Pure Maths Seminar Series

Only events in Spring 2020 are listed below.

• 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).
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  Past events can be found on this page, and past events 2017 onwards can be found in the Pure Maths Seminar Archive.