For rings of integers in an extension of number fields there are classica l methods for detecting ramification and for identifying ramification as b eing tame or wild. Noether's theorem characterizes tame ramification in te rms of a normal basis and tame ramification can also be detected via the s urjectivity of the norm map. We take the latter fact and use the Tate coho mology spectrum to detect wild ramification in the context of commutative ring spectra. I will discuss several examples in the context of topologica l K-theory and modular forms.
\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Du Pei (Harvard) DTSTART:20210526T130000Z DTEND:20210526T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/19 DESCRIPTION:Title: Hidden algebraic structures in geometry from fivebranes\nb y Du Pei (Harvard) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract \nThe existence of quantum field theories in higher dimensions predicts ma ny hidden algebraic structures in geometry and topology. In this talk\, I will survey some recent developments where such algebraic structures lead to new insights into 1) the quantization of moduli spaces of Higgs bundles \, 2) the categorification of quantum invariants of 3-manifolds\, and 3) n ovel types of TQFTs in four dimensions.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Kurinczuk (University of Sheffield) DTSTART:20211103T140000Z DTEND:20211103T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/20 DESCRIPTION:Title: Local Langlands in families for classical groups\nby Rober t Kurinczuk (University of Sheffield) as part of Sheffield Pure Maths Coll oquia\n\n\nAbstract\nThe conjectural local Langlands correspondence connec ts representations of p-adic groups to certain representations of Galois g roups of local fields called Langlands parameters. In recent joint work w ith Dat\, Helm\, and Moss\, we have constructed moduli spaces of Langlands parameters over Z[1/p] and studied their geometry. We expect this geomet ry is reflected in the representation theory of the p-adic group. Our mai n conjecture “local Langlands in families” describes the GIT quotient of the moduli space of Langlands parameters in terms of the centre of the category of representations of the p-adic group generalising a theorem of Helm-Moss for GL(n). I will give an introduction to this picture and expl ain how after inverting the "non-banal primes" one can prove this conjectu re for the local Langlands correspondence for classical groups of Arthur a nd others.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Ananyo Dan (University of Sheffield) DTSTART:20211110T140000Z DTEND:20211110T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/21 DESCRIPTION:Title: McKay correspondence for isolated Q-Gorenstein singularities\nby Ananyo Dan (University of Sheffield) as part of Sheffield Pure Math s Colloquia\n\n\nAbstract\nThe McKay correspondence is a (natural) corresp ondence between the (non-trivial) irreducible representations of a finite subgroup G of $SL(2\,\\C)$ and the irreducible components of the exception al divisor of a minimal resolution of the associated quotient singularity $\\C^2//G$. A geometric construction for this correspondence was given by González-Sprinberg and Verdier\, who showed that the two sets also corres pond bijectively to the set of indecomposable reflexive modules on the quo tient singularity. This was generalized to higher dimensional quotient sin gularities (i.e.\, quotient of $\\C^n$ by a finite subgroup of $SL(n\,\\C) $) by Ito-Reid\, where the above sets were substituted by certain smaller subsets. It was further generalized to more general quotient singularities by Bridgeland-King-Reid\, Iyama-Wemyss and others\, using the language of derived categories. In this talk\, I will survey past results and discuss what happens for the isolated Q-Gorenstein singularities case (not necess arily a quotient singularity). If time permits\, I will discuss applicatio ns to Matrix factorization. This is joint work in progress with J. F. de B obadilla and A. Romano-Velazquez.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Oscar Randal-Williams (University of Cambridge) DTSTART:20211124T140000Z DTEND:20211124T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/22 DESCRIPTION:Title: Homeomorphisms of $\\R^d$\nby Oscar Randal-Williams (Unive rsity of Cambridge) as part of Sheffield Pure Maths Colloquia\n\n\nAbstrac t\nThe group Top(d) of homeomorphisms of d-dimensional Euclidean space is a basic object in geometric topology\, with its quotient Top(d)/O(d) by th e subgroup of linear isometries completely controlling the difference betw een smooth and topological manifolds in all dimensions (except 4). I will explain some of the classical methods for studying the topology of this gr oup\, and report on some recent advances.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Sven Raum (Stockholm University) DTSTART:20220511T123000Z DTEND:20220511T133000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/23 DESCRIPTION:Title: Simple operator algebras associated with groups and group-like structures\nby Sven Raum (Stockholm University) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nOne of the original motivations of Mur ray and von Neumann introducing operator algebras was to study the unitary representation theory of groups. This naturally leads to the question of studying building blocks of representation theory\, that is simple operato r algebras associated with groups. From a modern point of view\, not only groups but also other group-like structures such as groupoids should be in vestigated. This talk introduces the audience to group and groupoid operat or algebras and tells the story of how our point of view on their simplici ty changed dramatically over the past 10 years. At the end of the talk\, I will present some results on simple groupoid C*-algebras that were obtain ed in joint work with Kennedy\, Kim\, Li and Ursu.