BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Arkadiy Skopenkov (Independent University of Moscow\, and Moscow I nstitute of Physics and Technology) DTSTART:20211008T160000Z DTEND:20211008T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/2 DESCRIPTION:Title: Invariants of graph drawings in the plane\nby Arkadiy Skopenkov (Independent University of Moscow\, and Moscow Institute of Phy sics and Technology) as part of Selected Topics in Mathematics - Online Ed ition\n\n\nAbstract\nWe present a simplified exposition of some classical results on graph drawings in the plane. These results are chosen so that t hey illustrate some spectacular recent higher-dimensional results on the b order of topology and combinatorics. In particular\, we define a $\\bmod 2 $−valued self−intersection invariant (i.e. the van Kampen number) and its generalizations. We present elementary formulations and arguments acce ssible to mathematicians not specialized in any of the areas mentioned. Mo tivated by algorithmic\, combinatorial and geometric problems\, we introdu ce starting ideas of algebraic topology.\n\nThis talk is based on work in the following paper https://arxiv.org/abs/1805.10237\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Polina Vytnova (University of Warwick) DTSTART:20211015T160000Z DTEND:20211015T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/3 DESCRIPTION:Title: Hausdorff dimension of Gauss−Cantor sets and their appl ications to the study of classical Markov spectrum\nby Polina Vytnova (University of Warwick) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe classical Lagrange and Markov spectra are subs ets of the real line which arise in connection with some problems in theor y Diophantine approximation theory. In 1921 O. Perron gave a definition in terms of continued fractions\, which allowed to study the Markov and Lagr ange spectra using limit sets of iterated function schemes.\n\nIn this tal k we will see how an upper bounds on the Hausdorff dimension of the differ ence of the Markov and Lagrange spectra can be computed by the means of es timating Hausdorff dimension of the certain Gauss-Cantor sets.\n\nThe talk is based on a joint work with C. Matheus\, C. G. Moreira and M. Pollicott .\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexey Sossinsky (Independent University of Moscow) DTSTART:20211022T140000Z DTEND:20211022T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/4 DESCRIPTION:Title: Energy Functionals and the Normal Forms of Knots and Plan e Curves\nby Alexey Sossinsky (Independent University of Moscow) as pa rt of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nFor a ny $\\varepsilon\\in[0\,1]$\, I will define the notion of $\\varepsilon$-t hin knot\, which for $\\varepsilon\\in[0\,1]$ is a mathematical model of a thin flexible rope of fixed length and\, for $\\varepsilon=0$\, is a clas sical smooth knot of fixed length. The aim is to construct an energy funct ional that would classify $\\varepsilon$-thin knots with a fixed positive $\\varepsilon$\nby bringing them to “form”\, i.e.\, to a position corr esponding to a local minimum of the functional.\n\nThe talk will begin wit h the demonstration of mechanical experiments with wire knots\, showing ho w they automatically switch from any position to normal form. I will then show videos of computer experiments with thin knots (from joint work with my former pupils O. Karpenkov and S. Avvakumov)\, showing how they evolve from some initial position to their normal form. In the videos\, we deal w ith polygonal knots rather than smooth thin knots\, and the functional con sists of a summand that tries to straighten out the curve locally and a su mmand that forbids self-intersections (and the subsequent crossing of one part of the curve by another). In practice\, the algorithm always terminat es\, which is actually amazing\, because its termination is not a determin stic fact (i.e.\, not a theorem) − it is due to probabilistic laws of na ture.\n\nAn unexpected result of our study is that the normal form is not always unique − for the eight knot there are two: they depend on the mec hanical characteristics of the wire\, and on the initial shape of the knot .\n\nAfter that I will discuss other possible energy functionals\, in part icular involving not only the minimization of curvature\, but also that of torsion (physically\, this means we are dealing with elastic wire knots t hat resist twisting).\n\nTime permitting\, I will briefly explain our work on plane curves\, a by-product of which was our solution of the Euler ela sticae problem\, and formulate some conjectures.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Lenny Fukshansky (Claremont McKenna College) DTSTART:20211029T140000Z DTEND:20211029T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/5 DESCRIPTION:Title: Representing integers by multilinear polynomials\nby Lenny Fukshansky (Claremont McKenna College) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nGiven a homogeneous multiline ar polynomial $F(x)$ in $n$ variables with integer coefficients\, we obtai n some sufficient conditions for it to represent all the integers. Further \, we derive effective results\, establishing bounds on the size of a solu tion $x$ to the equation $F(x)=b$\, where $b$\nis any integer. For a speci al class of polynomials coming from determinants of rectangular matrices w e are able to obtain necessary and sufficient conditions for such an effec tive representation problem. This result naturally connects to the problem of extending a collection of primitive vectors to a basis in a lattice\, where we present counting estimates on the number of such extensions.\n\nT he talk is based on joint works with A. Boettcher and with M. Forst.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Slilaty (Wright State University) DTSTART:20211105T150000Z DTEND:20211105T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/6 DESCRIPTION:Title: Representing Matroids using Partial Fields\nby Daniel Slilaty (Wright State University) as part of Selected Topics in Mathemati cs - Online Edition\n\nAbstract: TBA\n\nThe abstract of the talk is availa ble on the seminar website\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Irene Pasquinelli (University of Bristol) DTSTART:20211119T150000Z DTEND:20211119T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/7 DESCRIPTION:Title: Mapping class group orbit closures for non-orientable sur faces\nby Irene Pasquinelli (University of Bristol) as part of Selecte d Topics in Mathematics - Online Edition\n\n\nAbstract\nThe study of the a symptotic growth of the number of closed geodesics on a hyperbolic surface dates back to Huber (1961) and has implications in various fields of math ematics. In her thesis\, Mirzakhani proved that for an orientable hyperbol ic surface of finite area\, the number of simple closed geodesics of lengt h less than L is asymptotically equivalent to a polynomial in L\, whose de gree only depends on the Euler characteristic. \n\nWhen looking at non-ori entable surfaces\, the situation is very different. One of the main differ ences in this framework is the behaviour of the action of the mapping clas s group on the space of measured laminations. \n\nIn a joint work with Erl andsson\, Gendulphe and Souto\, we characterised mapping class group orbit closures of measured laminations\, projective measured laminations and po ints in Teichmueller space.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Giovanni Panti (University of Udine) DTSTART:20211126T150000Z DTEND:20211126T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/8 DESCRIPTION:Title: Attractors of dual continued fractions\nby Giovanni P anti (University of Udine) as part of Selected Topics in Mathematics - Onl ine Edition\n\n\nAbstract\nWe identify continued fractions with piecewise- projective Markov maps on some interval\, provided that all branches are induced by elements in a fixed Hecke triangle group. Ordinary continued fr actions constitute a very special case\, relative to the (2\,3\,infinity) group. The branches of the map are expanding on the domain interval\, but contracting on some other part of the real projective line. As such\, they give rise to an IFS whose attractor is relevant\, since it determines the natural extension and several properties\, both algebraic and dynamical\, of the original system. We use an appropriate generalization of the class ical Minkowski Question Mark function to simultaneously linearize all maps resulting from the same triangle group\, and to prove that the above IFS satisfies the open set condition. We draw consequences -as well as open pr oblems- from these facts.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Felikson (Durham University) DTSTART:20211203T150000Z DTEND:20211203T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/9 DESCRIPTION:Title: Friezes for a pair of pants\nby Anna Felikson (Durham University) as part of Selected Topics in Mathematics - Online Edition\n\ n\nAbstract\nFrieze patterns are numerical arrangements that satisfy a loc al arithmetic rule. Conway and Coxeter showed that frieze patterns are tig htly connected to triangulated polygons. Recently\, friezes were actively studied in connection to the theory of cluster algebras\, and the notion o f a frieze obtained a number of generalisations. In particular\, one can d efine a frieze associated with a bordered marked surface endowed with a de corated hyperbolic metric.\n\nWe will review the construction and will sho w that some nice properties can be extended friezes associated to a pair o f pants. This work is joint with Ilke Canakci\, Ana Garcia and Pavel Tumar kin.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/9/ END:VEVENT BEGIN:VEVENT SUMMARY:John Hunton (Durham University) DTSTART:20211210T150000Z DTEND:20211210T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/10 DESCRIPTION:Title: What is the space group of an aperiodic pattern?\nby John Hunton (Durham University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe space groups/crystallographic groups/ Bieberbach groups (depending on what dimension you are interested in) have played a classical role in the understanding of periodic patterns in d-di mensional space for many generations: the space group of a periodic patter n $P\\subset\\mathbf{R}^d$ is the subgroup of the full isometry group of $ \\mathbf{R}^d$ that fixes $P$. In some sense the space group determines up to local decoration the pattern itself. \n \nBut what of aperiodic patter ns\, such as the Penrose tiling\, or the objects used to model quasicrysta ls\, or more generally any aperiodic pattern? By their nature they are rat her short of symmetries\, but on the other hand\, they have a lot of appro ximate symmetries\, structure that can be accessed by topological methods. \n \nIn this talk I will sketch some methods developed with Jamie Walton to define the corresponding "space groups" for such objects.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexey Sossinsky (Independent University of Moscow) DTSTART:20211217T150000Z DTEND:20211217T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/11 DESCRIPTION:Title: Thickened Knots: Energy Functionals and Normal Forms \nby Alexey Sossinsky (Independent University of Moscow) as part of Select ed Topics in Mathematics - Online Edition\n\n\nAbstract\nFor any $ \\varep silon \\in [0\, 1]$\, I will define the notion of $ \\varepsilon$-thickene d knot\, which for $ \\varepsilon \\in (0\, 1]$ is a mathematical model of a thin flexible rope of fixed length with extremities glued together. The aim is to construct an energy functional that would classify $ \\varepsil on$-thickened knots with a fixed positive $ \\varepsilon$ by bringing them to "normal form" by gradient descent\, i.e.\, to a position corresponding to a local minimum of the functional.\n\nThe talk will begin with the dem onstration of mechanical experiments with wire knots\, showing how they au tomatically switch from any position to normal form. I will then show vide os of computer experiments with thickened knots (from joint work with my f ormer pupils O. Karpenkov and S. Avvakumov)\, showing how they evolve from some initial position to their normal form. In the videos\, we deal with polygonal knots rather than smooth thickened knots\, and the functional co nsists of a summand that tries to straighten out the curve locally and a s ummand that forbids self-intersections (and the subsequent crossing of one part of the curve by another). In practice\, the algorithm always termina tes\, which is actually amazing\, because its termination is not a determi nistic fact (i.e.