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# Seminars in Pure Mathematics

Pure Mathematics Seminars are usually held at 2:00pm on Tuesdays in room 119 of the Michael Atiyah building, preceded by lunch at 12:30 (in the Charles Wilson building) and followed by tea and biscuits (venue to be determined).

## Actual events:

Events mathematics Web page:

## Past events:

• 2020-01-14: Raphael Bennett-Tennenhaus (University of Leeds)

Title: Purity and positive-primitive formulas in triangulated categories: with examples from (and applications to) gentle algebras.

Abstract: In mathematical logic any theory comes equipped with sentences in a fixed language, and a model of that theory is a structure satisfying these sentences. Model theoretic algebra entails studying structures which are algebraic, such as modules over a fixed ring. By Baur's famous elimination of quantifiers, any sentence in the language of modules is equivalent to a Boolean combination of `positive-primitive' (pp) formulas. This motivated the study of pure-monomorphisms: those reflecting the existence of solutions to pp-formulas with constants from the domain. Indeed, any module is elementary equivalent to a so-called pure-injective module: one which is injective with respect to pure-monomorphisms.

These ideas have been generalised beyond the scope of modules. Crawley-Boevey defined notions of purity for any locally finitely presented category, such as various functor categories. Krause then used the Yoneda embedding to inherit these definitions to compactly generated triangulated categories. Since then Garkusha and Prest introduced the appropriate multi-sorted language for these triangulated categories. Examples of such categories include the homotopy category of complexes of projective modules over a finite-dimensional algebra. In my talk I will discuss recent work [arxiv: 1911.07691] in which the gentle algebras of Assem and Skowronski were considered, and indecomposable sigma-pure-injective complexes in these homotopy categories were classified. The proof uses the aforementioned canonical language to exploit ideas and results developed in my PhD thesis.

• 2020-01-21: Daniel Labardini-Fragoso (UNAM, Mexico)

Title: Algebraic and combinatorial descomposition of Fuchsian groups.

Abstract: The discrete subgroups of PSL_2(R) are often called 'Fuchsian groups'. For Fuchsian groups \Gamma whose action on the hyperbolic plane H is free, the orbit space H/\Gamma has a canonical structure of Riemann surface with a hyperbolic metric, whereas if the action of \Gamma is not free, then H/\Gamma has a structure of 'orbifold'. In the former case, there is a direct and very clear relation between \Gamma and the fundamental group \pi_1(H/\Gamma,x): a theorem of the theory of covering spaces states that they are isomorphic. When the action of \Gamma is not free, the relation between \Gamma and \pi_1(H/\Gamma,x) is subtler. A 1968 theorem of Armstrong states that there is a short exact sequence 1->E->\Gamma->\pi_1(H/\Gamma,x)->1, where E is the subgroup of \Gamma generated by the elliptic elements. For \Gamma finitely generated, non-elementary and with at least one parabolic element, I will present full algebraic and combinatorial decompositions of \Gamma in terms of \pi_1(H/\Gamma,x) and a specific finitely generated subgroup of E, thus improving Armstrong's theorem.

This talk is based on an ongoing joint project with Sibylle Schroll and Yadira Valdivieso-Díaz that aims at describing the bounded derived categories of skew-gentle algebras in terms of curves on surfaces with orbifold points of order 2.

• 2020-01-28: Mason Pember (Politecnico di Torino, Italy)
Title: Weierstrass type representations

Abstract: In the 19th century, Weierstrass and Enneper developed a representation formula for minimal surfaces in Euclidean 3-space. This formula has been fundamental in the creation of many interesting examples of minimal surfaces. Since then several similar representations have been developed for other classes of surfaces in 3 dimensional space forms such as constant mean curvature 1 surfaces in hyperbolic 3-space. All of these representations have the same basic ingredients: a meromorphic function and a holomorphic 1-form. It is therefore natural to ask if these representations can be unified in some way. In this talk we will outline a unified geometric way of viewing these representations via the transformation theory of Omega surfaces.

