#Dublin Mathematics Colloquium, Geometry Seminar ### [School of Mathematics](http://www.maths.tcd.ie/), [Trinity College Dublin](http://www.tcd.ie/) #### Location: * Mathematics Colloquium takes place in the New Seminar Room (Hamilton Building) at 4pm on Thursday.

Michaelmas Term 2022

Date Speaker Title Abstract Note
22.09.22 Jack Kelly (TCD) How (and why) to derive analytic geometry In this talk I will present a model of derived analytic geometry (over either the complex numbers or a non-Archimedean Banach field) as derived geometry, in the sense of Toën-Vezzosi, relative to the monoidal category of simplicial complete bornological vector spaces. In particular I will explain how this setup provides a solution to the long-standing issue of defining a well-behaved category of quasi-coherent sheaves on an analytic space which satisfies descent. I will also explain how these constructions permit a definition of analytic geometry over any Banach ring, including `universal' analytic geometry over the Banach ring of integers (the integers equipped with the absolute value norm). This is joint work with Oren Ben-Bassat and Kobi Kremnizer, and also uses work of Federico Bambozzi. 4pm, New Seminar Room
06.10.22 Anton Alekseev (University of Geneva) Virasoro Hamiltonian spaces We develop a theory of Hamiltonian actions of the canonical central extension of the group of diffeomorphisms of the circle. It turns out that Virasoro Hamiltonian spaces (this is another name for such Hamiltonian actions) are in bijective correspondence with group valued Hamiltonian spaces with moment map taking values in (a certain part of) the universal cover of the group SL(2, R). Among other things, this correspondence allows to recover the classical result of Lazutkin-Pankratova, Kirillov, Segal, Witten (and others) on classification of coadjoint orbits of the Virasoro algebra. Interesting examples of Virasoro Hamiltonian spaces arise as moduli spaces of conformally compact hyperbolic metrics on oriented surfaces with boundary. The talk is based on a joint work in progress with Eckhard Meinrenken. 4pm, New Seminar Room
13.10.22 Jan Pulmann (University of Edinburgh) On quantizations of moduli spaces of flat connections on marked surfaces The moduli space of flat G-connections on a surface S, being the phase space of the Chern-Simons theory on S×Ɪ, has a natural symplectic structure. With some boundary conditions, we get a so-called quasi-Poisson structure, which was deformation quantized by Li-Bland and Ševera. I will describe the combinatorial data necessary to specify their quantization uniquely and the relations between all such quantizations. The answer will use the Konstevich integral and the Drinfeld associator, relating the pentagon equation of the associator with the pentagon equation for triangulations of the surface S. Finally, I will describe some simple consequences of this description of the quantization. 4pm, New Seminar Room
20.10.22 Konstantin Wernli (University of Southern Denmark) Generalized Hamilton-Jacobi actions and Chern-Simons partition functions This talk is based on the papers 2012.13270 and 2012.13983 which are joint with A. Cattaneo and P. Mnev. To any constrained system, we associate a generalized Hamilton-Jacobi (HJ) action, which is a generating function for the evolution relation. One can quantize this constrained system in the BV-BFV formalism, and we show that the generalized Hamilton-Jacobi action is the leading order of the corresponding effective action. Applied to Chern-Simons theory, this gives a precise effective action explanation of the CS-WZW correspondence (in 3d) and the CS-BCOV correspondence (in 7d). These results suggest that holography is at heart an effective phenomenon. 4pm, New Seminar Room
3.11.22 Veronica Fantini (IHES) Wall-Crossing structures in mirror symmetry Mirror symmetry is one of the most fascinating conjectures in mathematical physics, in a nutshell it is a duality between the complex and the symplectic geometry of Calabi-Yau varieties. Since it was first conjectured in string theory, it has been explained within different geometric approaches such as the Strominger-Yau-Zaslow conjecture, Kontsevich homological mirror symmetry, etc. In addition, mathematicians got interested in the so called "reconstructing problem" aimed to understand how to reconstruct the mirror of a given Calabi-Yau. Among the main contributions, I will briefly recall ideas from Fukaya's multivalued Morse theory, Kontsevich-Soibleman non-Archimedean approach and the Gross-Siebert program, whose common point is to use "scattering diagrams". The latter encode geometric data (such as Gromov-Witten invariants) through a precise algebraic structure. In particular, the algebraic structure of scattering diagrams which conjecturally govern the reconstruction of the mirror of holomorphic pairs (a holomorphic vector bundle over a complex manifold) is the extended tropical vertex group (introduced in arXiv:1912.09956). I will then discuss a possible approach to understand the reconstruction of the mirror of holomorphic pairs based on the relationship between scattering diagrams in the extended tropical vertex group and 2d-4d wall-crossing structures. 4pm, New Seminar Room
10.11.22 Alexander Shapiro (University of Edinburgh) Cluster algebra and quantum Teichmüller theory Given a surface and a group one can consider the corresponding character variety. Quantum Teichmüller theory uses cluster algebra to quantise the character variety, and construct its representations, equivariant under the action of the mapping class group of the surface. Moreover, it is conjectured that the resulting construction respects cutting and gluing of surfaces. In this talk, I will discuss this conjecture, and its relation to representation theory of quantum groups and 3-manifold invariants. 4pm, New Seminar Room
17.11.22 Kasia Rejzner (University of York) Algebraic structures in quantization of gauge theories In this talk I will present the framework of perturbative algebraic quantum field theory and explain how it applies to quantization of physically realistic gauge theories, including QED and the Standard model of particle physics. The main mathematical tools are in homological algebra, where one uses a version of the BV (Batalin-Vilkovisky) formalism, combined with some functional analysis. 4pm, New Seminar Room
24.11.22 Manuel Araújo (University of Cambridge) String diagrams for semistrict n-categories String diagrams are a powerful computational tool, most commonly used in the context of monoidal categories and bicategories. I will talk about extending this to higher dimensions. The natural setting for n-dimensional string diagrams should be some form of semistrict n-category, where composition operations are strictly associative and unital, but the interchange laws hold only up to coherent equivalence. One idea is to define a semistrict n-category as something which admits composites for labelled string diagrams, much as one can define a strict n-category as something that admits composites for labelled pasting diagrams. The first step in this program, which we have carried out, is to develop a theory of n-sesquicategories based on simple string diagrams. These are like strict n-categories without the interchange laws. The second step, which is still work in progress, is to add operations implementing weak versions of the interchange laws, obtaining the desired notion of semistrict n-category. In dimension 3, this recovers the notion of Gray 3-category. (I will start with a gentle introduction to strict n-categories and the idea of weak n-categories). 4pm, New Seminar Room