\, not a theorem) - it is due to probabilistic laws of na ture.\n\nAn unexpected result of our study is that the normal form is not always unique - for the eight knot there are two: they depend on the mecha nical characteristics of the wire\, and on the initial shape of the knot.\ nI will then state some theorems about $ \\varepsilon$-thickened knots and formulate a conjecture about their complete and effective classification. Time permitting\, I will briefly explain our work on plane curves\, a by- product of which was our solution of the Euler elasticae problem\, formula te some conjectures\, and discuss other possible energy functionals\, in p articular involving not only the minimization of curvature\, but also that of writhe (physically\, this means we are dealing with elastic wire knots that resist twisting).\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Ivan Izmestiev (TU Wien) DTSTART:20220204T150000Z DTEND:20220204T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/12 DESCRIPTION:Title: Regge symmetry\nby Ivan Izmestiev (TU Wien) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe Regg e symmetry is a set of remarkable relations between two tetrahedra whose e dge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give an elementary geometric proof of Regge symmetries in Euclidean\, spherica l\, and hyperbolic geometry.\nThe talk is based on a joint work with Arsen iy Akopyan.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Fatemeh Mohammadi (Ghent University) DTSTART:20220304T150000Z DTEND:20220304T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/14 DESCRIPTION:Title: Matroid stratifications of hypergraph varieties and thei r realization spaces\nby Fatemeh Mohammadi (Ghent University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nI will p rovide an introductory talk to hypergraph varieties\, focusing on the comb inatorial aspect. The main themes of the talk are (1) connecting the geome tric properties of hypergraphs to their minimal matroids\; (2) reducing th e geometric invariants of these matroids to grid matroids\; and (3) unders tanding the realizability of these matroids. Finally\, I will mention the application to conditional independence models in statistics and will pres ent some geometric questions and computational challenges around this prob lem. This is based on joint works with Kevin Grace\, Oliver Clarke\, and Harshit J Motwani.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Alan Haynes (University of Houston) DTSTART:20220211T150000Z DTEND:20220211T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/15 DESCRIPTION:Title: Gap theorems for linear forms and for rotations on highe r dimensional tori\nby Alan Haynes (University of Houston) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThis talk i s based on joint work with Jens Marklof\, and with Roland Roeder. The thre e distance theorem states that\, if $x$ is any real number and $N$ is any positive integer\, the points $x\, 2x\, \\ldots \, Nx \\mod 1$ partition t he unit interval into component intervals having at most 3 distinct length s. We will present two higher dimensional analogues of this problem. In th e first we consider points of the form $mx+ny \\mod 1$\, where $x$ and $y$ are real numbers and $m$ and $n$ are integers taken from an expanding set in the plane. This version of the problem was previously studied by Geele n and Simpson\, Chevallier\, Erdős\, and many other people\, and it is cl osely related to the Littlewood conjecture in Diophantine approximation. T he second version of the problem is a straightforward generalization to ro tations on higher dimensional tori which\, surprisingly\, has been mostly overlooked in the literature. For the two dimensional torus\, we are able to prove a five distance theorem\, which is best possible. In higher dimen sions we also have bounds\, but establishing optimal bounds is an open pro blem.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Bernd Schulze (Lancaster University) DTSTART:20220429T140000Z DTEND:20220429T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/16 DESCRIPTION:Title: Group-theoretic methods for the rigidity analysis of dis crete structures\nby Bernd Schulze (Lancaster University) as part of S elected Topics in Mathematics - Online Edition\n\n\nAbstract\nGeometric ri gidity theory is concerned with the rigidity and flexibility analysis of b ar-joint frameworks and related constraint systems of geometric objects.\n In the beginning of this talk\, we will give a brief introduction to this area\, which has a rich history that can be traced back to classical work of Euler\, Cauchy and Maxwell on the rigidity of polyhedra and skeletal fr ames. One of the major recent research directions in this field is to stud y the impact of symmetry on the rigidity of bar-joint frameworks. We show how group representation theory can be used to reveal `hidden' infinitesim al motions and states of self-stress in symmetric frameworks that cannot b e detected with Maxwell's basic counting rule from 1864. We then show how this symmetry-adapted counting rule\, which was originally discovered by t he engineer Simon Guest and the chemist Patrick Fowler\, can be used to de rive an efficient new method for constructing symmetric frameworks with a large number of `fully-symmetric' or `anti-symmetric' states of self-stres s. Maximizing the number of independent states of self-stress of a planar framework\, as well as understanding their symmetry properties\, has impor tant practical applications\, for example in the design and construction o f gridshells. We show the usefulness of our method by applying it to some practical examples.\n\nThis is joint work with Cameron Millar (SOM)\, Arek Mazurek (Mazurek Consulting) and William Baker (SOM).\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Padraig Ó Catháin (Dublin City University) DTSTART:20220325T150000Z DTEND:20220325T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/17 DESCRIPTION:Title: Sequencing of Steiner Triple Systems\nby Padraig Ó Catháin (Dublin City University) as part of Selected Topics in Mathematic s - Online Edition\n\n\nAbstract\nSteiner triple systems (STSs) are the si mplest and best understood infinite family of block designs.\nThey are equ ivalent to decompositions of complete graphs into edge disjoint triangles. \nA sequencing of an STS is a bijection of the points with the integers $[ 1\, \\ldots\, v]$.\n\nIn 2019\, Stinson and Kreher introduced L-good seque ncings\, in which no block is contained in an\ninterval of length L\, that is $[d\, d+1\, \\ldots\, d+L]$. Stinson and Veitch gave an algorithmic pr oof\nthat an STS with v points has an L-good sequencing for $v \\geq L^{6} /16 + O(L^{5})$. This was later improved\nto $v \\geq L^{4}/2 + O(L^{3})$ by Blackburn and Linial with an explicit greedy algorithm.\n\nUsing the Lo vasz Local Lemma\, we prove that every STS with $v \\geq 121L^{2}$ admits an L-good\nsequencing. The exponent $2$ is optimal\, in the sense that cou nterexamples are known without L-good\nsequencings for $v = \\Theta(L^{2-\ \epsilon})$ for any $\\epsilon > 0$. In fact\, we prove a more general res ult on\ndecompositions of general Steiner systems into sufficiently large (as a function of L) disjoint independent sets.\nThis is joint work with D aniel Horsley of Monash University.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Krishnendu Gongopadhyay (IISER Mohali) DTSTART:20220311T150000Z DTEND:20220311T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/18 DESCRIPTION:Title: Reversibility of isometries\nby Krishnendu Gongopadh yay (IISER Mohali) as part of Selected Topics in Mathematics - Online Edit ion\n\n\nAbstract\nIn this talk\, I shall review recent work on reversibil ity of isometries of Hermitian spaces over the complex numbers and over th e quaternions. I shall explain what I mean by reversibility in the talk an d how it has been classified in some Lie groups.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Sinai Robins (Universidade de São Paulo) DTSTART:20220506T140000Z DTEND:20220506T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/19 DESCRIPTION:by Sinai Robins (Universidade de São Paulo) as part of Select ed Topics in Mathematics - Online Edition\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Sabir Gusein-Zade (Lomonosov Moscow State University) DTSTART:20220401T140000Z DTEND:20220401T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/20 DESCRIPTION:Title: Non-commutative analogue of the Berglund–Hübsch–Hen ningson duality and symmetries of the orbifold Euler characteristic (I)\nby Sabir Gusein-Zade (Lomonosov Moscow State University) as part of Sel ected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe Berglund –Hübsch–Henningson (BHH–) duality is a particular case of the mirro r symmetry. It is described as a duality on the set of pairs $(f\, G)$ con sisting of an invertible polynomial and a subgroup $G$ of diagonal symmetr ies of $f$. Symmetries of invariants of BHH-dual pairs are related to the mirror symmetry. There is a method to extend the BBH-duality to the set of pairs $(f\, G')$\, where $G'$ is the semidirect product of a group $G$ of diagonal symmetries of $f$ and a group $S$ of permutations of the coordin ates preserving $f$. The construction is based on ideas of A. Takahashi an d therefore is called the Berglund-Hübsch-Henningson-Takahashi (BHHT-) du ality. Invariants of BHHT-dual pairs have symmetries similar to mirror one s only under some restrictions on the group $S$: the so-called parity cond ition (PC). Under the PC-condition it is possible to prove symmetries of t he orbifold Euler characteristic and of some other orbifold invariants for actions on the Milnor fibers of dual pairs. The talk is based on joint re sults with W. Ebeling.\n\nThis is the first half of a two part talk.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Sabir Gusein-Zade (Lomonosov Moscow State University) DTSTART:20220408T140000Z DTEND:20220408T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/21 DESCRIPTION:Title: Non-commutative analogue of the Berglund–Hübsch–Hen ningson duality and symmetries of the orbifold Euler characteristic (II)\nby Sabir Gusein-Zade (Lomonosov Moscow State University) as part of Se lected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe Berglund –Hübsch–Henningson (BHH–) duality is a particular case of the mirro r symmetry. It is described as a duality on the set of pairs $(f\, G)$ con sisting of an invertible polynomial and a subgroup $G$ of diagonal symmetr ies of $f$. Symmetries of invariants of BHH-dual pairs are related to the mirror symmetry. There is a method to extend the BBH-duality to the set of pairs $(f\, G')$\, where $G'$ is the semidirect product of a group $G$ of diagonal symmetries of $f$ and a group $S$ of permutations of the coordin ates preserving $f$. The construction is based on ideas of A. Takahashi an d therefore is called the Berglund-Hübsch-Henningson-Takahashi (BHHT-) du ality. Invariants of BHHT-dual pairs have symmetries similar to mirror one s only under some restrictions on the group $S$: the so-called parity cond ition (PC). Under the PC-condition it is possible to prove symmetries of t he orbifold Euler characteristic and of some other orbifold invariants for actions on the Milnor fibers of dual pairs. The talk is based on joint re sults with W. Ebeling.\n\nThis is the second half of a two part talk.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Simon Kristensen (Aarhus University) DTSTART:20220617T140000Z DTEND:20220617T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/22 DESCRIPTION:by Simon Kristensen (Aarhus University) as part of Selected To pics in Mathematics - Online Edition\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/22/ END:VEVENT BEGIN:VEVENT SUMMARY:James Cruickshank (University of Galway) DTSTART:20221019T150000Z DTEND:20221019T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/23 DESCRIPTION:Title: Global Rigidity of Triangulated Manifolds\nby James Cruickshank (University of Galway) as part of Selected Topics in Mathemati cs - Online Edition\n\n\nAbstract\nIn this talk I will begin with an intro duction to global rigidity of frameworks and graphs and then describe rece nt joint work with Bill Jackson and Shin-ichi Tanigawa in which we prove t he following.\n

\n