• 2020-02-04: Pedro Luis del Ángel Rodríguez (CIMAT, Mexico)
Title: Quasi-Compact Group schemes and their representations

Abstract: We will start with a brief review of the classical representation theory for finite groups and recall the notion of topological group as well as the notion of group scheme and affine group scheme. We develop a representation theory for extensions of an abelian variety by an affine group scheme. We characterize the categories that arise as such a representation theory, generalizingin this way the classical theory of Tannaka Duality established for affine group schemes.

• 2020-02-11: Ana García-Elsener (UNMdP, Argentina)
Title: Frieze mutations

Abstract: Frieze is a lattice of positive integers satisfying certain rules. Friezes of type A were first studied by Conway and Coxeter in 1970’s, but they gained fresh interest in the last decade in relation to cluster algebras, that are generated via a combinatorial rule called mutation. Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of algebras. In this new theory a frieze is an array of positive integers on the Auslander-Reiten quiver of a cluster tilted algebra such that entries on a mesh satisfy a unimodular rule. In this talk, we will discuss friezes of type A and D, and their mutations

This is joint work with K. Serhiyenko (UC Berkeley).

• 2020-02-18: Double talk
• Alfredo Nájera (UNAM, Mexico)

Title: Cluster varieties and toric varieties associated to fans

Abstract: Cluster varieties are algebraic varieties over the complex numbers constructed gluing algebraic tori using distinguished birational maps called cluster transformations. In a seminal work Gross-Hacking-Keel-Kontsevich introduced the scattering diagram associated to a cluster variety. Every such diagram is a fan in a real vector space endowed with certain additional information attached to each co-dimension 1 cone (the wall crossing automorphisms). The purpose of this talk is to explain how scattering diagrams can be used to construct partial compactifications of cluster Poisson varieties and toric degenerations of such. In this talk we will provide various of the definitions needed for the next talk by T. Magee.

• Timothy Magee (University of Birmingham)

Title: Cluster varieties and toric varieties associated to polytopes

Abstract: Cluster varieties are a relatively new, broadly interesting class of geometric objects that generalize toric varieties.  In the previous talk Alfredo discussed how cluster Possion varieties partially compactify by a fan construction.  In this talk we will see that their mirrors-- cluster A-varieties-- compactify by a polytope construction.  Projective toric varieties are defined by convex lattice polytopes.  I'll explain how convexity generalizes to the cluster world, where "polytopes" live in a tropical space rather than a vector space and "convex polytopes" define projective compactifiactions of cluster varieties.  Time permitting, I'll conclude with two exciting applications of this more general notion of convexity: 1) an intrisic version of Newton-Okounkov bodies and 2) a possible cluster version of a classic toric mirror symmetry construction due to Batyrev.  Based on joint work with Man-Wai Cheung and Alfredo Nájera Chávez.

• 2020-02-25: Olga Maleva (University of Birhmingham)

Title: Sets where a typical Lipschitz function is somewhere differentiable

Abstract: This talk is devoted to the study of properties of Lipschitz functions, mainly between finite-dimensional spaces, and especially their differentiability properties. The classical Rademacher Theorem guarantees that every Lipschitz function between finite-dimensional spaces is differentiable almost everywhere. This means that every set of positive Lebesgue measure in a Euclidean space contains points of differentiability of every Lipschitz function defined on the whole space.

A major direction in geometric measure theory research of the last two decades was to explore to what extent this is true for Lebesgue null sets. Even for real-valued Lipschitz functions, there are null subsets S of R^n (with n>1) such that every Lipschitz function on R^n has points of differentiability in S; one says that S is a universal differentiability set (UDS).