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Oliver Lorscheid (IMPA / University of Groeningen) DTSTART:20211215T140000Z DTEND:20211215T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/24 DESCRIPTION:Title: The moduli space of matroids\nby Oliver Lorscheid (IMPA / University of Groeningen) as part of Sheffield Pure Maths Colloquia\n\n\nA bstract\nMatroids are combinatorial gadgets that reflect properties of lin ear algebra in situations where this latter theory is not available. This analogy prescribes that the moduli space of matroids should be a Grassmann ian over a suitable base object\, which cannot be a field or a ring\; in c onsequence usual algebraic geometry does not provide a suitable framework. In joint work with Matt Baker\, we use algebraic geometry over F1\, the s o-called field with one element\, to construct such moduli spaces. As an a pplication\, we streamline various results of matroid theory and find simp lified proofs of classical theorems\, such as the fact that a matroid is r egular if and only if it is binary and orientable.\n\nWe will dedicate the first half of this talk to an introduction of matroids and their generali zations. Then we will outline how to use F1-geometry to construct the modu li space of matroids. In a last part\, we will explain why this theory is useful to simplify classical results in matroid theory.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Shahn Majid (Queen Mary University of London) DTSTART:20220518T130000Z DTEND:20220518T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/25 DESCRIPTION:Title: Quantum Riemannian geometry of the $A_n$ graph\, jets and geod esics\nby Shahn Majid (Queen Mary University of London) as part of She ffield Pure Maths Colloquia\n\n\nAbstract\nWe describe recent results in q uantum or noncommutative Riemannian geometry based on bimodule connections . Here\nthe coordinate algebra can be any unital algebra A equipped with a differential structure expressed as a \nbimodule $\\Omega^1$ of $1$-forms as part of a differential graded algebra with $A$ in degree $0$. The simp lest case\nis $A$ the commutative algebra of functions on the vertices of a directed graph with $\\Omega^1$ spanned by the arrows. \nWe show in this framework that the intrinsic quantum Riemannian geometry of the $A_n$ gra ph $\\bullet-\\bullet- …-\\bullet$ of $n$ vertices is necessarily $q$-de formed with $q^{2(n+1)}=1$. Its $q\\to1$ limit is the intrinsic quantum Ri emannian geometry of the natural numbers viewed as a half-line graph. We t hen discuss more generally how solutions of the Yang-Baxter or braid relat ions arise naturally from noncommutative differential geometry and relate both to quantum jet bundles and\nto the notion of a quantum geodesic.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Umut Varolgunes (Bogazici University) DTSTART:20220525T130000Z DTEND:20220525T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/26 DESCRIPTION:Title: Local-to-global methods in relative symplectic cohomology\ nby Umut Varolgunes (Bogazici University) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn my thesis\, I introduced a Floer theoretic inv ariant for compact subsets of symplectic manifolds called relative symplec tic cohomology. This invariant has already proved to be very useful in sym plectic rigidity questions and also opened the way to a fruitful reinterpr etation of mirror symmetry. Most of these applications rely on an analogue of Mayer-Vietoris property from topology that holds for relative symplect ic cohomology under well-understood geometric assumptions. I will briefly introduce the invariant\, discuss the Mayer-Vietoris property and present some computations relevant to mirror symmetry. I will try to make the talk accessible to a more diverse audience by mainly sticking to dimension two \, where a symplectic form is nothing but an area form.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Vesna Stojanoska (University of Illinois Urbana-Champaign) DTSTART:20220309T150000Z DTEND:20220309T160000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/27 DESCRIPTION:Title: Duality for some Galois groups in stable homotopy theory\n by Vesna Stojanoska (University of Illinois Urbana-Champaign) as part of S heffield Pure Maths Colloquia\n\n\nAbstract\nIn classical algebra\, the in teger primes p help decompose objects as well as problems into their p-pri mary parts\, which may be easier to study. The same is true in homotopy th eory\, but the situation is more interesting since for each integer prime p\, there are infinitely many nested homotopical primes. For each of those homotopical primes\, there is an (unramified) Galois group that governs t he local story and encodes the symmetries of chromatic homotopy theory. Th ese Galois groups turn out to be particularly nice profinite groups\, know n as compact p-adic analytic. Such groups and their fascinating duality pr operties within algebra were studied by Lazard. I will try to explain a ne wer result\, which shows that their homotopical duality properties are eve n better\, giving powerful implications for the chromatic Galois extension s that they govern.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Hossein Movasati (IMPA) DTSTART:20220316T140000Z DTEND:20220316T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/28 DESCRIPTION:Title: Periods of families of curves in threefolds\nby Hossein Mo vasati (IMPA) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nCle mens' conjecture states that the the number of rational curve in a generic quintic threefold is finite. If it is false we prove that certain periods of rational curves in such a quintic threefold must vanish. Our method is based on a generalization of a proof of Max Noether's theorem using infi nitesimal variation of Hodge structures and its reformulation in terms of integrals and Gauss-Manin connection.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Bachir Bekka (Université de Rennes 1) DTSTART:20220608T130000Z DTEND:20220608T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/29 DESCRIPTION:Title: The spectral gap property for group actions\nby Bachir Bek ka (Université de Rennes 1) as part of Sheffield Pure Maths Colloquia\n\n \nAbstract\nA measure preserving action of a group G on a measure space X gives rise to a unitary representation of G on the Hilbert space $L^2(X)$. This action may or may not have the spectral gap property which is a v ery strong form of ergodicity. For instance\, groups with Kazhdan's proper ty T always have this property. We will survey the importance of the spec tral gap property in various problems arising in graph theory\, dynamical systems or operator algebras. In the case where X is a homogeneous space arising from an algebraic group\,\nwe will show that the absence of the spectral gap property is often related to amenability.