\n Theorem. For \\(d\\geq 3\\) the graph of a connected triangulated \\((d-1)\\)-manifold is generically globally rigid in \\(\\mathbb R^{d}\\) if and only if the graph is \\((d+1 )\\)-connected and\, if \\(d=3\\)\, not planar.\n

\n

\nThis proves and generalises a conjecture of Connelly. I will also discuss some applic ations of this result and of the techniques we use in the proof. We prove the generic case of a conjecture of Kalai on the reconstructability of a p olytope from its space of stresses. We also use our methods to generalise parts of the Lower Bound Theorem to a larger class of simplicial complexes .\n

\n

\n\n Some context for a general audience.\n \nA graph is said to be globally rigid in \\(\\mathbb R^d\\) if a generic embedding of the vertex set in \\(\\mathbb R^d\\) is determined\, up to is ometry of \\(\\mathbb R^d\\)\, by the distances between adjacent vertices. There is a weaker local version of rigidity in which the embedding is on ly determined within some neighbourhood. More detail\, and examples\, will be given in the talk. \n

\n

\nJackson and Jordán\, following e arlier work of Connelly\, have characterised graphs that are globally rigi d in \\(\\mathbb R^2\\) in terms of the 2-dimensional rigidity matroid. H owever extending this characterisation to higher dimensions is a very chal lenging open problem. Indeed there are very few examples known of natural ly interesting infinite families of graphs for which the global rigidity p roblem in \\(\\mathbb R^d\\)\, for \\(d\\geq 3\\)\, has been settled. \n < /p>\n

\nOne interesting family of graphs in this context are those aris ing as graphs of triangulations of manifolds. Fogelsanger showed that the graph of a triangulated \\(d\\)-manifold is locally rigid in \\(\\mathbb R^{d+1}\\)\, but the global rigidity problem for such graphs remained open . Connelly conjectured that \\(4\\)-connected triangulations of non-spheri cal surfaces are globally rigid in \\(\\mathbb R^3\\). In higher dimensio ns\, even the global rigidity of graphs of simplicial polytopes remained a n open question. Kalai\, Tay and others have used the local rigidity theo ry of graphs to prove important results concerning face numbers of pseudom anifolds\, but global rigidity theory remains relatively unexplored in tha t context.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Igor Potapov (University of Liverpool) DTSTART:20221026T150000Z DTEND:20221026T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/24 DESCRIPTION:Title: Reachability Problems in Matrix Semigroups\nby Igor Potapov (University of Liverpool) as part of Selected Topics in Mathematic s - Online Edition\n\n\nAbstract\nMatrices and matrix products play a cruc ial role in the representation and analysis of various computational proce sses\, The central decision problem in matrix semigroups is the membership problem: "Decide whether a given matrix $M$ belongs to a finitely generat ed matrix semigroup". By restricting $M$ to be the identity (zero) matrix the problem is known as the identity (mortality) problem. \n\nUnfortunatel y\, many simply formulated and elementary problems for matrices are inhere ntly difficult to solve even in dimension two\, and most of these problems become undecidable in general starting from dimension three or four. For example\, the identity problem for $3\\times 3$ matrices of integers is th e long-standing open problem.\n\nIn this talk I will provide an overview a bout various decision problems in matrix semigroups such as membership\, v ector reachability\, freeness\, scalar reachability\, etc. Also\, I will f ocus on the number of state-of-the-art theoretical computer science techni ques as well as decidability\, undecidability and computational complexity results.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Anton Lukyanenko (George Mason University) DTSTART:20221102T160000Z DTEND:20221102T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/25 DESCRIPTION:Title: Heisenberg continued fractions: overview and recent resu lts\nby Anton Lukyanenko (George Mason University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nContinued fraction theory over the real numbers has a long connection to real hyperbolic geom etry.\nAbout 10 years ago\, Joseph Vandehey and I proposed a new CF algori thm over the non-commutative\nHeisenberg group\, which is designed to take advantage of complex hyperbolic theory\,\nand connects directly to the wo rk of Falbel-Francsics-Lax-Parker\, Hersonsky-Paulin\, Series\, \nKatok-Ug arkovici\, Nakada\, Hensley\, and others.\n\nWe have since connected the t heory to Diophantine approximation\, established ergodicity\nof the Gauss- type map (for a folded variant of the CF)\, and developed a broader framew ork of\nIwasawa CFs\, which include many real\, higher-dimensional\, and n on-commutative CF algorithms.\nMore recently\, we returned to the Euclidea n setting to explore the dynamics of CFs over the complex \nnumbers\, quat ernions\, octonions\, as well as defining new CF algorithms in \\(\\mathbb {R}^3\\).\n\nIn this talk\, I will start by discussing the A. Hurwitz comp lex CFs as a motivating higher-dimensional algorithm\,\nthen discuss the H eisenberg group and Heisenberg CFs\, and then provide an overview of my wo rk with\nVandehey\, finishing with this year's results in Euclidean space. \n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Valentin Ovsienko (University of Reims Champagne-Ardenne) DTSTART:20221116T160000Z DTEND:20221116T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/26 DESCRIPTION:Title: Shadows of numbers: supergeometry with a human face\ nby Valentin Ovsienko (University of Reims Champagne-Ardenne) as part of S elected Topics in Mathematics - Online Edition\n\n\nAbstract\nIn this elem entary and accessible to everybody talk\, I will explain an attempt to app ly supersymmetry and supergeometry to arithmetic. The following general id ea looks crazy. What if every integer sequence has another integer sequenc e that follows it like a shadow? I will demonstrate that this is indeed th e case\, though perhaps not for every integer sequence\, but for many of t hem. The main examples are those of the Markov numbers and Somos sequences .\n\nIn the second part of the talk\, I will discuss the notions of supers ymmetric continued fractions and the modular group\, and arrive at yet a m ore crazy idea that every rational and every irrational has its own shadow . The second part of the talk is a joint work with Charles Conley.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Karin Baur (University of Leeds) DTSTART:20221123T160000Z DTEND:20221123T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/27 DESCRIPTION:Title: Frieze patterns and cluster theory\nby Karin Baur (U niversity of Leeds) as part of Selected Topics in Mathematics - Online Edi tion\n\n\nAbstract\nCluster categories and cluster algebras can be describ ed via triangulations of surfaces or via Postnikov diagrams. \nIn type A\ , such triangulations lead to frieze patterns or SL\\(_2\\)-friezes in the sense of Conway and Coxeter. \nWe explain how infinite frieze patterns a rise and how Grassmannians or Plücker coordinates give rise to SL\\(_k \\)-friezes.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Matteo Raffaelli (TU Wien) DTSTART:20221207T160000Z DTEND:20221207T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/28 DESCRIPTION:Title: Curvature-adapted submanifolds of semi-Riemannian groups \nby Matteo Raffaelli (TU Wien) as part of Selected Topics in Mathemat ics - Online Edition\n\n\nAbstract\nGiven a semi-Riemannian hypersurface $ M$ of a semi-Riemannian manifold $Q$\, one says that $M$ is $\\textit{curv ature-adapted}$ if\, for each $p \\in M$\, the normal Jacobi operator and the shape operator of $M$ at $p$ commute. The first operator measures the curvature of the ambient manifold along the normal vector of $M$\, whereas the second describes the curvature of $M$ as a submanifold of $Q$. This c ondition can be generalized to submanifolds of arbitrary codimension.\n\nI n this talk I will present joint work with Margarida Camarinha addressing the case where the ambient manifold is a Lie group equipped with a bi-inva riant metric. In particular\, we will see that\, if the normal bundle of $ M$ is $\\textit{closed under the Lie bracket}$ (i.e.\, if each normal spac e corresponds\, under the group's left action\, to a Lie subalgebra)\, the n curvature adaptedness can be understood geometrically\, in terms of left translations. Incidentally\, our analysis offers a new case-independent p roof of a well-known fact: every three-dimensional Lie group equipped with a bi-invariant semi-Riemannian metric has constant curvature.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/28/ END:VEVENT BEGIN:VEVENT SUMMARY:John Blackman (University of Liverpool) DTSTART:20221214T160000Z DTEND:20221214T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/29 DESCRIPTION:Title: Cutting Sequences and the p-adic Littlewood Conjecture\nby John Blackman (University of Liverpool) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nOne of the main themes of D iophantine approximation is the study of how well real numbers can be appr oximated by rational numbers. Classically\, a real number is defined to be well-approximable if the Markov constant is 0\, i.e. \\(M(x):=\\lim \\inf {q||qx||}=0\\). Otherwise\, the number is badly approximable\, with large r values of \\(M(x)\\) indicating worse rates of approximation. As a sligh t twist on this notion of approximability\, the \\(p\\)-adic Littlewood Co njecture asks if -- given a prime \\(p\\) and a badly approximable number \\(x\\) -- one can always find a subsequence of \\(xp^k\\) such that the M arkov constant of this sequence tends to \\(0\\)\, i.e. if \\(\\lim \\inf {M(xp^k)} =0\\).\n\nIn this talk\, I will outline a brief history of the \ \(p\\)-adic Littlewood Conjecture and discuss how hyperbolic geometry can be used to help understand the problem further. In particular\, I will dis cuss how one can represent integer multiplication of continued fractions b y replacing one triangulation of the hyperbolic plane with an alternative triangulation. Finally\, I will give a reformulation of pLC using infinite loops -- a family of objects that arise from this setting.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Andy Hone (University of Kent) DTSTART:20230208T160000Z DTEND:20230208T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/30 DESCRIPTION:Title: Continued fractions from hyperelliptic curves\nby An dy Hone (University of Kent) as part of Selected Topics in Mathematics - O nline Edition\n\n\nAbstract\nWe consider a family of nonlinear maps that a re generated from the continued fraction expansion of a function on a hype relliptic curve of genus \\(g\\)\, as originally described by van der Poor ten. Using the connection with the classical theory of \\(J\\)-fractions a nd orthogonal polynomials\, we show that in the simplest case \\(g=1\\) th is provides a straightforward derivation of Hankel determinant formulae fo r the terms of a general Somos-\\(4\\) sequence\, which were found in part icular cases by Chang\, Hu\, and Xin. We extend these formulae to the high er genus case\, and prove that generic Hankel determinants in genus \\(2\\ ) satisfy a Somos-\\(8\\) relation. Moreover\, for all \\(g\\) we show tha t the iteration for the continued fraction expansion is equivalent to a di screte Lax pair with a natural Poisson structure\, and the associated nonl inear map is a discrete integrable system\, connected with solutions of th e infinite Toda lattice. If time permits\, we will also mention the link t o (Stieltjes) \\(S\\)-fractions via contraction\, and a family of maps ass ociated with the Volterra lattice\, described in current joint work with J ohn Roberts and Pol Vanhaecke.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Iskander Aliev (Cardiff University) DTSTART:20230301T160000Z DTEND:20230301T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/31 DESCRIPTION:Title: Sparse integer points in rational polyhedra: bounds for the integer Caratheodory rank\nby Iskander Aliev (Cardiff University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\n We will give an overview of the recent results on sparse integer points (t hat is\, the integer points with a relatively large number of zero coordin ates) in the rational polyhedra of the form \\(\\{x: Ax=b\, x\\geq 0\\}\\) \, where \\(A\\) is an integer matrix\, and \\(b\\) is an integer vector. In particular\, we will discuss the bounds on the Integer Caratheodory ran k in various settings and proximity/sparsity transference results.