Moreover, for some sets T which are not UDS, meaning that some bad Lipschitz function is nowhere differentiable in T, it may happen that a typical Lipschitz function has many points of differentiability in T. In a recent joint work with M. Dymond we give a complete characterisation of such sets: these are the sets which cannot be covered by an F-sigma 1-purely unrectifiable set. We also show that for all remaining sets a typical 1-Lipschitz function is nowhere differentiable, even directionally, at each point. In this talk I will present the latest results on UDS and the dichotomy of typical differentiability sets.

• 2020-03-03: Michael Wemyss (University of Glasgow)
Title: Tits Cone Intersections and 3-fold flops

Abstract: The first half of the talk will give a brief overview of intersection hyperplane arrangements inside Tits cones of both finite and affine Coxeter groups.  Surprisingly, little seems to be known about them, and they exhibit quite strange behaviour!  For example, we obtain precisely 16 tilings of the plane, with only 3 being the "traditional" Coxeter tilings.

These new hyperplane arrangements turn out to be quite fundamental.  Algebraically, they (1) describe the exchange graph of certain "modifying modules", which should be thought of as maximal rigid objects, (2) classify all noncommutative resolutions for cDV singularities, and (3) classify tilting theory for (contracted) preprojective algebras.  They also have many geometric consequences, but I will talk about these less. The main point, and motivation, is that these hyperplane arrangements describe the stability manifold for an arbitrary 3-fold flop X --> Spec R, where X can have terminal singularities, and thus they describe the Stringy Kahler Moduli Space.

Parts of the talk are joint with Iyama, parts with Hirano, and parts with Donovan.

• 2020-03-10: Cancel
Title:

• 2020-03-17: Travis Schedler (Imperial College London) CANCEL
Title: Platonic solids, moduli spaces, and Calabi—Yau structures
Abstract: I will recall how Platonic solids are classified via their symmetry groups. This leads to ADE Dynkin diagrams and in turn to Lie algebras and groups. The link is made by certain moduli spaces of 2-Calabi-Yau algebras. As I will explain, these are local models for the rich geometric spaces appearing in geometry and physics. I will use these to understand singularities of moduli spaces and classify new kinds of symmetries.

• 2020-03-24: Konstanze Rietsch (King's College London) CANCEL
Title: Positive Laurent polynomials, critical points, and mirror symmetry
(joint work with Jamie Judd)
Abstract: In mirror symmetry a big role is played by Laurent polynomials. For example, consider the Laurent polynomial  W(x,y) = x+y+q 1/xy which arises as mirror dual to the complex projective plane X. This Laurent polynomial W is an example of the mirror superpotential of a toric symplectic manifold as constructed by Batyrev and Givental in the 1990s. This talk is about Laurent polynomials with coefficients in a field of (generalised) Puiseaux series in a variable q, such as this example W.  In joint work with Jamie Judd we show a positive Laurent polynomial has a unique positive critical point if and only if its Newton polytope has 0 in its interior. One application of this result, using mirror symmetry and work of Fukaya, Oh, Ohta, Ono, and Woodward, is to the construction of non-dispaceable Lagrangians in smoth toric manifolds (resp. orbifolds).

### FALL 2019 (Organiser Salvatore Stella)

Only events in Fall 2019 are listed below. Past events can be found on this page, and past events 2017 onwards can be found in the Pure Maths Seminar Archive.

