## Geometry and Physics Seminar 2014/2015

### General information

The Geometry and Physics seminar series is operated jointly between UAlberta and UBC. The seminars hosted by UBC are on Mondays 15:00-16:00 with coffee and cookies beforehand. The seminar hosted by UAlberta are Wednesdays 15:00-16:00. Exceptions to these dates do occur. The seminars at UBC take place in ESB 4127 (Earth Sciences Building).

### Term 1 (2014)

• 8/9/2014 - Sheldon Katz (UUIC) @UBC: Refined and motivic BPS invariants.

The virtual Poincare polynomials of the stable pair moduli spaces of a Calabi-Yau threefold are conjecturally equivalent to the refined BPS numbers of Gopakumar and Vafa. As an application, stable pair invariants of the del Pezzo surfaces dP_n determine BPS Hilbert spaces which are observed to be representions of the exceptional Lie algebra E_n, consistent with expectations of string theory. In another direction, string theory on K3 x T^2 leads to a reduced DT theory on K3, hence corresponding motivic and refined invariants. Work in progress on the rational elliptic surface dP_9 ("half K3") suggests that a blend of these two examples leads to a BPS Hilbert space with a representation of affine E_8. This talk includes separate joint works with Choi, Klemm, and Pandharipande.

• 8/9/2014 - Masoud Kamgarpour (Queensland) @UBC: Preservation of depth in local geometric Langlands program.

Local geometric Langlands program aims to establish a relationship between representations of the Galois group of a local field and irreducible representations of the dual group. It is expected that, under mild conditions, this correspondence preserves depths of representations. In this talk, I will explain the geometric analogue of this expectation, in the framework of Frenkel-Gaitsgory's local geometric Langlands correspondence. Based on a joint project with Tsao-Hsien Chen: http://arxiv.org/abs/1404.0598.

• 15/9/2014 - Helge Ruddat (Mainz) @UBC: Canonical Coordinates from Tropical Curves.

Morrison defined canonical coordinates near a maximal degeneration point in the moduli of Calabi-Yau manifolds using Hodge theory. Gross and Siebert introduced a logarithmic-tropical algorithm to provide a canonically parametrized smoothing of a degenerate Calabi-Yau. We show that the Gross-Siebert coordinate is a canonical coordinate in the sense of Morrison. The coordinates are given by period integrals which we compute explicitly integrating over cycles constructed using tropical geometry. This is joint work with Siebert.

• 22/9/2014 - Sarah Scherotzke (Bonn) @UBC: Graded quiver varieties and derived categories.

Nakajima's quiver varieties are important geometric objects in representation theory that can be used to give geometric constructions of quantum groups. Very recently, graded quiver varieties also found application to monoidal categorification of cluster algebras. Nakajima's original construction uses geometric invariant theory. In my talk, I will give an alternative representation theoretical definition of graded quiver varieties. I will show that the geometry of graded quiver varieties is governed by the derived category of the quiver. This is joint work with Berhard Keller.

• 25/9/2014 - Matt Ballard (South Carolina) @UAlberta: Wall crossing in moduli problems and semi-orthogonal decompositions.

We discuss how the derived category of a smooth algebraic stack of finite type changes as one removes certain types of closed substacks. As an application, we show how wall-crossing in moduli of stable sheaves and Bridgeland stable objects yields semi-orthogonal decompositions of relating their derived categories.

• 25/9/2014 - Colin Diemer (Miami) @UAlberta: Decomposing Landau-Ginzburg Models.

One version of homological mirror symmetry relates the algebraic geometry of certain varieties to the symplectic topology of a Lefschetz pencil. Extracting symplectic (i.e. Floer theoretic) invariants from these fibrations is quite difficult, even in simple examples. I'll review some recent proposals (particularly from Kapranov-Kontsevich-Soibelman and Diemer-Kerr-Katzarkov) for deforming symplectic fibrations into more tractable components. The corresponding mirror theory appears to be closely related to birational geometry and the Mori program.

• 29/9/2014 - Ravi Vakil (Stanford) @UBC: Stabilization of discriminants in the Grothendieck ring.

We consider the limiting behavior'' of {\em discriminants}, by which we mean informally the closure of the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the "ring of motives", as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization can be described in terms of the motivic zeta values. The results extend parallel results in both arithmetic and topology. I will also present a conjecture (on motivic stabilization of symmetric powers'') suggested by our work. Although it is true in important cases, Daniel Litt has shown that it contradicts other hoped-for statements. This is joint work with Melanie Wood.

• 6/10/2014 - Gabriel Kerr (Kansas) @UBC: Mirror symmetry for the punctured plane.

Homological mirror symmetry initially concerned Calabi-Yau 3-folds and, from that point, rapidly expanded to incorporate local Calabi-Yau's and Fano varieties. In this talk, I will discuss joint work with Ludmil Katzarkov and Maxim Kontsevich on extending this correspondence further to include quasi-affine toric varieties, the most basic example of which is a punctured plane. The complex side of the correspondence, or B-model, remains the derived category of coherent sheaves of the variety. On the mirror side, the A-model is a partially wrapped Fukaya category on the cotangent bundle of the torus. The key ingredient is the wrapping Hamiltonian which is defined as a distance^2 function away from a mirror non-compact Lagrangian skeleton. I will explain the geometric intuition for the case of the punctured plane and discuss elements of the proof for the general case.