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Wushi Goldring (Stockholm University) DTSTART:20220427T130000Z DTEND:20220427T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/31 DESCRIPTION:Title: Propagating algebraicity of automorphic representations via fu nctoriality\nby Wushi Goldring (Stockholm University) as part of Sheff ield Pure Maths Colloquia\n\n\nAbstract\nAutomorphic representations are s ome of the richest and most mysterious mathematical objects discovered to- date. They simultaneously generalize (i) infinite-dimensional representati ons of real Lie groups\, (ii) modular forms and (iii) the Hecke characters of class field theory. As such\, automorphic representations incorporate representation theory\, analysis and arithmetic. \n\nIn the late 1960's\, Robert Langlands laid out a program to unravel much of the seemingly hidde n structure of automorphic representations. To begin to understand the Lan glands program\, it is useful -- at least at first -- to distinguish two k inds of conjectures: Roughly\, Langlands' Functoriality Principle can be s een as intrinsic to automorphic representations -- revealing a myriad of r elations between different automorphic representations of different groups . By contrast\, the extrinsic Langlands correspondence explains how certai n automorphic representations should be related to Galois theory and algeb raic geometry. Every automorphic representation has associated numerical i nvariants called Hecke eigenvalues -- these are complex numbers. One of th e most interesting aspects of the Langlands program is that some automorph ic representations have Hecke eigenvalues which are algebraic numbers\, wh ile for others they are transcendental. At this time\, we seem to lack a c onceptual understanding for why this dichotomy exists. While the Langlands correspondence suggests that certain automorphic representations should h ave algebraic Hecke eigenvalues\, it remains unclear -- even at the level of conjectures -- wherein lies the watershed line between algebraic and tr anscendental. \n\nI will spend most of my talk introducing automorphic rep resentations\, their Hecke eigenvalues\, functoriality and the corresponde nce. The end goal of my talk is then to explain what can be said about the algebraicity of Hecke eigenvalues by combining (1) Previously known cases of algebraicity and (2) Langlands functoriality. On the one hand\, I will explain why the algebraicity of Hecke eigenvalues does propagate from som e cases to others via functoriality -- this gives new theorems and conject ures on algebraicity of Hecke eigenvalues. On the other hand\, I will expl ain why most cases -- including Maass forms -- are not reducible to known ones via functoriality.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Dan Ciubotaru (University of Oxford) DTSTART:20221116T140000Z DTEND:20221116T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/32 DESCRIPTION:Title: Unipotent conjugacy classes and group representations\nby Dan Ciubotaru (University of Oxford) as part of Sheffield Pure Maths Collo quia\n\n\nAbstract\nOne of the first theorems that we prove in the complex representation theory of a finite group is that the number of irreducible representations (up to isomorphism) equals the number of conjugacy classe s in the group. For example\, for the group of permutations of the set {1\ ,2\,...\,n}\, the conjugacy classes are parametrised by partitions of n (t he cycle decomposition) and so are the complex irreducible representations (via Young's construction from the 1890s). But this is not a natural bije ction\, just like there is not a natural isomorphism between a finite vect or space and its dual in general. However\, for certain classes of groups\ , that come with extra structure (like the ones appearing in Lie theory)\, one expects natural relations between the irreducible representations of the group\, on one hand\, and conjugacy classes in a *dual* group\, on the other. This happens for example\, when the group in question is a finite reflection crystallographic group\, or a connected algebraic group over a finite or local field. In these correspondences\, a particularly interesti ng role is played by the unipotent conjugacy classes in the dual group. I will give a survey of some of these connections and then emphasise the cas e of (infinite-dimensional) representations of reductive algebraic groups (like the general linear group of n by n matrices) with coefficients in a local field\, where I'll explain what the unipotent classes tell us about the growth of characters and the parametrisation of such representations. The new results in the talk are joint with Lucas Mason-Brown and Emile Oka da.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Luis Garcia (University College London) DTSTART:20221123T140000Z DTEND:20221123T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/33 DESCRIPTION:Title: Modular symbols\, linking numbers and the Euler class\nby Luis Garcia (University College London) as part of Sheffield Pure Maths Co lloquia\n\nLecture held in J-11 Hicks Building.\n\nAbstract\nModular symbo ls are a fundamental tool for the computation of the homology of certain l inear groups. It has been observed that\, surprisingly\, they also control the relations among certain trigonometric and elliptic functions. After i ntroducing modular symbols and their elementary properties I will explain why this is the case and give some arithmetic applications.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Fraser (Wichita State University) DTSTART:20230524T130000Z DTEND:20230524T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/34 DESCRIPTION:Title: (ONLINE) Two strategies for Fourier decay of measures in Dioph antine approximation\nby Robert Fraser (Wichita State University) as p art of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn 1980 and 1981\, Ka ufman constructed measures with polynomial Fourier decay on the set of bad ly-approximable numbers and the set of well-approximable numbers. The stra tegy for the badly-approximable numbers uses the continued fraction expans ion together with a change-of variables\, and the strategy for the well-ap proximable numbers uses the cancellation of an exponential sum. We will di scuss the application of both of these strategies to the set of numbers ap proximable to exact order introduced by Bugeaud. This talk is based on joi nt work with Reuben Wheeler.