Michaelmas Term 2020

Date Speaker Title Abstract Note
3.12.20 Jack Kelly (TCD) Koszul Duality in Exact Categories Toën and Vezzosi’s model of algebraic geometry as derived geometry relative to the monoidal model category of complexes of modules over a ring allows one to apply tools from homological algebra to study the geometry of derived stacks. A result of homological algebra which is of fundamental importance to derived algebraic geometry, particularly in the context of deformation theory and formal moduli problems, is Koszul Duality. Due to, amongst others, Quillen, Hinich, and Vallette, Koszul duality says that there is a Quillen equivalence between the category of coaugmented cocommutative differential graded coalgebras and the category of differential graded Lie algebras. The geometric interpretation is that a Lie algebra determines a unique formal space (i.e. a coalgebra) of which it is the shifted tangent Lie aglebra. Recent work with Bambozzi, Ben-Bassat, and Kremnitzer suggests that derived analytic geometry over a Banach field k of characteristic 0 can be modelled as derived geometry relative to the monoidal model category of chain complexes of bornological k-vector spaces. As is the case for derived algebraic geometry, one could in principle use techniques of homological algebra (internal to the category of bornological spaces) to understand analytic geometry. In this talk I will present a generalisation of Koszul duality to so-called monoidal elementary exact categories. The category of bornological spaces over k is such a category, so this result has implications for derived analytic geometry. 3pm, Zoom

Hilary Term 2020

Date Speaker Title Abstract Note
20.02.20 Balazs Szendroi (Oxford University) Hilbert schemes of points on singular surfaces: geometry, combinatorics and representation theory Given a smooth algebraic surface S over the complex numbers, the Hilbert scheme of points of S is the starting point for many investigations, leading in particular to generating functions with modular behaviour and Heisenberg algebra representations. I will explain aspects of a similar story for surfaces with rational double points, with links to algebraic combinatorics and the representation theory of affine Lie algebras. 4pm, Synge

Michaelmas Term 2019

Date Speaker Title Abstract Note
25.09.19 Yoshiaki Goto (Otaru University of Commerce) Contiguity relations for hypergeometric integrals of type (k,n) There are several generalizations of the Gauss hypergeometric function. The Aomoto-Gelfand hypergeometric function is one of them, which are introduced in the view point of integral representations. To study such hypergeometric integrals, twisted homology and cohomology groups are useful. In this talk, I would like to talk about these hypergeometric integrals and twisted cohomology groups mainly. I will give a short introduction to twisted cohomology groups and the intersection form on them, and explain derivations of contiguity relations (difference equation) as their application. Slides. 2pm, Seminar Room
25.09.19 Saiei-Jaeyeong Matsubara-Heo (Kobe University) Evaluating cohomology intersection numbers from twisted period relations After the pioneering work of Kazuhiko Aomoto in 70's, the study of hypergeometric integrals is now regarded as a twisted analogue of that of period integrals. Through Poincare duality, it is straightforward to define (co)homology intersection numbers, but their exact evaluation is not an easy problem to solve except for some special integrals such as Aomoto-Gelfand hypergeometric integral. In this talk, we investigate the possibilities of evaluating cohomology intersection numbers (c.i.n.) through the twisted analogue of period relations. One way is to characterize the c.i.n. as a solution of differential equations. This part is the joint work with Nobuki Takayama. The other way is to expand the c.i.n. at toric infinity which can be obtained by using GKZ hypergeometric systems. 3pm, Seminar Room
27.09.19 Maxim Kontsevich
Andrei Okounkov
Yuri Tschinkel
HMI mini-conference
30.10.19 Francis Brown (HMI Simons Visiting Professor) Motivic hypergeometric functions 2:15pm, Seminar Room