• 6/10/2014 - Charlie Beil (Bristol) @UBC: Nonlocality and the central geometry of dimer algebras.

A dimer algebra is a type of quiver algebra whose quiver embeds in a torus, with homotopy-like relations. Dimer algebras with the cancellation property are Calabi-Yau algebras, and their centers are 3-dimensional Gorenstein singularities. Non-cancellative dimer algebras, on the other hand, are not Calabi-Yau, and their centers are nonnoetherian. In contrast to their cancellative counterparts, very little is known about these algebras, despite the fact that almost all dimer algebras are non-cancellative. I will describe how their centers are also 3-dimensional singularities, but with the strange property that they contain positive dimensional 'smeared-out' points. Furthermore, I will describe how this nonlocal geometry is reflected in the homology of certain vertex simple representations.

• 20/10/2014 - Jim Bryan (UBC) @UBC: Donaldson-Thomas theory of local elliptic surfaces via the topological vertex.

Donaldson-Thomas (DT) invariants of a Calabi-Yau threefold X are fundamental quantum invariants given by (weighted) Euler characteristics of the Hilbert schemes of X. We compute these invariants for the case where X is a so-called local elliptic surface --- it is the total space of the canonical line bundle over an elliptic surface. We find that the generating functions for the invariants admit a nice product structure. We introduce a new technique which allows us to use the topological vertex in this computation --- a tool which previously could only be used for toric threefolds. As a by product, we discover surprising new identities for the topological vertex. This is joint work with Martijn Kool, with an assist from Ben Young.

• 22/10/2014 - David Favero (UAlberta) @UAlberta: Equivalences of derived categories of double mirrors.

Given a Calabi-Yau complete intersection in a toric Fano variety, there are various ways to construct the mirror. Sometimes these mirrors are isomorphic and sometimes they are not. These distinct 'double' mirrors should be equivalent in some way if they all have a shot at being the 'correct' mirror in some setting of mirror symmetry. We will discuss the Batyrev-Borisov and Berglund-Hübsch-Krawitz construction and the double mirrors which arise, as well as their relationship through variation of geometric invariant theory quotients, Landau-Ginzburg models, and derived equivalence. This is joint work with Tyler Kelly.

• 27/10/2014 - Ping Xu (PennState) @UBC: Rozansky--Witten-type invariants from symplectic Lie pairs.

In 1997, Rozansky and Witten built new finite-type invariants of 3-manifolds from hyperkahler manifolds. It was later shown by Kontsevich and Kapranov that those invariants only depend on the holomorphic symplectic structure of the hyperkahler manifolds. Indeed Kapranov proved that these invariants may be considered as an analogue of Chern-Simons type invariants, where the Atiyah class of the underlying complex manifold plays the role of Lie bracket. In this talk, we introduce symplectic structures on "Lie pairs"; of (real or complex) algebroids, encompassing homogeneous symplectic spaces, symplectic manifolds with a $\mathfrak g$-action and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky-Witten-type invariants of three-manifolds. This is a joint work with Yannick Voglaire.

• 7/11/2014 - Vivek Shende (Berkeley) @UBC: Moduli spaces from micolocal geometry.

Time: 1.30-2.30pm.>

• 10/11/2014 - Zinovy Reichstein (UBC) @UBC: Versal actions with a twist.

The term “versal” is best understood by subtracting “unique” from both sides of the formula Universal = unique + versal. In this talk based on joint work with Alex Duncan, I will discuss competing notions of versality for the action of an algebraic group G on an algebraic variety X and relate these notions to properties (such as existence and density) of rational points on twisted forms of X. I will then present examples, where this relationship can be used to prove that certain group actons are versal or, conversely, that certain varieties have rational points.

• 10/11/2014 - Vincent Bouchard (UAlberta) @UAlberta: Quantum is airy, but is Airy quantum?

According to Merriam-Webster, “airy” means “having a light or careless quality that shows a lack of concern”. That describes pretty accurately quantum physics. But Airy was also a mathematician and physicist that did a lot of things, and somehow got his name attached to a very simple complex curve. It turns out that this so-called Airy curve encapsulates intersection numbers on the moduli space of curves via topological recursion. Moreover, the Airy curve can be quantized; the resulting Schrodinger differential operator recursively constructs intersection numbers through WKB analysis. A natural question then is to ask whether this circle of ideas holds in a much more general setting; given a complex curve that encapsulates some nice enumerative invariants via topological recursion, does there exist a (unique?) quantization of the complex curve that reconstructs the invariants recursively via WKB analysis? This question has closed connections with many fundamental conjectures in enumerative geometry and other areas of mathematics, such as the AJ conjecture in knot theory, and Witten’s conjecture for intersection numbers. In recent work with B. Eynard we construct such a quantization in a number of different settings; in this talk I will focus on the case of the r-Airy curve, which generates intersection numbers on the moduli space of r-spin curves.