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Nikolay Moshchevitin (Israel Institute of Technology (Technion)) DTSTART:20230308T160000Z DTEND:20230308T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/32 DESCRIPTION:Title: Geometry of Diophantine Approximation\nby Nikolay Mo shchevitin (Israel Institute of Technology (Technion)) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nWe discuss some cla ssical and modern results related to the geometry of Diophantine Approxima tion\, in particular some multidimensional generalizations of continued fr actions algorithm related to patterns of the best approximations. Importan t tools for the study of the properties of approximations are related to i rrationality measure functions. We will give a brief introduction into the theory and explain a recent conjecture by Schmidt and Summerer and its so lution.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Bob Connelly (Cornell University) DTSTART:20230315T160000Z DTEND:20230315T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/33 DESCRIPTION:Title: Global Rigidity of Braced Convex Polygons\nby Bob Co nnelly (Cornell University) as part of Selected Topics in Mathematics - On line Edition\n\n\nAbstract\nA framework in the plane is globally rigid if any other realization of the framework with corresponding edges the same l ength is congruent. For example\, a collection of triangles placed end-to -end without overlap such that a bar connecting the first triangle to the last\, intersecting the interior of each triangle\, is globally rigid. W e would like to tell when a convex polygon with braces inside connecting t he vertices so that for \\(n\\) vertices there are \\(n-2\\) internal bra ces\, then this framework is always globally rigid. But we can’t do tha t yet. However\, we have some interesting classes of braced convex polygo nal frameworks that are always globally rigid.\n\n \n\nThis is joint work with Bob Connelly\, Bill Jackson\, Shin-ichi Tanagawa\, and Albert Zhen\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Garrity (Williams College) DTSTART:20230322T160000Z DTEND:20230322T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/34 DESCRIPTION:Title: On Partition Numbers and Multi-dimensional Continued Fra ctions\nby Thomas Garrity (Williams College) as part of Selected Topic s in Mathematics - Online Edition\n\n\nAbstract\nThis talk will link parti tion numbers from combinatorics with a certain multi-dimensional continued fraction algorithm from number theory and dynamical systems.\n\nAndrew an d Eriksson's Introduction to Integer Partitions starts with discussing Eul er's identity\, Every number has as many integer partitions into odd p arts as into distinct parts. As they state\, this is quite surprising if you have never seen it before. There are\, though\, many other equally if not more surprising partition identities. For all there are two basic questions. First\, how to even guess the existence of any potential partit ion identities. Then\, once a possible potential identity is conjectured\, how to prove it.\n\nIn joint work with Bonanno\, Del Vigna and Isola\, th ere was developed a link between traditional continued fractions and the s low triangle map (a type of multi-dimensional continued fraction algorithm ) with integer partitions of numbers into two or three distinct parts\, wi th multiplicity. These maps were initially introduced for number theoretic reasons but have over the years exhibited many interesting dynamical prop erties. In work with Wael Baalbaki\, we will see that the slow triangle ma p\, when extended to higher dimensions\, will provide a natural map (an al most internal symmetry) from the set of integer partitions to itself.\n\nT hus we will allow us to create a new technique for generating any number o f partition identities.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Pavel Tumarkin (Durham University) DTSTART:20230329T150000Z DTEND:20230329T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/35 DESCRIPTION:Title: Farey graph and ideal tetrahedra\nby Pavel Tumarkin (Durham University) as part of Selected Topics in Mathematics - Online Edi tion\n\n\nAbstract\nWe construct a 3-dimensional analog of the Farey tesse lation and show that it inherits many properties of the usual 2-dimensiona l Farey graph. As a by-product\, we get a 3-dimensional counterpart of the Ptolemy relation. The talk is based on an ongoing work joint with Anna Fe likson\, Oleg Karpenkov and Khrystyna Serhiyenko.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Egon Schulte (Northeastern University) DTSTART:20230405T150000Z DTEND:20230405T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/36 DESCRIPTION:Title: Skeletal Polyhedral Geometry and Symmetry\nby Egon S chulte (Northeastern University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe study of highly symmetric structures in Euclidean \\(3\\)-space has a long and fascinating history tracing back to the early days of geometry. With the passage of time\, various notions of polyhedral structures have attracted attention and have brought to lig ht new exciting figures intimately related to finite or infinite groups of isometries. A radically different\, skeletal approach to polyhedra was pi oneered by Grunbaum in the 1970's building on Coxeter's work. A polyhedron is viewed not as a solid but rather as a finite or infinite periodic geom etric edge graph in space equipped with additional polyhedral super-struct ure imposed by the faces. Since the mid 1970's there has been a lot of act ivity in this area. Much work has focused on classifying skeletal polyhedr a and complexes by symmetry\, with the degree of symmetry defined via dist inguished transitivity properties of the geometric symmetry groups. These skeletal figures exhibit fascinating geometric\, combinatorial\, and algeb raic properties and include many new finite and infinite structures.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Valérie Berthé (IRIF) DTSTART:20230412T150000Z DTEND:20230412T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/37 DESCRIPTION:Title: Balanced words and symbolic dynamical systems\nby Va lérie Berthé (IRIF) as part of Selected Topics in Mathematics - Online E dition\n\n\nAbstract\nThe chairman assignment problem can be stated as fol lows: \\(k\\) states are assumed to form a union and each year a union cha irman must be selected so that at any time the cumulative number of chairm en of each state is proportional to its weight.\nIt is closely related to the (discrete) apportionment problem\, which has its origins in the questi on of allocating seats in the house of representatives in the United State s\, in a proportional way to the population of each state.\nThe richness o f this problem lies in the fact that it can be reformulated both as a sequ encing problem in operations research for optimal routing and scheduling\, and as a symbolic discrepancy problem\, in the field of word combinatoric s\, where the discrepancy measures the difference between the number of oc currences of a letter in a prefix of an infinite word and the expected val ue in terms of frequency of occurrence of this letter. \nWe will see in th is lecture how to construct infinite words with values in a finite alphabe t having the smallest possible discrepancy\, by revisiting a construction due to R. Tijdeman in terms of dynamical systems.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Jörg Thuswaldner (Montanuniversität Leoben) DTSTART:20230419T150000Z DTEND:20230419T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/38 DESCRIPTION:Title: Multidimensional continued fractions and symbolic coding s of toral translations\nby Jörg Thuswaldner (Montanuniversität Leob en) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstra ct\nThe aim of this lecture is to find symbolic codings for translations o n the $d$-dimensional torus that enjoy many of the well-known properties o f Sturmian sequences (like low complexity\, balance of factors\, bounded r emainder sets of any scale). Inspired by the approach of G. Rauzy we const ruct such codings by the use of multidimensional continued fraction algori thms that are realized by sequences of substitutions. This is joint work w ith V. Berthé and W. Steiner.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Gekhtman (University of Notre Dame) DTSTART:20230510T150000Z DTEND:20230510T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/40 DESCRIPTION:Title: Unified approach to exotic cluster structures in simple Lie groups\nby Michael Gekhtman (University of Notre Dame) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nWe present a construction for cluster charts in simple Lie groups compatible with Poi sson structures in the Belavin-Drinfeld classification. The key ingredient is a birational Poisson map from the group to itself that transform a Poi sson bracket associated with a nontrivial Belavin-Drinfeld data into the s tandard one. It allows us to obtain a cluster chart as a pull-back of the Berenstein-Fomin-Zelevinsky cluster coordinates on the open double Bruhat cell. This is a joint work with M. Shapiro and A. Vainshtein.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Nathan Uricchio (WPI) DTSTART:20230426T150000Z DTEND:20230426T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/41 DESCRIPTION:by Nathan Uricchio (WPI) as part of Selected Topics in Mathema tics - Online Edition\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Ekaterina Shemyakova (University of Toledo) DTSTART:20231018T150000Z DTEND:20231018T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/42 DESCRIPTION:Title: On super cluster algebras based on super Plü\;cker and super Ptolemy relations\nby Ekaterina Shemyakova (University of To ledo) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbst ract\nI will speak about super exterior powers and our results on super an alogs of Plü\;cker embedding for the Grassmann manifold.\n\nThe probl em was motivated by the search for the definition of super cluster algebra s. Based on the obtained super Plü\;cker relations (which we have for the general case)\, we propose a super cluster structure for super Grass mannians \\(\\mathrm{Gr}_{2|0}(n|1)\\). The exchange graph structure is n ow understood. \n\nWe show how to simplify the super Plü\;cker relat ions for \\(\\mathrm{Gr}_{r|1}(n|1)\\)\, which can be seen as dual to $\\ mathrm{Gr}_{n-r|0}(n|1)$. We also present how super Ptolemy relations of P enner-Zeitlin for decorated super Teichmü\;ller space --- the basis o f Musiker-Ovenhouse-Zhang's super cluster algebra definition --- can be re -written as super Plü\;cker relations.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Derek Kitson (Mary Immaculate College) DTSTART:20231206T160000Z DTEND:20231206T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/43 DESCRIPTION:Title: Rigid graphs in dimension 3\nby Derek Kitson (Mary I mmaculate College) as part of Selected Topics in Mathematics - Online Edit ion\n\n\nAbstract\nA graph is rigid in \\(d\\)-\;dimensional Euclidean space if there is an embedding of the vertices which admits no non-\;tr ivial edge-\;length preserving continuous motion. Rigid graphs in dimen sions \\(1\\) and \\(2\\) are characterised by simple counting rules\, but currently no such rules are available in higher dimensional Euclidean spa ces. We will provide a gentle introduction to graph rigidity and report on recent progress in characterising rigid graphs for a class of cylindrical normed spaces of dimension \\(3\\). This is joint work with Sean Dewar (U niversity of Bristol).\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Sophie Morier-Genoud (University of Reims) DTSTART:20231011T150000Z DTEND:20231011T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/44 DESCRIPTION:Title: q-Analogs of rational numbers and the Burau representati on of the braid group B3.\nby Sophie Morier-Genoud (University of Reim s) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstrac t\nThe most popular \\(q\\)-\;analogs of numbers are certainly the \\(q \\)-\;integers and the \\(q\\)-\;binomial coefficients of Gauss whic h both appear in various areas of mathematics and physics. Classical seque nces of integers often have interesting \\(q\\)-\;analogs. With Valenti n Ovsienko we recently suggested a notion of \\(q\\)-\;analogs for rati onal numbers. Our approach is based on combinatorial properties and contin ued fraction expansions of the rationals. The subject can be developed in connections with various topics such as enumerative combinatorics\, cluste r algebras\, homological algebra\, knots invariants... I will give an over view of the theory and present an application to the problem of classific ation of faithful specialisations of the Burau representation of B3. This last part is joint work with V. Ovsienko and A. Veselov.