Hilary Term 2019

Date Speaker Title Abstract Note
24.01.19 Adam Keilthy (Oxford University) Relations and Filtration on Multiple Zeta Values Multiple zeta values are a class of numbers arising naturally in algebra, geometry and even QFT. They have a rich algebraic structure, and satisfy many relations, such as the double shuffle relations, and the associated relations. We will discuss these relations and their connection to the motivic theory of $\mathbb P^1-\{0,1,\infty\}$ along side two naturally arising filtrations on the algebra of MZVs. We shall then introduce a new filtration and derive two new sets of relations, including Charlton's cyclic insertion, that hold in the associated graded algebra with respect to this filtration 4pm, Synge
20.03.19 Francis Brown (HMI Simons Visiting Professor) New perspectives on periods and integration, with applications to physics Mini-course 2pm, Seminar Room
21.03.19 Marton Hablicsek (University of Copenhagen) Derived intersections and interactions with deformation theory One purpose of derived algebraic geometry is to deal with bad situations in geometry. In intersection theory, bad situations arise when the intersection is not of the expected dimension or when the intersection is highly singular. In my talk, I will use the theory of derived intersections to compare intersections of Lagrangians in symplectic varieties (and higher analogues) with their deformation quantizations. As applications, I will show how one can obtain the Hodge theorem and how one can obtain extra structures on deformation quantizations in positive characteristics. 4pm, Synge
22.03.19 Marton Hablicsek (University of Copenhagen) Extra structures on deformation quantizations of shifted symplectic derived stacks in positive characteristics Bezrukavnikov and Kaledin showed that convergent deformation quantizations of symplectic varieties over a perfect field of positive characteristic give rise to sheaves of Azumaya algebras. We investigate the structure of the convergent quantization of shifted symplectic derived stacks, especially the 1-shifted cotangent bundle $S$ of a smooth scheme $X$ over a perfect field of positive characteristic. We show that the quantization is an $E_2$-algebra over the Frobenius twist $S'$ of the 1-shifted cotangent bundle which restricted to the zero section $X'\to S'$ is weakly $E_2$-Morita equivalent to the structure sheaf of the Frobenius twist $X'$ of $X$. 4pm, Seminar Room
28.03.19 Alex Fink (QMU London) Some tropical geometry Tropical mathematics is the study of the tropical semifield, $\mathbb R\cup\{\infty\}$ with minimum as addition and plus as multiplication. It congealed as a field of study in the mid-1990 as several existing lines of research came to be recognised as related, especially in optimisation and in algebraic geometry. There are ways to "change base" to the tropical semifield from the real numbers or from fields with nonarchimedean valuation; the tropical varieties that result retain a surprising amount of the properties of the originals. I'll give a general introduction to the area, perhaps making it to some of my own contributions regarding tropical linear spaces. 4pm, Synge
3.04.19 Matija Tapuskovic (Oxford University) Motivic Galois coaction on 1-loop motivic Feyman amplitudes I will present some algebro-geometric tools and examples for computing the coaction of a Hopf algebra corresponding to the action of the motivic Galois group on 1-loop motivic Feynman amplitudes following Francis Brown. We will remain in the relam of 1-loop Feyman graphs because their associated motives can be proved, using the same techniques we will see during the talk, to be of Mixed-Tate type which is understood quite well in algebraic geometry. This will enable us to go into the subtleties of the computations, and of relating the Galois conjugates of a Feynman amplitude associated to a graph to periods of motives of its sub-quotient graphs. 11am, Seminar Room
4.04.19 Johan Leray (University Paris 13) Double Poisson algebras up to homotopy The notion of double Poisson algebra, introduced by Van den Bergh in 2008, is the noncommutative analogous of Poisson structure. After recalling what is a Poisson structure and presenting and motiving this new algebraic structure, I will define the notion of (pr)operads, which are algebraic objects used to encode many algebraic structures. I will show how we can use this notion to understand what is double Poisson algebras up to homotopy. 4pm, Synge
8-10.05.19 Analysis, Geometry and Algebra Conference
13-17.05.19 HMI Workshop Gauge theory and virtual invariants
27-31.05.19 HMI Workshop Homotopy meets homology

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) Squashed Toric Manifolds and Higher Depth Mock Modular Forms In 1980’s, a three-way relation between compact Calabi-Yau manifolds, two dimensional (2,2) superconformal field theories and modular forms was established. These objects can be tied together through the corresponding Gauged Linear Sigma Models and the computation of elliptic genera via such models. In this talk, we will focus on a class of sigma models that describe toric Calabi-Yau manifolds which are squashed. In the simplest one-dimensional example previously studied, the supersymmetric partition function that computes the elliptic genus is known to produce a mixed mock Jacobi form including its modular completion. I will describe the automorphic nature for the general case of squashed toric sigma models and show that they yield higher-depth mock modular forms that have been recently formulated. Finally, I will discuss further refinements and possible applications to physics and geometry. 5pm, Synge

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
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
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
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
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
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
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

[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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