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# Pure Maths Seminar Series

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. All are welcome to these seminars. For any questions or comments please contact Salvatore Stella, s.stella@leicester.ac.uk

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)
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• 2019-12-10: Thomas Cottrell (University of Bath)
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University Rankings

#### University of Leicester

• 29th in the Complete University Guide 2018
• 34th in the Times/Sunday Times Good University Guide 2018
• 43rd in the Guardian University Guide 2018
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Department of Mathematics
University of Leicester
Leicester LE1 7RH
United Kingdom

Tel.: +44 (0)116 229 7407

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