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Fran Burstall (University of Bath) DTSTART:20231129T160000Z DTEND:20231129T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/45 DESCRIPTION:Title: Isothermic surfaces and Noether's theorem\nby Fran B urstall (University of Bath) as part of Selected Topics in Mathematics - O nline Edition\n\n\nAbstract\nIsothermic surfaces were intensively studied in the late 19th century and have seen a recent revival of inte rest due to links with soliton theory. In this talk\, I will describe this classical theory and the modern integrable systems approach via a pencil of flat connections. I will explain how this connections arise from a vari ational characterisation of isothermic surfaces\, due to Bohle-Peters-Pink all\, together with Noether's theorem. This gives a puzzling link to the c onservations laws for CMC surfaces discovered by Korevaar-Kusner-Solomon.\ n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Vladlen Timorin (Higher School of Economics) DTSTART:20231213T160000Z DTEND:20231213T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/46 DESCRIPTION:Title: Aperiodic points for outer billiards\nby Vladlen Tim orin (Higher School of Economics) as part of Selected Topics in Mathematic s - Online Edition\n\n\nAbstract\nThis is a joint project with A. Kanel 5\;Belov\, Ph. Rukhovich\, and V. Zgurskii. A Euclidean outer billiard on a convex figure in the plane is the map sending a point outside the figure to the other endpoint of a segment touching the figure at the middle. Ite rating such a process was suggested by J. Moser as a crude model of planet ary motion. Polygonal outer billiards are arguably the principal examples of Euclidean piecewise rotations\, which serve as a natural generalization of interval exchange maps. They also found applications in electrical eng ineering. Previously known rigorous results on outer billiards on regular \\(N\\)-\;polygons are\, apart from trivial cases of \\(N=3\,4\, 6\\)\, based on dynamical self-\;similarities (this approach was origin ated by S. Tabachnikov). Dynamical self-\;similarities have been found so far only for \\(N=5\,7\,8\,9\,10\,12\\). In his ICM 2022 address\, R. S chwartz asked whether outer billiard on the regular \\(N\\)-\;gon ha s an aperiodic orbit if \\(N\\) is not \\(3\\)\, \\(4\\)\, \\(6\\). We answer this question in affirmative for \\(N\\) not divisible by \\(4\\). Our methods are not based on self-\;similarity. Rather\, scissor congr uence invariants (including that of Sah-\;Arnoux-\;Fathi) play a key role.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Victor Kleptsyn (Université de Rennes) DTSTART:20231108T160000Z DTEND:20231108T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/47 DESCRIPTION:Title: From the percolation theory to Fuchsian equations and Ri emann-\;Hilbert problem\nby Victor Kleptsyn (Université de Rennes) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\ nConsider the critical percolation problem on the hexagonal lattice: each of (tiny) hexagons is independently declared «\; open »\; or &la quo\; closed »\; with probability (\\(1/2\\)) &mdash\; by a fair coin tossing. Assume that on the boundary of a simply connected domain four po ints A\,B\,C\,D are marked. Then either there exists an «\; open &raq uo\; path\, joining AB and CD\, or there is a «\; closed »\; pat h\, joining AD and BC (one can recall the famous «\; Hex »\; gam e here).\n\n\n\n

\n

\n\nCardy's formula\, rigorously proved by S. S mirnov\, gives an explicit value of the limit of such percolation probabil ity\, when the same smooth domain is put onto lattices with smaller and sm aller mesh. Though\, a next natural question is: what if more than four po ints are marked? And thus that there are more possible configurations of o pen/closed paths joining the arcs? \n\n\n

\n

\n\nIn our joint work with M. Khristoforov we obtain the answer as an explicit integral for the case of six marked points on the boundary\, passing through Fuchsian diffe rential equations\, Riemann surfaces\, and Riemann-\;Hilbert problem. W e also obtain a generalisation of this answer to the case when one of the marked points is inside the domain (and not on the boundary).\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Iván Rasskin (Aix-Marseille University) DTSTART:20231115T160000Z DTEND:20231115T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/48 DESCRIPTION:Title: On the arithmetic and geometric properties of regular po lytopal sphere packings and their connection to knot theory\nby Iván Rasskin (Aix-Marseille University) as part of Selected Topics in Mathemati cs - Online Edition\n\n\nAbstract\nhe Apollonian Circle Packing (ACP) is a classic geometric fractal with diverse applications across various domain s\, particularly in number theory. This is due to its ability to be realiz ed as an integral packing\, where the curvatures of all the circles are in tegers. The ACP is constructed iteratively\, beginning with an initial pac king whose combinatorial structure is encoded by a tetrahedron. By changin g the initial configuration\, the ACP can be generalized for any polyhedro n. However\, not every polyhedron is integral in the sense that it can gen erate an integral packing. Moreover\, in higher dimensions\, not every pol ytope is crystallographic\, meaning that it can generate an Apollonian-lik e sphere packing. In this talk\, we will study the case of regular polytop es in any dimension to determine whether they are integral and crystallogr aphic. Additionally\, we will explore how the symmetry inherent in the pol ytope can be leveraged to extract special cross-sections of the packings. Furthermore\, we will demonstrate how a specific section of an orthoplicia l/hyperoctahedral Apollonian sphere packing can be utilized as a geometric framework to establish an upper bound on a knot invariant for rational li nks.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Herman Servatius (Worcester Polytechnic Institute) DTSTART:20231101T160000Z DTEND:20231101T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/49 DESCRIPTION:Title: Rigidity and movability of configurations in the project ive plane\nby Herman Servatius (Worcester Polytechnic Institute) as pa rt of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nConfi gurations of points and lines in the plane have a long history. The Theore m of Pappus from the fourth century begins a classical theory that has bee n advanced by Desargues\, Pascal\, Cayley\, Steinitz\, Grassman and many o thers. The study of such objects and their generalizations has deep roots in algebra\, geometry\, topology and combinatorics.\n\nIn this talk we dis cuss recent work which is the result of regarding these classical structur es as geometric constraint systems. The objects of interest then become th e topology\, geometry\, and parameterizations of the space of realizations of a configuration. Some of the tools derive from those developed by civi l and mechanical engineers in the analysis of the statics of structures an d the kinematics of linkages.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Pär Kurlberg (KTH) DTSTART:20231004T150000Z DTEND:20231004T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/50 DESCRIPTION:Title: Repulsion in number theory and physics\nby Pär Kurl berg (KTH) as part of Selected Topics in Mathematics - Online Edition\n\n\ nAbstract\nZeros of the Riemann zeta function and eigenvalues of quantized chaotic Hamiltonians appears to have something in common. Namely\, they both seem to be ruled by random matrix theory and consequently should exhi bit "repulsion" in the sense that small gaps between elements are very rar e. More mysteriously\, while zeros of different L-\;functions (i.e.\, generalizations of the Riemann zeta function) are "mostly independent" the y also exhibit subtle repulsion effects on zeros of other L-\;fu nctions.\n\nWe will give a survey of the above phenomena. Time permitting we will also discuss repulsion between eigenvalues of "arithmetic Seba bi lliards"\, a certain singular perturbation of the Laplacian on the 3D toru s \\(R^3/Z^3\\). The perturbation is weak enough to allow for arithmetic features from the unperturbed system to be brought into play\, yet strong enough to probably induce repulsion.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/50/ END:VEVENT BEGIN:VEVENT SUMMARY:Jeffrey Shallit (University of Waterloo) DTSTART:20231122T160000Z DTEND:20231122T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/51 DESCRIPTION:Title: Proving results in combinatorics on words and number the ory using a decidable logic theory\nby Jeffrey Shallit (University of Waterloo) as part of Selected Topics in Mathematics - Online Edition\n\n\n Abstract\nDavid Hilbert's dream\, of a deterministic finite procedure that could decide if a given theorem statement is true or false\, was killed o ff by Gä\;del and Turing. Nevertheless\, there are some logical theori es\, such as Presburger arithmetic\, that are decidable.\n\nIn this talk I will discuss one such theory\, Bü\;chi arithmetic\, and its implement ation in a computer system called Walnut. With this free software\, one ca n prove non-trivial theorems in combinatorics on words and number theory\; it suffices to state the desired result in first-order logic\, type it in to the system\, and wait. So far\, it has been used in over 70 papers in t he literature\, and has even detected errors in some published papers.\n\n My recent book\, The Logical Approach to Automatic Sequences\, pu blished by Cambridge University Press\, has more details.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Esterov (London Institute for Mathematical Sciences) DTSTART:20231025T150000Z DTEND:20231025T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/52 DESCRIPTION:by Alexander Esterov (London Institute for Mathematical Scienc es) as part of Selected Topics in Mathematics - Online Edition\n\nAbstract : TBA\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicholas Ovenhouse (Yale University) DTSTART:20240214T150000Z DTEND:20240214T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/53 DESCRIPTION:Title: Higher Continued Fractions from Dimer Models and Plane P artitions\nby Nicholas Ovenhouse (Yale University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThere is a well-kno wn relation between ordinary continued fractions and certain matrix produc ts in \\(\\text{SL}(2\,\\mathbb{Z})\\). There is also a theorem of Schiffl er and Canakci that the entries of these matrix products count the perfect matchings on certain planar graphs called ''snake graphs". Together with Musiker\, Schiffler\, and Zhang\, we studied the enumeration of ''\\(m\\)& #45\;dimer covers" on these snake graphs (these are combinatorial generali zations of perfect matchings)\, and obtained formulas in terms of products of \\(\\text{SL}(m+1\,\\mathbb{Z})\\) matrices. This led to a definition of ''higher continued fractions". I will discuss these higher continued fr actions\, their properties\, and their combinatorial interpretations (incl uding perfect matchings\, lattice paths\, plane partitions\, and more). Ti me permitting\, I will mention work-\;in-\;progress about \\(q\\) 5\;analogs of these notions.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Dmitry Badziahin (Univesity of Sydney) DTSTART:20240221T120000Z DTEND:20240221T130000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/54 DESCRIPTION:Title: Simultaneous Diophantine approximation on the Veronese c urve\nby Dmitry Badziahin (Univesity of Sydney) as part of Selected To pics in Mathematics - Online Edition\n\n\nAbstract\nMeasuring the set of s imultaneously well approximable points on manifolds is one of the most int ricate problems in metric theory of Diophantine approximation. Unlike the dual case of well approximable linear forms\, the results here are known t o depend on a manifold. For example\, some of the manifolds do not contain simultaneously very well approximable points at all\, while for the other s the set of such points always has positive Hausdorff dimension. In this talk\, we will closely look at the Veronese curve \\(\\{x\, x^2\, x^3\, \\ ldots\, x^n\\}\\)\, discuss what is known about the sets of simultaneously well approximable points on it and provide several new results. In partic ular\, for \\(n=3\\) we provide the Hausdorff dimension of the set of \\(x \\) such that \\(\\lambda_3(x) \\le \\lambda\\) where \\(\\lambda\\le \\fr ac25\\) or \\(\\lambda\\ge \\frac79\\).