#Dublin Mathematics Colloquium, Geometry Seminar ### [School of Mathematics](http://www.maths.tcd.ie/), [Trinity College Dublin](http://www.tcd.ie/) #### Location: * Geometry Seminar takes place in the SNIAM Lecture Room at 3pm on Thursday. * Mathematics Colloquium takes place in the Synge Lecture Theatre (Hamilton Building) at 5pm on Thursday.

Michaelmas Term 2018

Date Speaker Title Abstract Note
20.9.18 James Griffin (Coventry) The space of circles To define a Euclidean circle in three dimensional space you need six parameters, three for the coordinates of its centre, one for its radius and two for its spatial orientation. As such the space of circles, Circ is a six dimensional manifold; we investigate its algebra and geometry. For example the set of osculating circles to a three dimensional path is a path in Circ. Such paths may be characterised as those that are "doubly light-like" using a complexified version of Minkowski space. The configuration space of k pairwise unlinked circles has fundamental group isomorphic to a subgroup of automorphisms of the free group with k generators. The homology groups of this subgroup was computed by Jensen, McCammond, and Meier, and their ranks suggest there may be an topological operad structure on the configuration space of unlinked circles. It turns out there is and in joint work with Allen Hatcher we describe the operad and give a presentation. 5pm, Synge
10.10.18 Timothy Logvinenko (Cardiff) Bar category of modules and homotopy adjunction I will describe joint work with Rina Anno in which we introduce the bar category of modules Modbar(A) over a DG category A. It is a DG-enhancement of the derived category D(A) of A which is isomorphic to the category of DG A-modules with A-infinity morphisms between them. However, it is defined intrinsically in the language of DG-categories and requires no complex machinery or sign conventions of A-infinity categories. We define for these bar categories Tensor and Hom bifunctors, dualisation functors, and a convolution of twisted complexes. The intended application is to working with DG-bimodules as enhancements of exact functors between triangulated categories. As a demonstration, we develop homotopy adjunction theory for tensor functors between derived categories of DG-categories. 3pm, NSR
11.10.18 Timothy Logvinenko (Cardiff) Generalised braid category Ordinary braid group $Br_n$ is a well-known algebraic structure which encodes configurations of n non-touching strands (“braids”) up to continious transformations (“isotopies”). A classical result of Khovanov and Thomas states that this group acts categorically on the space $Fl_n$ of complete flags in $C^n$. I will begin by reviewing the basics on braid group and flag varieties, and then give a sketch of the geometry involved in the Khovanon-Thomas construction of the categorical action of $Br_n$ on $T^* Fl_n$. I will then describe a longstanding work-in-progress with Rina Anno: the categorification of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. A decade old conjecture states that generalised braids act categorically on the spaces of full and partial flags in $C^n$. I will describe our present progress towards it and future expectations. 5pm, Synge
25.10.18 Francis Brown (HMI Simons Visiting Professor) Single-valued integration Single-valued functions are ubiquitous in mathematics and physics, since a well-defined problem has a well-defined answer. On the other hand, the solution to such a problem is often given by an integral, which is usually a multi-valued function of its parameters. The reason is that integration is a pairing between differential forms and chains of integration, and the latter are ambiguously defined. In this talk, which is joint work with Clément Dupont, I will describe a way to pair differential forms with `duals of differential forms'. This defines a theory of integration which satisfies the usual rules, but is automatically single-valued. Many well-known constructions in mathematics and physics are examples of such objects, and I will illustrate the theory with examples depending on the interests of the audience. 5pm, Synge
8.11.18 David Carchedi (MPIM Bonn) The Universal Property of Derived Manifolds Given two smooth maps of manifolds $f:M \to L$ and $g:N \to L$, if they are not transverse, the fibered product $M \times_L N$ may not exist, or may not have the correct cohomological properties. Thus lack of transversality obstructs many natural constructions in topology and differential geometry. Derived manifolds generalize the concept of smooth manifolds to allow arbitrary (iterative) intersections to exist as smooth objects, regardless of transversality. We will discuss the universal property of derived manifolds, and how this naturally gives rise to concrete models. 5pm, Synge
29.11.18 Caner Nazaroglu (University of Cologne)

