of this setting beyond the first order context.

Teresa Conde: Given an arbitrary algebra $A$, we may associate to it a special endomorphism algebra $R_A$, introduced by Auslander, and further studied by Smalø. The algebra $R_A$ contains $A$ as an idempotent subalgebra and is quasihereditary with respect to a heredity chain constructed by Dlab and Ringel. In this talk we will discuss the nice properties of $R_A$ that stem from this heredity chain and mention some questions that arise from this setting.

Alexei Vernitski: Recently I introduced knot semigroups as a semigroup-theory construction inspired by knot groups. Then I started studying a number of questions which can be asked about knot semigroups. Here are some. Is it true that a semigroup of a knot is merely the set of 'positive' elements of its knot group? Can every knot semigroup be embedded in a group? Do all knot semigroups have a solvable word problem? Is a knot semigroup a knot invariant? Can we define semigroups of braids, links and tangles? How are knot semigroups related to Garside monoids (this is another knot-inspired semigroup construction)? How are knot semigroups related to the semigroup of knots with the operation of connected sum? How are knot semigroups related to graph semigroups?

András Pongrácz: The significance of algebraic invariants, such as the automorphism group, endomorphism monoid and polymorphism clone, in the study of countable omega-categorical structures was observed long ago, notably by Ryll-Nardzewski, who showed that omega-categoricity itself only depends on the permutation group structure of Aut(F). Later on, Ahlbrandt and Ziegler have shown that a countable omega-categorical F is uniquely determined up to first-order bi-interpretability by the topological group structure of Aut(F). Here, the topology is the one induced by pointwise convergence: closed sets are subsets of the full symmetric group Sym(F) that are closed under pointwise convergence. It was investigated in the 80's and 90's whether the condition of the Ahlbrandt-Ziegler theorem can be relaxed to the abstract group structure. We say that a group of the form Aut(F) has reconstruction if whenever it is isomorphic to another closed subgroup H of Sym(F) (as abstract group), it is also isomorphic to H as a topological group. Surprisingly, many subgroups of Sym(F) have this property, and it is in fact consistent with ZF that for all countable structures F the group Aut(F) has reconstruction.

With Manuel Bodirsky and Michael Pinsker, we initiated the study of the analogous concept of reconstruction for endomorphism monoids and polymorphism clones of omega-categorical structures. This is in particular motivated by generalizations of the Ahlbrandt-Ziegler theorem for these invariants, but also has potential applications in theoretical computer science. We have shown that the polymorphism clone of the random graph has reconstruction. In my talk, I will present the most important notions and results related to reconstruction, and briefly explain the connection of these results with theoretical computer science.

Philip Welch: There has been much talk (some of it rather loose) in the last few years concerning the possibility of "hypercomputation".  This can start out from discussions as to whether Turing's machine model is appropriate for "modern" computation, or from claims that we can - perhaps only theoretically - compute "beyond the Turing limit", that is somehow fix up a device that enables us to 'compute' the Turing halting problem. On closer inspection most of such devices are seen to be formally cheating in some way or other, usually by assuming infinite precision measurement of some kind.
We look at the mathematics of one model that at least is not cheating in that sense: these are computations in Malament-Hogarth spacetimes. We investigate as a purely logical exercise the limits of computation in such manifolds.
Discrete transfinite machine models can be defined that run through ticks of time that are transfinitely ordered. These are purely logico-mathematical models, but are of interest as the associated recursion theory for such models ties in with work of Moschovakis, Harrington, Kechris, Gandy and Normann from the 1970's in generalized recursion theories. If time permits we shall discuss these too.

Stefanie Zegowitz We study the relationship between pairs of topological dynamical systems (X, T) and (X', T') where (X',T') is the quotient of (X, T) under the action of a finite group G. We describe three phenomena concerning the behaviour of closed orbits in the quotient system, and the constraints given by these phenomena. We find upper and lower bounds for the extremal behaviour of closed orbits in the quotient system in terms of properties of G and show that any growth rate in between these bounds can be achieved. Further, we study the asymptotic behaviour of the dynamical analogue of the Prime Number Theorem and Mertens' Theorem in the context of quotient systems.