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Cecilia Salgado (University of Groeningen) DTSTART:20221207T140000Z DTEND:20221207T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/35 DESCRIPTION:Title: CANCELLED\nby Cecilia Salgado (University of Groeningen) a s part of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Tyler Kelly (University of Birmingham) DTSTART:20230426T130000Z DTEND:20230426T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/36 DESCRIPTION:Title: CANCELLED!\nby Tyler Kelly (University of Birmingham) as p art of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Markus Szymik (University of Sheffield) DTSTART:20230215T140000Z DTEND:20230215T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/37 DESCRIPTION:Title: CANCELLED DUE TO UCU STRIKE\nby Markus Szymik (University of Sheffield) as part of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Johannes Girsch (University of Sheffield) DTSTART:20230208T140000Z DTEND:20230208T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/38 DESCRIPTION:Title: Introduction to the representation theory of $p$-adic groups\nby Johannes Girsch (University of Sheffield) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nThe Langlands program is a set of wide-rea ching conjectures with great importance to many aspects of number theory. One kind of objects that are being studied in this setting are representat ions of $p$-adic groups. I will explain what these groups are and mention some rather strange properties they possess. Then I will mention some aspe cts of their representation theory and mention some recent results.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Evgenios Kakariadis (University of Newcastle) DTSTART:20230301T140000Z DTEND:20230301T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/39 DESCRIPTION:Title: Morita equivalence for operator systems\nby Evgenios Kakar iadis (University of Newcastle) as part of Sheffield Pure Maths Colloquia\ n\n\nAbstract\nIn ring theory\, Morita equivalence preserves many properti es of the objects\, and generalizes the isomorphism equivalence between co mmutative rings. A strong Morita equivalence for selfadjoint operator alge bras was introduced by Rieffel in the 60s\, and works as a correspondence between their representations. In the past 30 years there has been an inte rest to develop a similar theory for nonselfadjoint operator algebras and operator spaces with much success. Taking motivation from recent work of C onnes and van Suijlekom\, we will present a Morita theory for operator sys tems. We will give equivalent characterizations of Morita equivalence via Morita contexts\, bihomomoprhisms and stable isomorphism\, while we will h ighlight properties that are preserved in this context. Time permitted we will provide applications to rigid systems\, function systems and non-comm utative graphs. This is joint work with George Eleftherakis and Ivan Todor ov.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Samuele Anni (Aix-Marseille University) DTSTART:20230308T140000Z DTEND:20230308T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/40 DESCRIPTION:Title: (ONLINE) Isomorphisms of modular Galois representations and gr aphs\nby Samuele Anni (Aix-Marseille University) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn this talk\, I will explain how to t est efficiently and effectively whether two odd modular Galois representat ions of the absolute Galois group of the rational numbers are isomorphic. In particular\, I will present new optimal bounds on the number of traces to be checked (joint work with Peter Bruin\, University of Leiden). I will also briefly discuss graphs of isomorphisms associated to such objects\, related results on Hecke algebras\, and a database of modular representati ons.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Min Lee (moved to Term 2) (University of Bristol) DTSTART:20221109T140000Z DTEND:20221109T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/41 DESCRIPTION:by Min Lee (moved to Term 2) (University of Bristol) as part o f Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Min Lee (University of Bristol) DTSTART:20230517T130000Z DTEND:20230517T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/42 DESCRIPTION:Title: Frobenius numbers and further - equidistribution of rational p oints on the expanding horospheres\nby Min Lee (University of Bristol) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nFix a finite set R of positive integers bigger than one with no common factors. The Froben ius number for R is the largest number that cannot be written as a linear combination of the integers in R with non-negative integral coefficients. \n\n\nIn general\, Frobenius numbers fluctuate. To study such things\, we search for structures. Here\, the given set of positive integers R can be a point in the lattices studied in the dynamics and number theory crossove r. We study the behaviour of these rational points on expanding closed hor ospheres in the space of lattices. The equidistribution of these rational points is proved by Einsiedler\, Mozes\, Shah and Shapira (2016). Their pr oof uses techniques from homogeneous dynamics and relies particularly on m easure-classification theorems\, due to Ratner. We pursue an alternative s trategy based on Fourier analysis\, Weil's bound for Kloosterman sums\, re cently proved bounds (by M. Erdélyi and Á. Tóth) for matrix Kloosterman sums\, Roger's formula\, and the spectral theory of automorphic functions .\n\n\nThis is a joint work with D. El-Baz\, B. Huang\, J. Marklof and A. Strömbergsson.