Hilary Term 2018

Date Speaker Title Abstract Note
9.2.18 L. Takhtajan
N. Nekrasov
Yu. Tschinkel
HMI mini-conference
22.2.18 Mikhail Kapranov (HMI Simons Visiting Professor) Triangulated categorification, perverse sheaves and perverse schobers The idea of categorification, that is replacing vector spaces by (triangulated) categories, has been an important guiding tool in mathematics and physics. For example, the Fukaya category of a symplectic manifold can be seen as categorification of the middle homology. The talk will explain applications of categorification to the theory of perverse sheaves, a central tool in topology of algebraic varieties. The corresponding categorified concept, perverse schobers, turns out to be related to the program of introducing coefficients into Fukaya categories as well as to the homological minimal model program in birational algebraic geometry. 4pm, Synge
22.2.18 Estanislao Herscovich (Grenoble) Double monoidal categorical structures appearing in Quantum Field Theory R. Borcherds has introduced a different point of view to formalise perturbative Quantum Field Theory (pQFT), making use of several objects that behave somehow like bialgebras and comodules over them, and which are essential in his definition of Feynman measure. The former objects don’t seem however to be bialgebras in the classical sense, for their product and coproduct are with respect to two different tensor products, and similarly for comodules. Moreover, following physical motivations, these objects are given as some symmetric constructions of geometric nature. The aim of this talk is on the one hand to show that the “bialgebras” and “comodules” introduced by Borcherds cannot “naturally” exist, and on the other side to provide a background where a modified version of the so-called “bialgebras” and “comodules” do exist. This involves a category provided with two monoidal structures satisfying some compatibility conditions. As expected, the modified version of the mentioned “bialgebras” and “comodules” are not so far from the original one, considered by Borcherds. Moreover, we remark that these new candidates allowed us to prove the main results stated by Borcherds in his article (see my manuscript "Renormalization in Quantum Field Theory (after R. Borcherds)"). 5pm, Synge
22.3.18 Herve Gaussier (Grenoble) About curvature of domains in the complex Euclidean space Different generalizations of the Gauss curvature were introduced on manifolds, in different contexts. We will try to explain, on simple examples, the links between them for domains in $\mathbb C^n$. 5pm, Synge