• 2019-09-24: Rachel Newton (University of Reading)
Title: Counting failures of a local-global principle
Abstract: Methods for solving polynomial equations in the integers and rationals have been sought and studied for more than 4000 years. Modern approaches try to piece together 'local' (meaning real and p-adic) information to decide whether a polynomial equation has a 'global' (meaning rational) solution. I will describe this approach and its limitations, with the aim of quantifying how often the local-global method fails within families of polynomial equations arising from the norm map between fields, as seen in Galois theory. This is joint work with Tim Browning.
• 2019-10-01: Alexandre Zalesski (University of East Anglia)
Title: Eigenvalue 1 of elements of finite groups of Lie type in their representations in the defining characteristic
Abstract: For various applications one needs information on existence and multiplicities of eigenvalues of individual elements of groups in their representations. In general the problem is untreatable, but many special cases were discussed in the literature. The eigenvalue 1 is more significant due to its geometric nature. For groups of Lie type and representations in defining characteristic the theory of weights is a very helpful tool. In the talk I shall discuss the problem on the occurrence of eigenvalue 1 of all elements in a group representation, and state some recent results.
• 2019-10-08: Vincent Pilaud (CNRS)
Title: On type cones of g-vector fans
Abstract: In a recent preprint, Bazier-Matte - Douville - Mousavand - Thomas - Yildirim constructed all polytopal realizations of the g-vector fans of finite type cluster algebras with respect to an acyclic initial seed. This talk will explain the geometric reason hidden behind their construction and will propose an alternative proof of their result. More generally, we will discuss the type cone (in other words, the space of all polytopal realizations) of g-vector fans of finite type cluster-like complexes (finite type cluster complexes, non-kissing complexes of gentle algebra, graph associahedra). Joint work with Arnau Padrol, Yann Palu and Pierre-Guy Plamondon.
• 2019-10-15: Andrea Appel (University of Edinburgh)
Title: Meromorphically braided module categories
Abstract: The category Mod(H) of finite-dimensional modules over a quasi-triangular Hopf algebra H (that is, a Hopf algebra with an R-matrix R) is the prototype of a braided monoidal category. Similarly, the category of finite-dimensional modules over a coideal subalgebra in H is automatically endowed with a categorical action of Mod(H) and is the prototype of a "module" category.

The compatibility of this action with the braiding in Mod(H) is encoded by an additional datum, called a K-matrix, which provides, together with R, a universal solution of the reflection equation. Coideal subalgebras with a K-matrix are also called quasi-triangular and their finite-dimensional modules form a braided module category. While braided monoidal categories are associated with braid groups (Artin groups of type A), braided module categories corresponds to cylindrical braid groups (Artin groups of type B).

In this talk, I will review the construction of K-matrices for quantum groups of finite type due to Balagovic and Kolb. I will then describe alternative approaches to this construction, which are more suitable to generalizations and apply in particular to quantum affine algebras, where they produce examples of “meromorphically” braided module category. This is based on joint ongoing works with David Jordan and Bart Vlaar.
• 2019-10-22: Chris Wood (University of York)
Title: Energy, volume and what lies between
Abstract: In differential geometry, and particularly Riemannian geometry, the energy and volume functionals form the basis of the extensively studied variational theories of harmonic maps and minimal immersions, respectively. In fact there is a discrete family of energy-type functionals that “interpolate” between energy and volume. We briefly describe these “higher-power energy” functionals and indicate why they present interesting possibilities. By way of example, we analyse their behaviour on certain classes of Lie groups: semi-abelian, and 3-dimensional unimodular.
• 2019-10-29: Double talk
• Sonia Trepode (Universidad Nacional de Mar del Plata)
Title: On the representation dimension of artin algebras and Auslander's generators.
Abstract: Our objective in this talk is to explore the relation between the representation theory of an algebra and its homological invariants. We are in particular interested here in the representation dimension of an algebra, introduced by Auslander, which measures in some way the complexity of the morphisms of the module category.

There were several attempts to understand, or compute, this invariant. Special attention was given to algebras of representation dimension three. The reason for this interest is two-fold. Firstly, it is related to the finitistic dimension conjecture: Igusa and Todorov have proved that algebras of representation dimension three have a finite finitistic dimension. Secondly, because Auslander's expectation was that the representation dimension would measure how far an algebra is from being representation-finite. He proved in the 70's that representation finite algebras are characterized as the ones having representation dimension two. Based in on this result and the fact that all the examples known having representation dimension greater than three are wild algebras, there is a standing conjecture that the representation dimension of a tame algebra is at most three. Indeed, while there exist algebras of arbitrary, but finite, representation dimension, most of the best understood classes of algebras have representation dimension three.