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Eleonore Faber (University of Leeds) DTSTART:20221102T140000Z DTEND:20221102T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/43 DESCRIPTION:Title: From the magic square of rotations and reflections to the McKa y correspondence\nby Eleonore Faber (University of Leeds) as part of S heffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Celine Maistret (University of Bristol) DTSTART:20221026T130000Z DTEND:20221026T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/44 DESCRIPTION:Title: The Birch and Swinnerton-Dyer conjecture and the Parity conjec ture\nby Celine Maistret (University of Bristol) as part of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Vlad Bavula (University of Sheffield) DTSTART:20221019T130000Z DTEND:20221019T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/45 DESCRIPTION:Title: Holonomic modules and 1-generation in the Jacobian Conjecture< /a>\nby Vlad Bavula (University of Sheffield) as part of Sheffield Pure Ma ths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Andre Henriques (University of Oxford) DTSTART:20221012T130000Z DTEND:20221012T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/46 DESCRIPTION:Title: 2d QFTs as objects of mathematics\nby Andre Henriques (Uni versity of Oxford) as part of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Haluk Sengun (University of Sheffield) DTSTART:20221005T130000Z DTEND:20221005T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/47 DESCRIPTION:Title: Local theta correspondence via $C^*$-algebras\nby Haluk Se ngun (University of Sheffield) as part of Sheffield Pure Maths Colloquia\n \nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Neil Strickland (University of Sheffield) DTSTART:20230503T130000Z DTEND:20230503T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/48 DESCRIPTION:by Neil Strickland (University of Sheffield) as part of Sheffi eld Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Polyxeni Spilioti (University of Göttingen) DTSTART:20230329T140000Z DTEND:20230329T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/49 DESCRIPTION:Title: (ONLINE) On the spectrum of twisted Laplacians and the Teichm üller representation\nby Polyxeni Spilioti (University of Göttingen) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn this talk\, w e will present some results concerning the spectrum of Laplacians with non unitary twists acting on sections of flat vector bundles over compact hyp erbolic surfaces. These non self-adjoint Laplacians have discrete spectrum inside a parabola in the complex plane. For representations of the fundam ental group of the base surface which are of Teichmüller type\, we invest igate the high energy limit and give a precise description of the bulk of the spectrum where Weyl’s law is satisfied in terms of critical exponent s of the representation which are completely determined by the Manhattan c urve associated to the Teichmüller deformation. This is joint work with F rédéric Naud.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Tyler Kelly (University of Birmingham) DTSTART:20231004T130000Z DTEND:20231004T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/50 DESCRIPTION:Title: Moduli of genus-zero higher spin curves and their invariants\nby Tyler Kelly (University of Birmingham) as part of Sheffield Pure Ma ths Colloquia\n\n\nAbstract\nIn mathematics\, we like classifying objects. A moduli space is a space where each point represents a(n isomorphism cla ss of a) space satisfying certain criteria\, giving a geometric answer to a classification problem. Often the geometry of such spaces are interestin g in our own right and their corresponding enumerative information has ric h structure. We will study the case of genus-zero n-pointed curves and a g eneralisation where they are further equipped with an r-spin structure. En umerative invariants built from their characteristic classes have rich str ucture due to generalisations of predictions of Witten confirmed by Kontse vich. We will explain approaches to understanding these invariants on a ve ry concrete level through combinatorial structures like recursion and trop ical geometry.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/50/ END:VEVENT BEGIN:VEVENT SUMMARY:Markus Szymik (University of Sheffield) DTSTART:20231011T130000Z DTEND:20231011T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/51 DESCRIPTION:Title: From rings to algebraic K-theory and back\nby Markus Szymi k (University of Sheffield) as part of Sheffield Pure Maths Colloquia\n\n\ nAbstract\nAlgebraic K-theory is a conceptual tool for the classification of mathematical objects. A typical scenario comes from linear algebra: the classification of vector spaces and\, more generally\, modules over a giv en ring. In this colloquium talk\, I will advertise this tool and its use in non-linear algebra. The focus will be on examples\, and I will discuss groups\, rings\, and many other more or less exotic algebraic structures.\ n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Lassina Dembélé (King's College London) DTSTART:20231018T130000Z DTEND:20231018T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/52 DESCRIPTION:Title: (Black History Month Special Colloquium) How mentorship could help fight underrepresentation in STEM\nby Lassina Dembélé (King's C ollege London) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nTh ere is no denial that certain visible minorities are severely underreprese nted in STEM. I hear people often say that the best way to fight underrepr esentation is to have more role models from those minority groups. That is true\, perhaps. However\, I believe that there needs to be an intermediat e solution until we reach that point when we have enough role models to ha ve an impact. Based on my own personal experience\, I want to explain how an innovative approach to mentorship can help fight underrepresentation.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Reem Yassawi (Queen Mary University of London) DTSTART:20231025T130000Z DTEND:20231025T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/53 DESCRIPTION:Title: Automatic sequences in dynamics and number theory\nby Reem Yassawi (Queen Mary University of London) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nAn infinite sequence $a = (a_n)_{n\\geq 0}$ is $ q$-automatic if an is a finite-state function of the base-$q$ expansion of $n$. This means that there exists a deterministic finite automaton that t akes the base-q expansion of n as input and produces the symbol an as outp ut for each $n \\in \\mathbb{N}$.\n\nAutomatic sequences appear in diverse fields of mathematics\, such as algebra\, logic\, number theory\, and top ological dynamics. They have the advantage of lend- ing themselves to comp utation\, so that in each area there arise specific problems concerning au tomatic sequences\, and much of the time\, constructive solutions.\n\nI wi ll give a background of their characterisations in algebra and dynamics\, via Furstenberg’s\, Cobham’s and Christol’s theorems. I will then ta lk about joint work with Eric Rowland and Manon Stipulanti\, concerning au tomatic sequences in number theory\, and also about joint work with Johann es Kellendonk\, concerning automatic sequences in topological dynamics\, e nding with a topological invariant which seems to defy computation.