Michaelmas Term 2017

Date Speaker Title Abstract Note
19.10.17 Yuuji Tanaka (Oxford University) On the singular sets of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on compact Kahler surfaces I'll talk about some observations on singular sets, which appear in some asymptotic analysis of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on closed four-manifolds. Rather recently, Cliff Taubes made a great breakthrough about the analysis of them by proving that the real two-dimensional Hausdorff measures of these singular sets are finite. In this talk, we analyse the singular sets more in some cases, in particular, we figure out that they have the structure of an analytic subvariety when the underlying manifold is a compact Kahler surface. Seminar, 3pm, SNIAM
19.10.17 Yuuji Tanaka (Oxford University) Vafa-Witten invariants for projective surfaces This talk describes studies of the Vafa-Witten theory on projective surfaces using techniques in algebraic geometry. These are joint work with Richard Thomas. After mentioning some backgrounds and motivation, we define invariants by (virtual) $\mathbb C^*$ localisations from the moduli space of Higgs pairs, which is the algebraic counterpart of the moduli space of solutions to the Vafa-Witten equations. We then describe calculations of them in examples, which match with Vafa-Witten's original conjectures raised more than 20 years ago. Colloquium, 5pm, Synge
Oct 23-27 HMI Workshop Geometry and Combinatorics of Associativity
2.11.17 Yan Soibelman (Kansas State University) Riemann-Hilbert correspondence, Fukaya categories and periodic monopoles Classical Riemann-Hilbert correspondence gives a categorical equivalence of connections with regular singularities on curves and representations of the fundamental groups of these curves (via the monodromy). One can state the same problem in the case of connections with irregular singularities, or more generally, in the cases of q-difference and elliptic difference equations. It turns out that the underlying geometry in all cases is the geometry of complex symplectic manifolds (surfaces in one-dimensional case), which are partially compactified to Poisson manifolds. The structure which replaces the representations of the fundamental groups is the one of the Fukaya category of those symplectic manifolds (category of A-branes in the language of physics). I plan to discuss these ideas in the one-dimensional case. If time permits, I will discuss the relationship of doubly and triply periodic monopoles on $\mathbb R^3$ with twistor families, which generalize the twistor family giving the hyperkahler structure on the moduli space of stable Higgs bundles on a projective curve. Colloquium, 5pm, Synge
9.11.17 Thomas Poguntke (Bonn University) Higher Segal structures in algebraic K-theory One of the main results of Dyckerhoff-Kapranov's work on higher Segal spaces concerns the fibrancy properties of Waldhausen's simplicial construction of algebraic K-theory, which are in particular responsible for the associativity of various Hall algebras. We will explain their result and generalize it to a certain higher dimensional analogue, replacing short exact sequences by longer extensions, in the case of an abelian category. Finally, if time permits, we will indicate how to relate it back to (an iteration of) Waldhausen's construction, in particular showing how it recovers the whole K-theory spectrum. Colloquium, 5pm, Synge
16.11.17 Tobias Dyckerhoff (Bonn University) A categorified Dold-Kan correspondence Various recent developments, in particular in the context of topological Fukaya categories, seem to be glimpses of an emerging theory of categorified homotopical and homological algebra. The increasing number of meaningful examples and constructions make it desirable to develop such a theory systematically. In this talk, we discuss a step towards this goal: a categorification of the classical Dold–Kan correspondence. Colloquium, 5pm, Synge
23.11.17 Natalia Iyudu (University of Edinburgh) Sklyanin algebras via Groebner bases and finiteness conditions for potential algebras I will discuss how some questions on Sklyanin algebras can be solved using combinatorial techniques, namely, the theory of Groebner bases, and elements of homological algebra. We calculate the Poincaré series, prove Koszulity, PBW, Calabi-Yau, etc., depending on the parameters of the Sklyanin algebras. There was a gap in the Artin-Schelter classification of algebras of global dimension 3, where Koszulity and the Poincaré series for Sklyanin algebras were proved only generically. It was filled in the Grothendieck Festschrift paper of Artin, Tate and Van den Bergh, using the geometry of elliptic curves. Our point is that we recover these results by purely algebraic, combinatorial means.
We use similar methods for other potential algebras as well, including homology of moduli of pointed curves given by Keel relations, and contraction algebras arising in noncommutative resolution of singularities.
Colloquium, 5pm, Synge
Nov 27-Dec 1 HMI Workshop Hall algebras and related topics
7.12.17 Wiesław Pawłucki (Jagiellonian University) Hironaka Rectilinearization in general O-minimal structures PDF