\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Eleonore Faber (University of Leeds) DTSTART:20240228T190000Z DTEND:20240228T200000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/55 DESCRIPTION:Title: Friezes and resolutions of plane curve singularities \nby Eleonore Faber (University of Leeds) as part of Selected Topics in Ma thematics - Online Edition\n\n\nAbstract\nConway‐\;Coxeter friezes ar e arrays of positive integers satisfying a determinantal condition\, the s o‐\;called diamond rule. \n Recently\, these combinatorial objects ha ve been of considerable interest in representation theory\, since they enc ode cluster combinatorics of type A.\n\nIn this talk I will discuss a new connection between Conway‐\;Coxeter friezes and the combinatorics of a resolution of a complex curve singularity: via the beautiful relation be tween friezes and triangulations of polygons one can relate each frieze to the so‐\;called lotus of a curve singularity\, which was introduced by Popescu‐\;Pampu. \nThis allows to interpret the entries in the fri eze in terms of invariants of the curve singularity\, and on the other han d\, we can see cluster mutations in terms of the desingularization of the curve. \nThis is joint work with Bernd Schober.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Martin Henk (TU Berlin) DTSTART:20240306T150000Z DTEND:20240306T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/56 DESCRIPTION:Title: Polynomial bounds in Koldobsky's discrete slicing proble m\nby Martin Henk (TU Berlin) as part of Selected Topics in Mathematic s - Online Edition\n\n\nAbstract\nIn 2013\, Koldobsky posed the problem to find a constant \\(d_n\\)\,\n depending only on the dimension \\(n\\)\, s uch that for any\n origin-symmetric convex body \\(K\\subset\\mathbb{R}^n\ \) there exists an\n \\((n-1)\\)-dimensional linear subspace \\(H\\subset\ \mathbb{R}^n\\) with\n \\[\n |K\\cap\\mathbb{Z}^n| \\leq d_n\\\,|K\\cap H \\cap \\mathbb{Z}^n|\\\,\\text{vol}(K)^{\\frac 1n}.\n \\]\nIn this ar ticle we show that \\(d_n\\) is bounded from above by\n\\(c\\\,n^2\\\,\\o mega(n)/\\log(n)\\)\, where \\(c\\) is an absolute constant and \\(\\omega (n)\\) is\nthe flatness constant. Due to the recent best known upper bound on\n\\(\\omega(n)\\) we get a \\({c\\\,n^3\\log(n)^2}\\) bound on \\(d_n \\). This improves on former bounds. \n\n\n (Based on joint work s with Ansgar Freyer.)\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/56/ END:VEVENT BEGIN:VEVENT SUMMARY:Will Traves (United States Naval Academy) DTSTART:20240313T150000Z DTEND:20240313T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/57 DESCRIPTION:Title: Incidence results defining plane curves\nby Will Tra ves (United States Naval Academy) as part of Selected Topics in Mathematic s - Online Edition\n\n\nAbstract\nI'll explain Hermann Grassmann's approac h to the geometry of curves. In the mid-1800's\, he characterized points o n cubics using a clever incidence construction. I'll discuss ways to exten d Grassmann's results. In particular\, I will explain how to use a straigh tedge to find the ninth point of intersection of two cubics\, given just \ \(8\\) points common to the two curves and one extra point on each cubic.\ n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/57/ END:VEVENT BEGIN:VEVENT SUMMARY:Ilya Bogaevskii (University of Liverpool) DTSTART:20240320T150000Z DTEND:20240320T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/58 DESCRIPTION:Title: Discontinuous gradient ODEs\, trajectories in the minima l action problem\, and massive points in one cosmological model\nby Il ya Bogaevskii (University of Liverpool) as part of Selected Topics in Math ematics - Online Edition\n\n\nAbstract\nThe gradient of a concave function is discontinuous vector field but has well-\;defined trajectories. We formulate an existence and forward-\;uniqueness theorem and its general isation for non-\;stationary case. Using the latter we construct trajec tories in the minimal action problem and investigate how massive points ap pear. Their formation simulates the large-\;scale matter distribution i n one of the simplest cosmological models based on the Burgers equation.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Jarosław Kędra (University of Aberdeen) DTSTART:20241004T140000Z DTEND:20241004T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/59 DESCRIPTION:Title: Bi-invariant metric on groups\nby Jarosław Kędra ( University of Aberdeen) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nA bi-invariant metric on a group \\(G\\) is a metr ic such that both the right and the left action of \\(G\\) on itself is by isometries. Examples of such metrics include the Hofer metric on the grou p of Hamiltonian diffeomorphisms of a symplectic manifold\, the reflection length on a Coxeter group\, the commutator length and many others mostly in group theory and dynamics. A particularly interesting example is the ca ncellation length on free groups\, which was first discovered by biologist s investigating RNA folding.\n\nIn the talk\, I will discuss various examp les\, present a sample of results and open problems.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Boris Springborn (TU Berlin) DTSTART:20241018T140000Z DTEND:20241018T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/60 DESCRIPTION:Title: The hyperbolic geometry of numbers\nby Boris Springb orn (TU Berlin) as part of Selected Topics in Mathematics - Online Edition \n\n\nAbstract\nIn his 1880 thesis\, Markov classified the unimodular inde finite binary quadratic forms whose values on the integer lattice (minus t he origin) stay farthest away from zero. \nThis is closely linked to the c lassification of the worst approximable irrational numbers. \nThe primary tool in this theory have always been continued fractions.\nWell-known conn ections to hyperbolic geometry are based on the fact that continued fracti ons describe the symbolic dynamics of geodesics in the Farey triangulation of the hyperbolic plane. \nThis talk\, however\, will be about a new geom etric approach to the Markov theory that eliminates the complicated symbol ic dynamics of continued fractions by considering the set of all ideal tri angulations of the modular torus and not just the simplest and most symmet ric one. \nIn the end\, the problem boils down to the question: How far ca n a geodesic that crosses a triangle stay away from the vertices? \nThis g eometric approach can also be used to classify the worst approximable rati onal (sic!) numbers.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Miranda (UCLA) DTSTART:20241101T150000Z DTEND:20241101T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/61 DESCRIPTION:Title: Fully Flexible Periodic Polyhedral Surfaces\nby Robe rt Miranda (UCLA) as part of Selected Topics in Mathematics - Online Editi on\n\n\nAbstract\nA polyhedral surface has an $n$-dimensional flex if ther e exists a continuous family of realizations $\\{Q_t : t \\in [0\,1]^n\\}$ which are pairwise nonisomorphic. Gaifullin and Gaifullin showed that if a 2-periodic polyhedral surface in is homeomorphic to a plane\, then it ca n have at most a 1-dimensional flex which preserves periodicity. Glazyrin and Pak later found an example of a 2-periodic polyhedral surface\, not ho meomorphic to a plane\, which has a full 3-dimensional flex which preserve s periodicity. In this talk\, I will present a new construction of a fully flexible 3-periodic polyhedral surface using the universality of polyhedr al linkages and discuss generalizations to general periodic polyhedral sur faces and higher dimensions.\n\nThis talk is based on joint work with Alex ey Glazyrin (University of Texas Rio Grande Valley) and Igor Pak (Universi ty of California Los Angeles)\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Esther Banaian (UC Riverside) DTSTART:20241115T150000Z DTEND:20241115T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/62 DESCRIPTION:Title: Skein relations and bases for cluster algebras from punc tured surfaces\nby Esther Banaian (UC Riverside) as part of Selected T opics in Mathematics - Online Edition\n\n\nAbstract\nCluster algebras are commutative rings with a set of recursively-defined generators. Many clust er algebras with desirable properties arise from a surface with marked poi nts (S\,M) in the sense that they can be realized geometrically through th e decorated Teichmüller space of (S\,M). Kantarcı Oğuz-Yıldırım and Pilaud-Reading-Schroll have recently exhibited a method to use the set of order ideals of a poset to give a direct formula for any generator of a cl uster algebra of surface type. We use this construction to give "skein rel ations"\, which will be multiplication formulas for elements of the cluste r algebra which arise from resolving intersections of the corresponding cu rves on the surface. By working with surfaces which could have internal ma rked points\, called "punctures"\, we generalize previously known skein re lations from Musiker-Schiffler-Williams and Canakci-Schiffler\, who largel y only work in unpunctured surfaces. A corollary of our results is the abi lity to construct bases for cluster algebras from punctured surfaces. Thi s talk is based on joint work with Wonwoo Kang and Elizabeth Kelley.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/62/ END:VEVENT BEGIN:VEVENT SUMMARY:Khrystyna Serhiyenko (University of Kentucky) DTSTART:20241122T150000Z DTEND:20241122T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/63 DESCRIPTION:Title: SL\\(_k\\) tilings and paths in \\(\\mathbb{Z}^k\\)\ nby Khrystyna Serhiyenko (University of Kentucky) as part of Selected Topi cs in Mathematics - Online Edition\n\n\nAbstract\nAn SL\\(_k\\) frieze is a bi-\;infinite array of integers where adjacent entries satisfy a cer tain diamond rule. \nSL\\(_2\\) friezes were introduced and studied by Co nway and Coxeter. \nLater\, these were generalized to infinite matrix-\ ;like structures called tilings as well as higher values of \\(k\\). \nA r ecent paper by Short showed a bijection between bi-\;infinite paths of reduced rationals in the Farey graph and SL\\(_2\\) tilings. \nWe extend t his result to higher \\(k\\) by constructing a bijection between SL\\(_k\\ ) tilings and certain pairs of bi-\;infinite strips of vectors in \\(Z^ k\\) called paths. \nThe key ingredient in the proof is the relation to Pl uecker friezes and Grassmannian cluster algebras. \nAs an application\, we obtain results about periodicity\, duality\, and positivity for tilings.\ n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/63/ END:VEVENT BEGIN:VEVENT SUMMARY:Bernd Schulze (Lancaster University) DTSTART:20241206T150000Z DTEND:20241206T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/64 DESCRIPTION:Title: A differential approach to Maxwell-Cremona liftings\ nby Bernd Schulze (Lancaster University) as part of Selected Topics in Mat hematics - Online Edition\n\n\nAbstract\nIn 1864\, James Clerk Maxwell int roduced a link between self-stressed frameworks in the plane and piecewise linear liftings to 3-space. This connection has found numerous applicatio ns in areas such as rigidity theory\, discrete and computational geometry\ , control theory and structural engineering. While there are some generali sations of this theory to liftings of d-complexes in d-space\, extensions for liftings of frameworks in d-space for d at least 3 have been missing. In this talk we introduce differential liftings on general graphs using di fferential forms associated with the elements of the homotopy groups of th e complements to the frameworks. Such liftings play the role of integrands for the classical notion of liftings for planar frameworks. These differe ntial liftings have a natural extension to self-stressed frameworks in hig her dimensions. As a result we generalise the notion of classical liftings to both graphs and multidimensional k-complexes in d-space (k=2\,…\,d). This is joint work with Oleg Karpenkov\, Fatemeh Mohammadi and Christian Mueller.