In this talk we consider several classes of algebras and discuss their representation dimension. We discuss the connection with the finitistic conjecture and the relation with tame algebras.

Finally, we present a work in progress with Heily Wagner, Edson Ribeiro Alvares and Clezio Braga. We consider a class of algebras having a minimal Auslander's generator and we call these algebras representation hereditary algebras. We derive some properties of these algebras and we discuss some characterizations of them.
• Elsa Fernández (Universidad Nacional de Mar del Plata)
Title: Isomorphic Trivial Extensions
Abstract: In the first part of this talk se describe the ideal of relation for the Trivial extension of a finite dimensional álgebra over an algehraically closed field. In the second part wey study the isomorphism problem of trivial extensions for different classes of algebras.
• 2019-11-05: Aaron Chan (University of Nagoya)
Title: Torsion classes of gentle algebras
Abstract: We classify torsion classes of (not necessarily finite dimensional) gentle algebras using string-and-band combinatorics. Interpretation via laminations will be discussed if time permits. This is a joint work in progress with Laurent Demonet.
• 2019-11-12: Chris Heunen (University of Edinburgh)
Title: Tensor topology
Abstract: Any braided monoidal category comes with a built-in notion of space, namely, a meet-semilattice of so-called idempotent subunits. For a classical example, in a sheaf category this recovers the lattice of open sets of the base space. But the theory also captures non-cartesian examples such as Hilbert modules and quantales. There is an accompanying notion of support, showing 'where' a morphism acts, that satisfies a universal property. The meet-semilattice can be completed to a commutative quantale via the Yoneda embedding. Finally, there is an appropriate notion of restriction or localisation. Time permitting I will sketch example applications.
• 2019-11-19: No meeting because of overlapping events:
• 2019-11-26: Valentina Di Proietto (University of Exeter)
Title: A non-abelian algebraic criterion for good reduction of curves
Abstract: For a family of proper hyperbolic complex curves $f: X \rightarrow \Delta ^*$ over a puntured disc $\Delta^*$ with semistable reduction at the center, Oda proved, with transcendental methods, that the outer monodromy action of $\pi_1(\Delta^*)\cong\mathbb{Z}$ on the classical unipotent fundamental group of the generic fiber of $f$ is trivial if and only if $f$ has good reduction at the center. In this talk I explain a joint work with B. Chiarellotto and A. Shiho in which we give a purely algebraic proof of Oda's result.
• 2019-12-03: Celeste Damiani (University of Leeds)
Title: Loop braid groups and a new lift of Artin's representation
Abstract: Artin's representation for braid groups lifts to Dahm's homomorphism in the case of (extended) loop braid groups. We give by an injection of the (extended) loop braid group into the group of automorphisms of a triple, composed by a group, an abelian group, and an action of the first on the second one. We show this injection is both an extension of Artin's representation for braid groups and of Dahm's homomorphism for (extended) loop braid groups.
• 2019-12-10: Thomas Cottrell (University of Bath)
Title: Enriched and Internal Categories: An Extensive Relationship
Abstract: Enrichment and internalisation are two standard methods of generalising the notion of category, which arise by extending two different -- but equivalent -- forms of the definition of category. The relationship between these two generalisations is the topic of this talk.

Both generalisations require a suitable choice of base category, and our leading example takes the base category to be the category of small categories. Here, the enriched categories are 2-categories, and the internal categories are double categories, which enjoy a close, well-understood relationship. The key result of this talk is that this relationship holds more generally, provided the base category satisfies certain conditions. I will explain this relationship, and explore the conditions under which it holds. Most notably, these conditions include extensivity, which ensures disjointness of coproducts.

This is joint work with Soichiro Fujii and John Power.

Seminars 2018-2019

Seminars 2017-2018

Seminars 2016-2017

Seminars 2013-2014

Seminars 2012-2013

Seminars 2011-2012

Seminars 2010-2011

Seminars 2009-2010

Seminars 2008-2009

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University of Leicester
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