Hilary Term 2017

Date Speaker Title Abstract Note
26.1.17 Hans Franzen (Bochum University) Donaldson-Thomas invariants of quivers via Chow groups of quiver moduli We use a presentation of Chow rings of (semi-)stable quiver moduli to show that the primitive part of the Cohomological Hall algebra of a symmetric quiver is given by Chow groups of moduli of simple representations. This implies that the DT invariants are determined by these Chow groups. Seminar, 3pm, OSR
26.1.17 Hans Franzen (Bochum University) Geometric properties of quiver Grassmannians Quiver Grassmannians are geometric objects which parametrize subrepresentations of quiver representations. They can ban be regarded as analogs of (the usual) Grassmannians. Caldero and Chapoton have shown that the cohomology of quiver Grassmannians plays an important role in the theory of cluster algebras. After giving a basic introduction to representations of quivers and quiver Grassmannians, we exhibit classes of representations for which the quiver Grassmannian has a very simple geometric structure. Colloquium, 5pm, Synge
2.2.17 Norbert Hoffmann (Mary Immaculate College, Limerick) Del Pezzo surfaces and universal torsors A homogenuous polynomial equation of degree three in four variables defines a cubic surface S in complex projective threespace. Manin's Conjecture relates the geometry of this complex surface S to integer solutions of the equation. A main tool to study such integer solutions are the universal torsors over S introduced by Colliot-Thelene and Sansuc. All this generalizes from cubic surfaces to more general del Pezzo surfaces. After explaining this, I will speak about recent joint work with Ulrich Derenthal on universal torsors over degenerating del Pezzo surfaces. Colloquium, 5pm, Synge
9.2.17 G. Gabadadze
Yu. Tschinkel
A. Okounkov
HMI mini-conference
10.2.17 M. Kontsevich
N. Nekrasov
HMI mini-conference
16.2.17 Antonio Giambruno (Palermo University) Polynomial identities and their growth A polynomial identity satisfied by an algebra A is a polynomial in non commuting indeterminates vanishing identically when evaluated in A. These polynomials form a T-ideal of the free algebra, i.e., an ideal invariant under the endomorphisms of the free algebra. In general one can attach to such a T-ideal a numerical sequence, called the sequence of codimensions of A. In characterstic zero this sequence gives a quantitative measure of the polynomial identities satisfied by the algebra A. The aim of this talk is to present some of the results obtained in recent years on the asymptotic computation of such sequence. As a by-product I shall describe a method based on the representation theory of the symmetric group leading to some classification results. Colloquium, 5pm, Synge
23.2.17 Sung Yeon Kim (CMC, KIAS) CR maps between boundaries of flag domains PDF Colloquium, 5pm, Synge
16.3.17 Leandro Vendramin (University of Buenos Aires) Nichols algebras and applications Nichols algebras appear in several branches of mathematics going from Hopf algebras and quantum groups, to Schubert calculus and conformal field theories. In this talk we review the main problems related to Nichols algebras and we discuss some classification theorems and some applications. Seminar, 2pm, OSR
16.3.17 Brent Pym (University of Edinburgh) Divergent series and the Stokes groupoids Near a point where an ordinary differential equation has a pole of order two or more, its solutions may have very complicated behaviour, including essential singularities and divergent power series expansions. Nevertheless, using natural objects of differential geometry known as Lie groupoids, one can construct a canonical domain on which the solutions are globally defined and holomorphic. I will describe this construction and explain how it gives a new geometric interpretation of the classical techniques for "resumming" divergent series. This talk is based on joint work with Marco Gualtieri and Songhao Li. Colloquium, 5pm, Synge
30.3.17 Franc Forstnerič (University of Ljubljana) Complete bounded submanifolds in different geometries In this lecture I will survey recent results on the construction of bounded metrically complete submanifolds in several different geometries: holomorphic submanifolds of complex Euclidean spaces (the problem of Paul Yang), null holomorphic curves and conformal minimal surfaces in Euclidean spaces (the Calabi-Yau problem), and complete Legendrian curves in contact complex manifolds. Colloquium, 5pm, Synge
7.4.17 Kobi Kremnitzer (Oxford University) Constructing geometries A possible approach to solving certain mathematical problems is to construct a new geometry which is suitable to the problem. The main motivating example is the attempt to construct a geometry in which the integers become a "continuous" object (over the field with one element) as an approach to solving the Riemann Hypothesis. In this talk I will explain what do I mean by geometry and different ways of constructing new geometries. I will also give some examples of applications to number theory, analytic geometry and quantum field theory. Colloquium, Friday 3pm, Salmon