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/64/ END:VEVENT BEGIN:VEVENT SUMMARY:Jeffrey Vaaler (University of Texas at Austin) DTSTART:20250213T150000Z DTEND:20250213T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/65 DESCRIPTION:Title: Sums of small fractional parts and a problem of Littlewo od\nby Jeffrey Vaaler (University of Texas at Austin) as part of Selec ted Topics in Mathematics - Online Edition\n\n\nAbstract\nWe prove estimat es for certain partial sums of which the simplest nontrivial examples are\ n\\[\n\\sum\\limits_{n=1}^N \\frac{1}{||\\alpha n||\\\,||\\beta n||}\,\\ \ \text{ and }\\ \\sum\\limits_{m=1}^M\\sum\\limits_{n=1}^N \\frac{1}{||\\al pha m+\\beta n||}.\n\\]\nHere $\\alpha$ and $\\beta$ are real numbers\, an d $||x||$ is the distance from the real number $x$ to the nearest integer. Such estimates are somewhat related to a notorious open problem of Little wood: is it true that for all pairs of real numbers $\\alpha$ and $\\beta$ we have\n\\[\n\\lim\\limits_{n\\to\\infty}\\inf n||\\alpha n||\\\,||\\bet a n|| =0\\\,?\n\\]\nWe also consider estimates for more general sums that contain products of many linear forms in many variables. If time permits w e also discuss an analogue of such problems in function fields. This is jo int work with Thá\;i Hoà\;ng Lê\;.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/65/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Evans (Loughborough University) DTSTART:20241025T140000Z DTEND:20241025T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/66 DESCRIPTION:Title: Arithmetic and geometry of Markov polynomials\nby Sa m Evans (Loughborough University) as part of Selected Topics in Mathematic s - Online Edition\n\n\nAbstract\nMarkov polynomials are the Laurent-polyn omial solutions of the Markov equation\n$$X^2+Y^2+Z^2=aXYZ\,$$\nwhich are the results of the cluster mutations applied to the initial triple $(x\,y\ ,z)$.\n\nThey were first discussed by Propp\, who proved that their coeffi cients are non-negative integers. This result relates to an interesting co mbinatorial interpretation of Markov numbers\, related to perfect matching s on snake graphs.\n\nWe present more results and formulate new conjecture s about the specific coefficients that appear in these Markov polynomials. Whilst many of the results remain conjecture\, based on observations from numerical investigations\, in specific cases the results can be proven. I n particular\, in subsets of Markov polynomials such as the Fibonacci pol ynomials\; in the sense of Caldero and Zelevinsky.\n\nThe talk is based on the ongoing joint work with Alexander Veselov and Brian Winn\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/66/ END:VEVENT BEGIN:VEVENT SUMMARY:Miloslav Torda (University of Liverpool) DTSTART:20241129T150000Z DTEND:20241129T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/67 DESCRIPTION:Title: Geometric Perspectives on the Crystallization of Molecul ar Crystals: Crystallographic Symmetry Group Packings\, Uniform Tessellati ons\, and Molecular Frameworks\nby Miloslav Torda (University of Liver pool) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbst ract\nThe mathematical theory of crystallization is still in its early sta ges\, even for simple mono-\;atomic systems. Molecular systems present additional challenges due to their intermolecular interactions. However\, by focusing on molecules with inherent symmetries\, the problem of periodi c ground state formation in the thermodynamic limit becomes more tractable . In this talk\, we consider an additive Hamiltonian at zero temperature\, involving pairwise atom-\;atom interaction potentials and crystallogra phic symmetry groups&ndash\;discrete isometry groups of Euclidean space th at include a lattice subgroup. We begin by investigating two-\;dimensio nal molecular systems where the pairwise interaction potential consists of repulsive and dispersion attraction terms. By examining the densest packi ngs of regular polygons within wallpaper groups\, Archimedean circle packi ngs\, and their corresponding dual frameworks\, we formulate a conjecture regarding the ground state configurations of molecules with six-\;fold rotational symmetry. We then consider two-\;dimensional molecular salt frameworks composed of negatively charged nodes connected by positively ch arged triangular linkers. For these purely electrostatic systems\, we enum erate possible ground states via sums of integer reciprocals. Building on these observations\, we extend our considerations to tetrahedral salt fram eworks and propose that the ground states of these systems can be represen ted by an absolutely symmetric quadratic form. This work is in progress jo intly with Roland Pú\;č\;ek (University of Jena) and Andrew I Coop er (University of Liverpool).\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/67/ END:VEVENT BEGIN:VEVENT SUMMARY:Anthony Nixon (Lancaster University) DTSTART:20250227T150000Z DTEND:20250227T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/68 DESCRIPTION:Title: Stable cuts\, NAC-colourings and flexible realisations o f graphs\nby Anthony Nixon (Lancaster University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nA (2-\;dimensiona l) realisation of a graph G is a pair \\((G\,p)\\)\, where \\(p\\) maps th e vertices of \\(G\\) to \\(\\mathbb{R}^2\\). A realisation is flexible if it can be continuously deformed while keeping the edge lengths fixed\, an d rigid otherwise. Similarly\, a graph is flexible if its generic realisat ions are flexible\, and rigid otherwise. We show that a minimally rigid gr aph has a flexible realisation with positive edge lengths if and only if i t is not a 2-\;tree. This confirms a conjecture of Grasegger\, Legersky and Schicho. Our proof is based on a characterisation of graphs with \\(n \\) vertices and \\(2n-3\\) edges and without stable cuts due to Le and Pf ender. We also strengthen a result of Chen and Yu\, who proved that every graph with at most \\(2n-4\\) edges has a stable cut\, by showing that eve ry flexible graph has a stable cut. Additionally\, we investigate the numb er of NAC-\;colourings in various graphs. A NAC-\;colouring is a typ e of edge colouring introduced by Grasegger\, Legersky and Schicho\, who s howed that the existence of such a colouring characterises the existence o f a flexible realisation with positive edge lengths. \n\nThis is joint wor k with Clinch\, Garamvolgyi\, Haslegrave\, Huynh and Legersky.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/68/ END:VEVENT BEGIN:VEVENT SUMMARY:Maxim Arnold (UT Dallas) DTSTART:20250313T150000Z DTEND:20250313T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/69 DESCRIPTION:Title: Circle patterns and ideal polygon folding\nby Maxim Arnold (UT Dallas) as part of Selected Topics in Mathematics - Online Edit ion\n\n\nAbstract\nFolding of the ideal polygon in its \\(j\\)&ndash\;th v ertex reflects the vertex in the corresponding short diagonal. We show tha t compositions of such foldings along any Coxeter element provide Liouvill e integrable system on the moduli space of ideal polygons. This result als o provides integrability for Shramm circle patterns. This is joint work wi th Anton Izosimov.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/69/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikhail Gabdullin (University of Illinois) DTSTART:20250320T150000Z DTEND:20250320T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/70 DESCRIPTION:Title: Primes with small primitive roots\nby Mikhail Gabdul lin (University of Illinois) as part of Selected Topics in Mathematics - O nline Edition\n\n\nAbstract\nLet \\(\\delta(p)\\) tend to zero arbitrarily slowly as \\(p\\to\\infty\\). We exhibit an explicit set \\(\\mathcal{S}\ \) of primes \\(p\\)\, defined in terms of simple functions of the prime f actors of \\(p-1\\)\, for which the least primitive root of \\(p\\) is at most \\( p^{1/4-\\delta(p)}\\) for all \\(p\\in \\mathcal{S}\\)\, where \\ (\\#\\{p\\leq x: p\\in \\mathcal{S}\\} \\sim \\pi(x)\\) as \\(x\\to\\infty \\). This is a joint work with Kevin Ford and Andrew Granville.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/70/ END:VEVENT BEGIN:VEVENT SUMMARY:Amanda Burcroff (Harvard University) DTSTART:20250410T140000Z DTEND:20250410T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/71 DESCRIPTION:Title: Combining Combinatorics and Mirror Symmetry in Cluster A lgebra Positivity\nby Amanda Burcroff (Harvard University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe theory of cluster algebras gives a combinatorial framework for understanding the previously opaque nature of certain algebraic and geometric spaces. Cluste r algebras are celebrated for their intriguing positivity properties\, whi ch unify positivity phenomena in many areas of math and physics. Two dist inct proofs of this positivity have emerged\, one combinatorial and the ot her using scattering diagrams from mirror symmetry. Combining these appro aches\, we give a directly computable\, manifestly positive\, and elementa ry (yet highly nontrivial) formula describing generalized cluster scatteri ng diagrams in rank \\(2\\). Using this\, we prove the Laurent positivity of generalized cluster algebras of all ranks\, resolving a conjecture of Chekhov and Shapiro from 2014. This is joint work with Kyungyong Lee and Lang Mou.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/71/ END:VEVENT BEGIN:VEVENT SUMMARY:Shin-ichi Tanigawa (University of Tokyo) DTSTART:20250417T140000Z DTEND:20250417T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/72 DESCRIPTION:Title: Identifying Generic Points from Non-Generic Measurements \nby Shin-ichi Tanigawa (University of Tokyo) as part of Selected Topi cs in Mathematics - Online Edition\n\n\nAbstract\nThe generic global rigid ity characterization by Gortler\, Healy\, and Thurston is one of the most significant results in graph rigidity theory. In particular\, this charact erization implies that global rigidity is a generic property of graphs: ei ther every generic realization of a graph in \\(d\\)-\;space is globall y rigid\, or none of them are. Although a few variations are known\, our u nderstanding of its extendability to other rigidity models remains limited .\n\nIn this talk\, we examine the generic rigidity problem within the fra mework of the point identifiability problem\, by Cruickshank\, Mohammadi\, Nixon\, and Tanigawa. We present several successful examples in \\(\\math scr{l}_p\\)-\;rigidity and tensor completion and discuss unsolved probl ems.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/72/ END:VEVENT BEGIN:VEVENT SUMMARY:Luke Jeffreys (University of Bristol) DTSTART:20250327T150000Z DTEND:20250327T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/73 DESCRIPTION:Title: On the complement of the Lagrange spectrum in the Markov spectrum\nby Luke Jeffreys (University of Bristol) as part of Selecte d Topics in Mathematics - Online Edition\n\n\nAbstract\nInitially studied by Markov around \\(1880\\)\, the Lagrange and Markov spectra are complica ted subsets of the real line that play a crucial role in the study of Diop hantine approximation and the study of binary quadratic forms. In the \\(1 920\\)s\, Perron gave an amazingly useful description of the spectra in te rms of continued fractions and\, in the \\(1960\\)s\, Freiman demonstrated that the Lagrange spectrum is a strict subset of the Markov spectrum. It still remains a difficult task to find points in the complement of the Lag range spectrum within the Markov spectrum and modern research is focussed on further developing our understanding of this complement.\n\nIn this tal k\, I will introduce these spectra\, discussing the historical results abo ve\, and speak about recent works with Harold Erazo\, Carlos Matheus and C arlos Gustavo Moreira finding new points in the complement and obtaining b etter lower bounds for its Hausdorff dimension.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/73/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Pratoussevitch (University of Liverpool) DTSTART:20250424T140000Z DTEND:20250424T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/74 DESCRIPTION:Title: Farey Bryophylla\nby Anna Pratoussevitch (University of Liverpool) as part of Selected Topics in Mathematics - Online Edition\ n\n\nAbstract\nThe construction of the Farey tessellation in the hyperboli c plane starts with a finitely generated group of symmetries of an ideal t riangle and induces a remarkable fractal structure on the boundary of the hyperbolic plane\, encoding every element by the continued fraction relate d to the structure of the tessellation. The problem of finding a generalis ation of this construction to the higher dimensional hyperbolic spaces has remained open for many years. In this paper we make the first steps towar ds a generalisation in the three-\;dimensional case. We introduce confo rmal bryophylla\, a class of subsets of the boundary of the hyperbolic \\( 3\\)-\;space which possess fractal properties similar to the Farey tess ellation. We classify all conformal bryophylla and study the properties of their limiting sets. This is joint work with Oleg Karpenkov.