Aaron Chan A ribbon graph is a combinatorial object one could use to construct bounded Riemann surfaces; it is a tool which is used in several branches of differential geometry. One can also construct finite dimensional algebras - the Brauer graph algebras out of ribbon graphs.  Thus, one could ask the following questions. Given a homological problem of a Brauer graph algebra, what is its translation in the combinatorial theory of the defining ribbon graph? Similarly, what is its translation in the geometric theory of the associated Riemann surface? In this talk, I will describe an on-going collaboration with Takahide Adachi and Takuma Aihara, where we found that many tilting complexes of a Brauer graph algebra can be translated into nice collections of curves on the associated Riemann surface.

Jeremie Guilhot: A Coxeter group is a group generated by involutions plus some relations. The two most important kind of Coxeter groups are Weyl groups (which are finite groups generated by reflections) and affine Weyl groups  (which are infinite groups generated by affine reflections). The aim of this talk is to introduce the notion of Kazhdan-Lusztig cells in a (affine) Weyl group. In the first part, we will concentrate on a small example (the symmetric group of order 3) and try to explicitly compute some cells. In the second part, we will discuss cells in affine Weyl groups.

## Autumn 2014 Autumn 2014

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

### Talks

• October 6, (LT1), Carlos De la Mora (UEA), L-functions Here and There
• October 13, 16:00, (SCI 0.31), Alex Zalesski (UEA), Restriction of the Steinberg of Orthogonal Groups to the Orthogonal Subgroup of One Less Dimension
• October 20, (LT1), Tom Ward (Durham), Attempting a Polya-Carlson Dichotomy in Algebraic Dynamics
• October 27, 15:00, (SCI 3.05), Pure and Applied Joint PhD Maths Student Seminars, Ruari Walker - KLR Algebras and VV Algebras and Davide Maestrini - Vortex Clustering and Negative Temperature States in a 2D Bose-Einstein Condensate
• November 3, (ARTS 3.07), Keith Brown (UEA), Properly Stratified Quotients of Khovanov-Lauda-Rouquier Algebras
• November 10, (LT3), Sarah Scherotzke (Bonn), Graded Quiver Varieties and Derived Categories
• November 17, (ARTS 0.100), Baiying Liu (Utah), On Fourier Coefficients of Automorphic Forms
• November 24, (EFRY 01.08), Rob Kuriinczuk (Bristol), Rankin-Selberg Local Factors Modulo-1
• December 8, (TPSC 0.1), Faculty joint pure and applied seminar, David Aspero (MTH), Extremely Large Cardinals in the Absence of Choice
• December 15, (SCI 3.05), Nicola Gambino (Leeds), From Type Theory and Homotopy Theory to Univalent Foundations of Mathematics

### Abstracts

Carlos De la Mora: Let F be a number field and let Ḟ be the algebraic closure of F. Given ρ a representation of Gal(Ḟ/F) we can form its Artin L-function. It is known that Artin's L-functions have a meromorphic continuation to the complex plane and that they have an Euler product. Since Artin's L-functions have an Euler product we can study one prime at the time. Indeed, to each prime ideal we can attach two canonical complex functions, these are the local L-factor and the local epsilon factor. We now change gears and consider G to be a connected reductive group defined over F, A be the ring of adeles over F and let π be an automorphic representation of G(A). If V is the set of valuations of F, for v ϵ V we can complete F, we denote by Fv the completion of F with respect to v. It is known that π = ¤' v ϵ Vπv where πv is a smooth representation of G(Fv) for all v. One of the conjectures by Langlands is that to every smooth representation πv of G(Fv) one should be able to attach two canonical complex functions, these should be a local L-factor and an epsilon factor. Moreover if ρ is a representation of Gal(Ḟ/F) there should exist an automorphic cuspidal representation π of G(A) such that the local L-factor and epsilon factor of ρ and the local L-factor and epsilon factor of π agree for each valuation. The aim of this talk is to explain all the above in more detail.

Alex Zalesski: I consider the Stenberg character of the special orthogonal group over a finite field and the Steinberg character of it. The problem under discussion is to find the irreducible constituents of the restriction of this character to orthogonal group of one less dimension. The main result is a rather satisfactory solution to this problem.

Tom Ward: A subtle and beautiful classical result in complex variables is the Polya-Carlson theorem, which gives a strong rigidity to the possible behaviour of complex power series with integer coefficients. Many examples - and a few theorems - suggest a similar dichotomy may exist for a collection of functions arising in dynamical systems. I will give an overview of what is known here, and explain a wider context into which this dichotomy should fit - if it is true. Many of the arguments amount to quite concrete calculations with power series.