Michaelmas Term 2016

Date Speaker Title Abstract Note
29.9.16 Emilio Franco (Unicamp, Brazil) Brane involutions and irreducible holomorphic symplectic manifolds We study natural brane-involutions on moduli space of sheaves over symplectic surfaces and their behaviour under Fourier-Mukai transform and lattice Mirror symmetry. This is joint work with M. Jardim (Campinas State University) and G. Menet (Campinas State University).
6.10.16 Victoria Lebed (TCD) Unexpected facets of the Yang-Baxter equation In this talk I propose to look at the good old Yang-Baxter equation from an unexpected viewpoint. We shall see that this equation generalizes basic algebraic laws: associativity, the Jacobi identity, self-distributivity, the axioms of a lattice. In spite of this generality, one can say non-trivial things valid for all solutions to the YBE - namely, concerning their representation and (co)homology theories. Moreover, any solution comes with a quadratic universal enveloping algebra, which for certain solution classes enjoys particularly nice algebraic properties. As an application, we shall outline how to construct resolutions of some algebras by interpreting them as the universal enveloping algebras of easily manipulable solutions to the YBE. Knot-theoretic applications will also be sketched. Mathematics Colloquium
Synge
5pm
13.10.16 Jonas Kaszian (University of Cologne) Indefinite Theta Functions arising in Gromov-Witten Theory In this talk, we consider a function occuring naturally in the open Gromov-Witten potential of an elliptic orbifold. To help illuminate its mysterious nature, we connect it with higher-depth mock modular forms by studying related indefinite theta functions of signature (1,3). The study of these functions was recently opened up by work of Alexandrov, Banerjee, Manschot, and Pioline discussing the case of signature (n,2).
20.10.16 Raf Bocklandt (University of Amsterdam) Moduli of Matrix factorizations We discuss how to construct moduli spaces of matrix factorizations using concepts from mirror symmetry and illustrate these ideas with examples coming from dimer models.
From the Freezer to the Tropics Following Goncharov and Kenyon one can define a dynamical system from a consistent dimer model. On the other hand one can also use this dimer model to resolve a singularity. We will explain how these two become related if we go to the absolute zero temperature in the dynanical system and to the tropical limit of the resolution. Mathematics Colloquium
Synge
5pm
27.10.16 Emmanuel Letellier
(Paris Diderot-Paris 7 University)
Higgs bundles and indecomposable parabolic vector bundles over the projective line In this talk we will count the number of isomorphism classes of geometrically indecomposable parabolic bundles over the projective line over a finite field. We will explain the relation between this counting and the moduli space of Higgs bundles with prescribed residues over the complex projective line.
Character varieties and representation theory The aim of this talk is to explain the relation between two problems in mathematics: the first one is about the representation theory of finite Lie groups and the second one is about the geometry of the so-called character varieties, namely the moduli space of representations of the fundamental groups of punctured compact Riemann surfaces into complex Lie groups. Mathematics Colloquium
Synge
5pm
17.11.16 Herve Gaussier (Grenoble) How can metrics explain some geometric properties of complex manifolds? The Poincaré distance is an example of a hyperbolic distance on the unit disk in C. It admits different generalizations in the context of complex geometry, such as the Bergman metric, example of a Kähler-Einstein metric, or the Kobayashi metric, example of a Finsler metric, defined on complex manifolds. We will explain how the properties of such metrics restrict the geometry of the manifold. Mathematics Colloquium
Synge
5pm
24.11.16 Tyler L. Kelly (University of Cambridge) Unifying Mirror Constructions Mirror symmetry uses dualities in string theory to predict that, given a symplectic variety, there should exist an algebraic variety known as the mirror such that various geometric and physical data are exchanged. Over the past 25 years, there have been many recipes that have been proposed to construct the mirror for certain symplectic varieties. However, in certain cases, for the same symplectic variety, different recipes can give different algebraic varieties as the mirror. In this talk, we will talk about how one can fix this discrepancy in the context of Kontsevich's Homological Mirror Symmetry. Mathematics Colloquium
Synge
5pm
1.12.16 Arnaud Mortier (DCU) Finite-type cohomology of the space of long knots The aim of this talk is to give an overview of Vassiliev's cohomology of the space of knots. Topics discussed will include the origins in singularity theory, several algebraic reformulations, the Kontsevich integral (a universal Vassiliev knot invariant), a systematic way to construct cocycles, and some examples.
Knots, virtual knots, and finite-type invariants In the classical sense, a knot is an embedding of a circle in R^3. This notion has known a multitude of variations, with as many purposes: trading the circle for another 1-manifold, increasing dimensions... Virtual knot theory is such a variation. It came up during the 90's after a period of enthusiasm towards a new kind of knot invariants, today widely understood except for the fact that it is still open whether they classify all knots: Vassiliev's 'finite-type' invariants. I will discuss the main ideas behind these notions. Mathematics Colloquium
New sem. room
4:30pm
9.12.16 Nikos Diamantis (University of Nottingham)
Values of L-functions and shifted convolutions A characterisation of the field containing the values of L-functions is proposed. This is described in terms of shifted convolution series of divisor sums and to establish it we use a double Eisenstein series we previously studied with C. O'Sullivan and a kernel function used by F. Brown in his study of multiple modular values. Mathematics Colloquium
Synge
4pm

[Travel information](travelinfo/), [Hamilton Building](https://www.maths.tcd.ie/about/map.php). ### Links: * [Dublin Area Mathematics Colloquium 2009-2016](http://www.maths.tcd.ie/~zaitsev/colloquium/) * [Dublin Area TP Colloquium](http://www.maths.tcd.ie/~dtpc) * [Archive of Events 1998-2012](http://www.maths.tcd.ie/seminars/index.php?name=indexarchive)
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