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Eli Hawkins (University of York) DTSTART:20231101T110000Z DTEND:20231101T120000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/54 DESCRIPTION:Title: Quantization of Multiply Connected Manifolds\nby Eli Hawki ns (University of York) as part of Sheffield Pure Maths Colloquia\n\n\nAbs tract\nGiven a compact Kähler manifold satisfying an integrality conditio n\, the Berezin-Toeplitz geometric quantization construction produces matr ix algebras\; these fit together into a fundamental example of strict defo rmation quantization. The integrality condition can be circumvented by pas sing to the universal covering space\, if the lift of the symplectic form is exact\; in this case\, the symplectic form determines a $2$-cocycle of the fundamental group. The key to analyzing this construction is to use Hi lbert $C^*$-modules\, which generalize Hilbert spaces. The resulting algeb ras are more interesting than matrix algebras and are partially determined by index theorems. The simplest example is the noncommutative torus\, and this gives higher-genus noncommutative Riemann surfaces as well.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Veronique Fischer (University of Bath) DTSTART:20231129T140000Z DTEND:20231129T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/55 DESCRIPTION:Title: Sub-Riemannian quantum limits\nby Veronique Fischer (Unive rsity of Bath) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nWe will start with a short discussion on semi-classical analysis to introduc e the concept of quantum limits. We will present an overview of sub-Rieman nian geometry and the recent developments of spectral geometry in this con text\, especially quantum limits on nilpotent Lie groups.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Paul Johnson (University of Sheffield) DTSTART:20231122T140000Z DTEND:20231122T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/57 DESCRIPTION:Title: From Orbifold Hilbert schemes to Sec(x)\nby Paul Johnson ( University of Sheffield) as part of Sheffield Pure Maths Colloquia\n\n\nAb stract\nThe Hilbert Scheme of points of n points in the plane is a smooth algebraic variety with a rich topology connected to partitions and represe ntation theory. If G acts on a C^2\, it also acts on the Hilbert scheme o f points. The question of when certain G fixed point sets are nonempty wi nds up having a connection to zig-zag permutations\, which are counted by the Taylor series coefficients of Tan(x) and Sec(x).\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/57/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Graves (University of Leeds) DTSTART:20231108T140000Z DTEND:20231108T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/58 DESCRIPTION:Title: Homology of diagram algebras\nby Daniel Graves (University of Leeds) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nDiagra m algebras\, such as the Brauer algebras and Temperley-Lieb algebras\, hav e been studied for many years. They appear in wide-ranging places such as statistical mechanics\, knot theory and representation theory. However\, t he study of the homology of these algebras is a very young field indeed\, having emerged over the course of last decade. In this talk I will give an introduction to these diagram algebras\, their homology and their connect ion to group homology and homological stability. Time permitting\, I will discuss some recent generalizations of these algebras.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Ozgur Bayindir (City University of London) DTSTART:20240228T140000Z DTEND:20240228T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/59 DESCRIPTION:Title: Algebraic K-theory and chromatic redshift\nby Ozgur Bayind ir (City University of London) as part of Sheffield Pure Maths Colloquia\n \n\nAbstract\nI will begin with an introduction to algebraic K-theory\, ri ng spectra and the chromatic redshift conjecture. After this\, I will talk about our new proof of the redshift conjecture for Lubin-Tate spectra and our algebraic K-theory computations.\n\nThis work is partially joint with Christian Ausoni and Tasos Moulinos.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Simon Willerton (University of Sheffield) DTSTART:20240306T140000Z DTEND:20240306T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/60 DESCRIPTION:Title: Instantaneous dimension of metric spaces via spread and magnit ude\nby Simon Willerton (University of Sheffield) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nSome spaces seem to have different di mensions at different\nscales. A long thin strip might appear one-dimensi onal at a distance\,\nthen two-dimensional when zoomed in on\, but when zo omed in on even\ncloser it is seen to be made of a finite array of points\ , so at that\nscale it seems zero-dimensional. I will present a way of qu antifying\nthis phenomenon using a couple of measures of the size of metri c spaces\,\nnamely magnitude and spread. I will show lots of examples fo r finite\nmetric spaces.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Evgeny Shinder (University of Sheffield) DTSTART:20240313T140000Z DTEND:20240313T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/61 DESCRIPTION:Title: Gromov's cancellation question in birational algebraic geometr y\nby Evgeny Shinder (University of Sheffield) as part of Sheffield Pu re Maths Colloquia\n\n\nAbstract\nI explain some cancellation and non-canc ellation phenomena in algebraic geometry and relate them to the structure of the Grothendieck ring of varieties and to the groups of birational self -maps of algebraic varieties\, in particular the Cremona groups.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Emine Yildirim (University of Leeds) DTSTART:20240320T140000Z DTEND:20240320T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/62 DESCRIPTION:Title: Why the Return to Pictures in Algebra\nby Emine Yildirim ( University of Leeds) as part of Sheffield Pure Maths Colloquia\n\n\nAbstra ct\nIn ancient Greece\, geometry was about points\, lines\, circles\, and communicated through pictures. The 17th Century marked a transformative sh ift\, connecting geometry with algebra\, and lead to working with equation s over visual representations. Algebraic geometry emerged as a magical ble nd of geometric intuition and algebraic methods. Commutative algebra\, mai nly the study of polynomial rings and their ideals\, dominated the field f or an extensive period. Then with the emergence of noncommutative algebras \, such as matrix algebras\, our unstoppable geometric intuition hit an im movable wall. The solution? A return to pictures as representations. In th is expository talk\, I will introduce a visual perspective on algebras\, e xploring path algebras and their captivating connections to different fiel ds.