Ruari Walker: A new family of graded algebras have been introduced by Khovanov, Lauda and independently by Rouquier, the representation theory of which is closely related to that of the affine Hecke algebra of type A. They are often called KLR algebras. More recently, Varagnolo and Vasserot have defined a new family of graded algebras whose representation theory is related to the representation theory of the affine Hecke algebras of type B. These algebras can be thought of as type B analogues of the KLR algebras. I plan to explain this in a little more detail, show how the KLR algebras relate to the VV algebras and compare their module categories via Morita equivalence.

Davide Maestrini: In this work we investigate the question of clustering of like signed vortices in a two-dimensional Bose-Einstein condensate. Such clustering can be understood in terms of negative temperature states of a vortex gas. Due to the long-range nature of the Coloumb-like interactions in point vortex flows, these negative temperature states strongly depend on the shape of the geometry in which this clustering phenomena is considered. We analyze the problem of clustering of portices in a number of different regions. We present a theory to uncover the regimes for which clustering of like signed vortices can occur and compare our predictions with numerical simulations of a point vortex gas. We also extend our results to the Gross-Pitaevskii model of a Bose gas by performing numerical simulations for a range of vortex configurations using parameters that are relevant to current experiments.

Keith Brown: Introduced in 2008 by Khovanov and Lauda, and independently by Rouquier, the KLR algebras, are a family of infinite dimensional graded algebras which categorify the negative part of the quantum group associated to a graph~$\Gamma$. These algebras are known to have nice homological properties, in particular they are affine quasi-hereditary. In this talk I'll explain what it means to be affine quasi-hereditary and how this relates to properties of finite dimensional algebras. I'll then introduce a finite dimensional quotient of the KLR algebra which preserves some of the homological structure of the original algebra and provide a bound on its finitistic dimension. If time permits I'll explore a particular example in which the homological algebra is particularly nice. This work will form part of my PhD thesis, supervised by Dr. Vanessa Miemietz.

Sarah Scherotzke (Bonn): Nakajima's quiver varieties are important geometric objects in representation theory that can be used to give geometric constructions of quantum groups. Very recently, graded quiver varieties also found application to monoidal categorification of cluster algebras. Nakajima's original construction uses geometric invariant theory. In my talk, I will give an alternative representation theoretical definition of graded quiver varieties. I will show that the geometry of graded quiver varieties is governed by the derived category of the quiver $Q$. This approach brings about some new results on geometric properties of quiver varieties.

Baiying Liu (Utah): Fourier coefficients play an important role in the study of modular forms. In this talk, mainly focusing on general linear groups, I briefly introduce how to obtain Fourier coefficients and what kinds of Fourier coefficients people usually consider in the setting of automorphic forms. More explicitly, consider the standard maximal unipotent subgroup Un of GLn, consisting of upper triangular matrices. The question is that how to obtain Fourier coefficients of an automorphic form \phi along Un. When n = 2, Un is abelian, we know that taking abelian Fourier expansion is enough. But, when n >= 3, Un is not abelian. First, I will introduce the idea of Piatetski-Shapiro and Shalika, to obtain non-degenerate Whittaker Fourier coefficients of cuspidal automorphic forms along Un, using the property that Un is solvable. Then, I will show how to apply the same idea to obtain Fourier coefficients of automorphic forms in the residual spectrum of GLn. If time permitting, I will also briefly introduce a general setting of attaching Fourier coefficients to nilpotent orbits.

Rob Kuriinczuk (Bristol): I will briefly explain what an L-function is, how Rankin-Selberg local factors arise in the theory of automorphic L-functions, and discuss some recent work defining l-modular Rankin-Selberg local factors. This is joint work with Nadir Matringe.

David Aspero (UEA): We will stroll through the upper reaches of the large cardinal hierarchy, especially in a non-AC context.

Nicola Gambino (Leeds): Voevodsky's Univalent Foundations of Mathematics programme is an ambitious, long-term, project that seeks to develop a new approach to the foundations of mathematics on the basis of recently-discovered connections between type theory and homotopy theory. This programme aims also at facilitating the use of computer systems for the verification of mathematical proofs. The talk will consist of two parts. In the first part, I will give an introduction to the Univalent Foundations of Mathematics programme, without assuming prior familiarity with type theory or homotopy theory. In the second part, I will explain how the type-theoretic counterpart of the topological notion of contractibility can be used to characterise certain free constructions by means of a universal property that circumvents the use of infinite sets of coherence conditions.