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/74/ END:VEVENT BEGIN:VEVENT SUMMARY:Michail Zhitomirskii DTSTART:20250926T160000Z DTEND:20250926T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/75 DESCRIPTION:Title: Local Classification Problems with Functional Moduli \nby Michail Zhitomirskii as part of Selected Topics in Mathematics - Onli ne Edition\n\n\nAbstract\nI will discuss the class of local classificatio n problems\, including classification of vector distributions\, Riemannian metrics\, and real hypersurfaces in \\(\\mathbb{C}^n\\)\, where the funct ional dimension of the space of objects is bigger than that of the transfo rmation group\, unlike the classification problem of singularity theory wh ere it is not so. I will explain that combining a coordinate-free approach with normal forms gives a nice explanation of known results and many new results.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/75/ END:VEVENT BEGIN:VEVENT SUMMARY:Rebecca Sheppard (University of Liverpool) DTSTART:20251008T150000Z DTEND:20251008T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/76 DESCRIPTION:Title: Not your Usual Circle: Geometry on the Integer Grid\ nby Rebecca Sheppard (University of Liverpool) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nInteger geometry explores o bjects whose vertices lie on the integer lattice \\(\\mathbb{Z}^2\\)\, wit h congruence defined by lattice-preserving affine transformations. In this project\, I introduced remarkable geometric objects called integer circle s: discrete analogues of Euclidean circle. These objects challenge our geo metric intuition regarding circles. Unlike their classical counterparts\, integer circles are unbounded\, exhibit nontrivial arithmetic structure\, and possess positive density in the plane.\n\nIn this talk\, I will define integer circles\, illustrate their unusual behaviour\, and demonstrate ho w to rigorously compute their densities and intersection patterns.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/76/ END:VEVENT BEGIN:VEVENT SUMMARY:James Dolan (University of Liverpool) DTSTART:20251029T160000Z DTEND:20251029T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/77 DESCRIPTION:Title: Integer Angles of Integer Polygons\nby James Dolan ( University of Liverpool) as part of Selected Topics in Mathematics - Onlin e Edition\n\n\nAbstract\nIn 2008\, the first formula expressing conditions on the geometric continued fractions for lattice angles of triangles was derived\, while the cases of n-gons for \\(n > 3\\) remained unresolved. I n this talk\, we introduce an integer geometric analogue to the classical sum of interior angles of a polygon theorem that will act as an ext ension to the above result in the \\(n>3\\) cases. I first will frame hist orical contributions in this area by drawing comparison to their Euclidean counterparts. This will provide background for a simplified overview of t he main results for the \\(n>3\\) case\, introducing novel notions in inte ger geometry such as chord curvature. Finally\, I will briefly touch on th e consequences of this work within the field of toric singularities.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/77/ END:VEVENT BEGIN:VEVENT SUMMARY:Thọ Nguyễn Phước (University of Ostrava) DTSTART:20251105T160000Z DTEND:20251105T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/78 DESCRIPTION:Title: On a Theorem of Nathanson on Diophantine Approximation\nby Thọ Nguyễn Phước (University of Ostrava) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nIn 1974\, M. B. Nat hanson proved that every irrational number \\(\\alpha\\) represented by a simple continued fraction with infinitely many elements greater than or eq ual to \\(k\\) is approximable by an infinite number of rational numbers \ \(p/q\\) satisfying \\(|\\alpha-p/q| < 1/(\\sqrt{k^2 + 4}q^2) \\). In this talk we refine this result.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/78/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrei Zabolotskii (The Open University) DTSTART:20251119T160000Z DTEND:20251119T170000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/79 DESCRIPTION:Title: Friezes\, Cluster Algebras\, and the Poly Phenomenon \nby Andrei Zabolotskii (The Open University) as part of Selected Topics i n Mathematics - Online Edition\n\n\nAbstract\nCoxeter friezes are related to closed paths in the Farey graph and triangulated polygons\, as well as to the most basic and important examples of cluster algebras. Cluster alge bras are certain algebras of rational functions\, which actually turn out to consist only of Laurent polynomials ‒\; a surprising fact known a s the Laurent phenomenon. We will introduce all these objects using an interactive demonstr ation\, outline the connections between them and introduce a surprisin g phenomenon in cluster algebras: specialising variables in a specific con trolled way turns Laurent polynomials into (non-Laurent) polynomials.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/79/ END:VEVENT BEGIN:VEVENT SUMMARY:Konstantin Khanin (University of Toronto) DTSTART:20251211T130000Z DTEND:20251211T140000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/80 DESCRIPTION:Title: Renormalization and Rigidity in Dynamical Systems\nb y Konstantin Khanin (University of Toronto) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nRenormalization ideas were int roduced in dynamics in the late 1970s. By now\, renormalization is one of the most important methods of asymptotic analysis in the theory of dynamic al systems. This talk serves as an introduction to dynamical renormalizati on. I'll also discuss closely connected rigidity theory and formulate some open problems. No previous knowledge of renormalization will be assumed.\ n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/80/ END:VEVENT BEGIN:VEVENT SUMMARY:Jane Coons (Worcester Polytechnic Institute) DTSTART:20260213T150000Z DTEND:20260213T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/81 DESCRIPTION:Title: Likelihood Geometry of Brownian Motion Tree Models\n by Jane Coons (Worcester Polytechnic Institute) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nBrownian motion tree model s are used to describe the evolution of a continuous trait along a phyloge netic tree under genetic drift. Such a model is obtained by placing linear constraints on a mean-zero multivariate Gaussian distribution according t o the topology of the underlying tree. We investigate the enumerative geom etry of the standard and dual maximum likelihood estimation problems in th ese models. In particular\, we study the number of complex critical points of the log-likelihood and dual log-likelihood functions\, known as the ML -degree and dML-degree\, respectively. We use the toric geometry of Browni an motion tree models to give a formula for the dML-degree for all trees. We also prove a formula for the ML-degree of a star tree and show that for general trees\, the ML-degree does not depend on the location of the root .\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/81/ END:VEVENT BEGIN:VEVENT SUMMARY:Dirk Siersma (Utrecht University) DTSTART:20260220T150000Z DTEND:20260220T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/82 DESCRIPTION:Title: Distance and critical points on PL-manifolds\nby Dir k Siersma (Utrecht University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThere is long history in the relation betwe en the critical points of distance function and concur- rent normals to a submanifold in Euclidean space. The study of caustics and counting the num ber of normals play a important role. In this talk we will give a general approach to the study of crit- ical points of the distance function to a P L submanifold X. Examples are: polygons in the plane and in space and poly gonal surfaces in 3-space (not necessarily convex)\, etc. What is the rela tion between normals and critical points ? Are generic singularities Morse and if so what is the index ? We will discuss the bifurcation set and sho w that for a knotted closed PL-curve there are at least 10 concurrent norm als. Also for a convex simple polytope there is a point at least 10 concur rent normals. What can be said about the ED-degree?\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/82/ END:VEVENT BEGIN:VEVENT SUMMARY:Camilla Hollanti (Aalto University) DTSTART:20260313T150000Z DTEND:20260313T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/83 DESCRIPTION:Title: Well-rounded lattices and applications to security\n by Camilla Hollanti (Aalto University) as part of Selected Topics in Mathe matics - Online Edition\n\n\nAbstract\nI will give a brief introduction to well-rounded lattices and to their utility in (post-quantum) security. We will see how the lattice theta series naturally arises in these contexts and discuss its connections to well-rounded lattices.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/83/ END:VEVENT BEGIN:VEVENT SUMMARY:Maciej P. Denkowski (Jagiellonian University do Ceará) DTSTART:20260417T140000Z DTEND:20260417T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/84 DESCRIPTION:Title: Medial axes and the reaching of singularities\nby Ma ciej P. Denkowski (Jagiellonian University do Ceará) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe medial axis of a closed subset X of the Euclidean space is the set of points admitting mor e than one closest point in X. It has been extensively studied since the l ate 60's as a central concept in pattern recognition. On the other hand\, its peculiar relation to the singularities of X was observed only a decade ago. Focusing on the question which singular points of X are `reached' by the medial axis (i.e. belong to the closure of the latter) we will give a n overview of what is known and discuss the latest developments in this ma tter.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/84/ END:VEVENT BEGIN:VEVENT SUMMARY:Sagar Kalane (The Institute of Mathematical Sciences (IMSc)) DTSTART:20260508T140000Z DTEND:20260508T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/85 DESCRIPTION:by Sagar Kalane (The Institute of Mathematical Sciences (IMSc) ) as part of Selected Topics in Mathematics - Online Edition\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/85/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexey Ustinov (HSE University) DTSTART:20260320T150000Z DTEND:20260320T160000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/86 DESCRIPTION:Title: On the periodicity of Somos sequences\nby Alexey Ust inov (HSE University) as part of Selected Topics in Mathematics - Online E dition\n\n\nAbstract\nFor integer \\(k\\geq4 \\) Somos--\\(k \\) sequence is a sequence generated by quadratic recurrence relation of the form $$s_{ n+k}s_n=\\sum_{j=1}^{[k/2]}\\alpha_js_{n+k-j}s_{n+j}\,$$ where \\(\\alpha_ j \\) are constants and \\(s_0 \, \\dots\, s_{k-1} \\) are initial data. A mong them exist an important class of sequences with many properties. This class consists of\nfinite rank sequences. \nThe sequence \\(\\{s_ n\\}_{n=-\\infty}^\\infty \\) has a (finite) rank \\(r \\) if maximal ra nk of two infinite matices\n$$\\left.\\vphantom{\\sum}(s_{m+n}s_{m-n})\\ri ght|_{m\,n=-\\infty}^\\infty\,\\qquad \\left.\\vphantom{\\sum}(s_{m+n+1}s_ {m-n})\\right|_{m\,n=-\\infty}^\\infty$$\nis \\(r \\). If \\(r=2 \\) then general term of Somos sequence can be expressed in terms of elliptic funct ion. One can consider a general finite rank sequence as a sequence admitti ng more complicated addition theorem.\n\nPresumably the following properti es are more or less equivalent: finitness of the rank\, Laurent phenomenon \, periodicity \\(\\pmod N \\)\, solvability in theta-functions. The talk will be devoted to periodicity \\(\\pmod N \\) of general integer finite r ank sequences.\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/86/ END:VEVENT BEGIN:VEVENT SUMMARY:Yefei Ma (Université de Montpellie) DTSTART:20260501T140000Z DTEND:20260501T150000Z DTSTAMP:20260422T225700Z UID:SelectedTopics-Liverpool/87 DESCRIPTION:by Yefei Ma (Université de Montpellie) as part of Selected To pics in Mathematics - Online Edition\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/87/ END:VEVENT END:VCALENDAR