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/62/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Samuel (University of Birmingham) DTSTART:20240417T130000Z DTEND:20240417T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/63 DESCRIPTION:by Tony Samuel (University of Birmingham) as part of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/63/ END:VEVENT BEGIN:VEVENT SUMMARY:Catherine Meusburger (University of Erlangen) DTSTART:20240424T130000Z DTEND:20240424T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/64 DESCRIPTION:Title: Dijkgraaf-Witten theory with defects\nby Catherine Meusbur ger (University of Erlangen) as part of Sheffield Pure Maths Colloquia\n\n \nAbstract\nWe use 3d defect TQFTs to give a gauge theoretical formulation of (untwisted) Dijkgraaf-Witten TQFT with defects. This leads to a simple description in terms of embedding quivers\, groupoids and their represent ations. Defect Dijkgraaf-Witten TQFTs is then formulated in terms of spans of groupoids and representations of spans. This is work in progress with João Faría-Martins\, University of Leeds.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/64/ END:VEVENT BEGIN:VEVENT SUMMARY:David Corfield (University of Kent) DTSTART:20240501T130000Z DTEND:20240501T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/65 DESCRIPTION:by David Corfield (University of Kent) as part of Sheffield Pu re Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/65/ END:VEVENT BEGIN:VEVENT SUMMARY:Rohini Ramadas (University of Warwick) DTSTART:20240508T130000Z DTEND:20240508T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/67 DESCRIPTION:Title: Complex dynamics via algebraic geometry (MOVED TO FALL)\nb y Rohini Ramadas (University of Warwick) as part of Sheffield Pure Maths C olloquia\n\n\nAbstract\nComplex dynamics began in the early 1900s with the study of iterating polynomial functions with complex coefficients. This s imple idea gives rise to beautiful fractal pictures such as the Mandelbrot set\, as well as interesting mathematical questions of many different fla vours (algebraic\, analytic\, topological\, arithmetic\, etc.). The field gained momentum in the 1980s due to work of Thurston\, Douady-Hubbard\, Su llivan\, and others\, connecting these dynamical questions to surface topo logy and the theory of 3-manifolds. The last decade has seen many breakthr oughs achieved via new tools from number theory\, measure theory and algeb raic geometry. I will discuss some of these recent developments\, highligh ting the interplay between topology on one hand and algebraic geometry/num ber theory on the other hand.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/67/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Magee (Durham University) DTSTART:20241009T130000Z DTEND:20241009T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/68 DESCRIPTION:Title: Convergence of unitary representations of discrete groups\ nby Michael Magee (Durham University) as part of Sheffield Pure Maths Coll oquia\n\n\nAbstract\nLet G be an infinite discrete group\; e.g. hyperbolic 3-manifold group.\nFinite dimensional unitary representations of G of fix ed dimension are usually very hard to understand. However\, there are inte resting notions of convergence of such representations as the dimension te nds to infinity. One notion — strong convergence — is of interest both from the point of view of G alone but also through recently realized appl ications to spectral gaps of locally symmetric spaces. For example\, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjectu re in obtaining existence of large area hyperbolic surfaces with near-opti mal spectral gaps. \n\nThe talk is a broadly accessible discussion on thes e themes\, based on joint works with W. Hide\, L. Louder\, D. Puder\, J. T homas.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/68/ END:VEVENT BEGIN:VEVENT SUMMARY:Herbert Gangl (Durham University) DTSTART:20241016T130000Z DTEND:20241016T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/69 DESCRIPTION:Title: The beauty of Zagier's Polylogarithm Conjecture\nby Herber t Gangl (Durham University) as part of Sheffield Pure Maths Colloquia\n\n\ nAbstract\nDirichlet related the residue at s=1 of the Dedekind zeta funct ion of a number field F (a slight generalisation of the famous Riemann zet a function) to two important arithmetical notions: the size of the ideal c lass group and the `volume' of the unit group in the number ring O_F of F. Generalising this surprising connection\, the special values of the Dedek ind zeta function of a number field F at integer argument n should\, accor ding to Zagier's Polylogarithm Conjecture\, be expressed via a determinant of F-values of the n-th polylogarithm function. Goncharov laid out a vast program incorporating this conjecture using properties of multiple polylo garithms and the structure of a motivic Lie coalgebra.\nIn this impression ist talk I intend to give a rough idea of the developments from the early days on\, avoiding most of the technical bits\, and also hint at a number of recent results for higher weight\, some in joint work with\, or develo ped by\, S.Charlton\, D.Radchenko as well as D.Rudenko and his collaborato rs.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/69/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Fink (Queen Mary University of London) DTSTART:20241023T130000Z DTEND:20241023T140000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/70 DESCRIPTION:Title: Matroid inequalities from algebraic geometry\nby Alex Fink (Queen Mary University of London) as part of Sheffield Pure Maths Colloqu ia\n\n\nAbstract\nMatroids are combinatorial structures that track ``indep endence'' relations on a set.\nA key example is linear independence of som e linear functions on a vector space.\nNot all matroids come from a vector space\,\nbut those that don't behave in surprising algebraic ways as if t hey do.\nBreakthroughs of the last decade have opened a kit of tools\nfrom \, and inspired by\, algebraic geometry to prove inequalities for matroids \,\namong them the ``matroid Hodge theory'' of June Huh and others.\nI'll start by motivating matroids\,\nand aim to end with enough about my work i n progress with Andy Berget\nto show how its central tool is different to matroid Hodge theory.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/70/ END:VEVENT BEGIN:VEVENT SUMMARY:Rohini Ramadas (University of Warwick) DTSTART:20241106T140000Z DTEND:20241106T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/71 DESCRIPTION:Title: Complex dynamics via algebraic geometry\nby Rohini Ramadas (University of Warwick) as part of Sheffield Pure Maths Colloquia\n\n\nAb stract\nComplex dynamics began in the early 1900s with the study of iterat ing polynomial functions with complex coefficients. This simple idea gives rise to beautiful fractal pictures such as the Mandelbrot set\, as well a s interesting mathematical questions of many different flavours (algebraic \, analytic\, topological\, arithmetic\, etc.). The field gained momentum in the 1980s due to work of Thurston\, Douady-Hubbard\, Sullivan\, and oth ers\, connecting these dynamical questions to surface topology and the the ory of 3-manifolds. The last decade has seen many breakthroughs achieved v ia new tools from number theory\, measure theory and algebraic geometry. I will discuss some of these recent developments\, highlighting the interpl ay between topology on one hand and algebraic geometry/number theory on th e other hand.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/71/ END:VEVENT BEGIN:VEVENT SUMMARY:Gavin Brown (University of Warwick) DTSTART:20241113T140000Z DTEND:20241113T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/72 DESCRIPTION:Title: Flops and noncommutative potentials\nby Gavin Brown (Unive rsity of Warwick) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\ nI give an overview of a project with Michael Wemyss to classify simple 3- fold flops. This amounts to understanding when a smooth rational curve (i. e. the Riemann sphere) inside a complex 3-dimensional manifold can be cont racted. (One might think of this as a 3d analogue of shrinking the central axis of a Moebius strip to a point\, and indeed one could do an unnecessa rily elaborate analysis of that situation by the same methods.) In fact\, this large family of surgery operations is central to 3d complex geometry\ , but has nevertheless resisted classification\, or even the construction of a set of representative examples.\n\nBeing a manifold\, one can descri be the situation by glueing together patches - and it is enough to glue to gether two copies of the affine space $\\mathbb{C}^3$ by a simple formula .. but with lots of free parameters\, most of which do not contract. Howev er finding good (i.e. contractible) glue functions (or even classifying th em) seems to be a bit needle-in-a-haystack. Instead\, we translate the pro blem to one of classifying noncommutative germs $f(x\,y)$ [or equivalently certain complete local algebras up to isomorphism]\, where the necessary criteria seem more amenable. That context feels much like the classical si ngularity theory of function germs in the style of Arnold (types ADE and a ll that)\, and we can solve enough of that problem to construct all flops and to provide a classification.\n\nFrom one point of view\, I’d like to give some idea of what Theorems 5.1 and 5.4 of the following expository t ea-time article mean:\nhttps://arxiv.org/abs/2410.21500\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/72/ END:VEVENT BEGIN:VEVENT SUMMARY:Cheuk Yu Mak (University of Sheffield) DTSTART:20241127T140000Z DTEND:20241127T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/73 DESCRIPTION:Title: From area preserving homeomorphism groups to symplectic Khovan ov homology and beyond\nby Cheuk Yu Mak (University of Sheffield) as p art of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn the first half of the talk\, I will explain some recent breakthroughs in the study of the ar ea preserving homeomorphism groups of surfaces using Floer theory. After t hat\, I will explain what happens when we try to generalize it to higher d imensions and the relation to Khovanov homology as well as the Hilbert sch emes of points. No prior knowledge on Floer theory or symplectic geometry is assumed.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/73/ END:VEVENT BEGIN:VEVENT SUMMARY:Jason Semeraro (University of Loughborough) DTSTART:20241120T140000Z DTEND:20241120T150000Z DTSTAMP:20260422T225723Z UID:SheffieldPureMaths/76 DESCRIPTION:Title: Fusion-stable representations of finite groups\nby Jason S emeraro (University of Loughborough) as part of Sheffield Pure Maths Collo quia\n\n\nAbstract\nThis is joint work with my PhD student Tom Lawrence.\n \nFor a prime p\, the p-decomposition matrix D of a finite group G records the way each irreducible ordinary representation of G breaks up into irre ducible p-Brauer characters under reduction modulo p. Multiplying D by its transpose yields the Cartan matrix\, whose determinant is well-known to b e a power of p. A representation of a Sylow p-subgroup S of G is fusion-st able if it is left invariant by the conjugation action of G. After first f ixing a basis B of fusion-stable representations of S one can consider an analogue of D for fusion-stable representations which records how each irr educible ordinary representation of G breaks up in B under restriction to S. It turns out this matrix has many properties analogous to those of the classical decomposition matrix\, and using them one can show that the modu lus square of the determinant of the fusion-stable character table (column s indexed by G-classes of p-elements\, rows by elements of B) is always a particular power of p independent of the choice of B. I conjectured that t he same result holds for any saturated fusion system on S and I'll provide some evidence for this by explicitly computing with some infinite familie s of exotic examples. If time permits I will also explain how this project fits within the larger framework of "exotic representation theory" whose aim to extend results about ordinary representations to the settings of fu sion systems\, spetses and other related structures.\n LOCATION:https://researchseminars.org/talk/SheffieldPureMaths/76/ END:VEVENT END:VCALENDAR