• 17/11/2014 - Bruno Kahn (Jussieu) @UBC: Applications of birational motives.

I will give the definition of birational motives and explain how they can be used in various areas of algebraic geometry: counting rational points over finite fields, defining the "Tate-Shafarevich motive" of an abelian variety over a function field, shedding a new light on Roitman's theorem on torsion 0-cycles.

• 24/11/2014 - Jochen Kuttler (UAlberta) @UBC: Modules of differentials for Lie algebras.

In this talk, I will attempt to introduce/discuss modules of differentials for Lie algebras modelled after the corresponding notion for rings. This is relevant to the structure of certain infinite dimensional Lie algebras. This is joint work with Arturo Pianzola.

• 26/11/2014 - Andrew Harder (UAlberta) @UAlberta: TBA.
• 1/12/2014 - Nicolò Sibilla (UBC) @UBC: The topological Fukaya category and mirror symmetry for toric Calabi-Yau threefolds.

The Fukaya category of open symplectic manifolds is expected to have good local-to-global properties. Based on this idea several people have developed sheaf-theoretic models for the Fukaya category of punctured Riemann surfaces: the name topological Fukaya category appearing in the title refers to the (equivalent) constructions due to Dyckerhoff-Kapranov, Nadler and Sibilla-Treumann-Zaslow. In this talk I will introduce the topological Fukaya category and explain applications to Homological Mirror Symmetry for toric Calabi-Yau threefolds. This is work in progress joint with James Pascaleff.

### Term 2 (2015)

• 5/1/2015 - Jeffrey Ginasiracusa (Swansea) @UBC: Scheme Theory in Tropical Geometry.

In the standard approach to tropicalization, an algebraic subset X of a toric variety over a non-archimedean valued field k is sent to a weighted polyhedral set Trop(X) which we think of as a combinatorial shadow of X.  The result depends only on the k-points of X.  A system of polynomial equations often contains more information than the set of its solutions over a field, and the philosophy of scheme theory is that we should treat the system of equations itself as a fundamental geometric object from which the solution set is derived. Scheme-theoretic tropicalization is about realizing Trop(X) as the solution set to an underlying system of polynomial equations over the idempotent semiring of tropical numbers - a system that is constructed in a canonical way from the equations defining X. The theory involves the field with one element, and with these ideas the Berkovich analytification appears as the universal tropicalization of X and as the moduli space of valuations on X.

• 12/1/2015 - Theo Johnson-Freyd (Northwestern) @UBC: Functorial axioms for Heisenberg-picture quantum field theory

The usual Atiyah--Segal "functorial" description of quantum field theory corresponds to the "Schrodinger picture" in quantum mechanics.  I will describe a slight modification that corresponds to the "Heisenberg picture", which I will argue is better physically motivated.  The example I am most interested in is a version of quantum Chern--Simons theory that does not require the level to be quantized; it provides a neat packaging of pretty much all objects of skein theory.

• 26/1/2015 - Francois Greer (Stanford) @UBC: Picard Groups of K3 Moduli Spaces

Polarized K3 surfaces of genus g can be thought of as families of canonical curves.  As such, their moduli space K_g has similar properties to M_g.  For instance, both are unirational for low values of g, and both have discrete Picard group.  In this talk, we will use the explicit unirationality of K_g to compute its Picard number in a few cases, which verifies the Noether-Lefschetz conjecture for genus up to 10.

• 02/2/2015 - Mattia Talpo (UBC) @UBC: Infinite root stacks of log schemes

I will talk about the notion of "infinite root stack" of a logarithmic scheme, introduced by myself and Angelo Vistoli as part of my PhD thesis. It is a "limit" version of the generalization to log schemes of the stack of roots of a divisor on a variety, and we show, among other things, that its "bare" geometry closely reflects the "log" geometry of the base log scheme. After giving some motivation, I will briefly define log schemes and describe this infinite root construction. I will then state the results we get about it, and their relevance to log geometry, also in view of (hopefully) upcoming applications.

• 24/2/2015 - Philippe Gille (IMAR and Université Claude Bernard) @UBC: Serre's conjecture II : beyond the de Jong-He-Starr's theorem

A 1962 conjecture of J.-P. Serre asserts that Galois cohomology set H^1(K, G) vanishes for every simply connected semisimple group G defined over a field K of cohomological dimension ≤ 2. In other words, every G-torsor over Spec(K) is split. In this talk I will survey recent progress on this conjecture.

• 2/3/2015 - David Carchedi (UBC) @UBC: Dg-manifolds as 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. In the world of derived manifolds, such a fibered product always exists as a smooth object, regardless of transversality. In this talk we will describe recent progress of ours with D. Roytenberg on giving an accessible geometric model for derived manifolds using differential graded manifolds.

• 9/3/2015 - Jim Bryan (UBC) @UBC: The Donaldson-Thomas theory of K3xE via motivic and toric methods

Donaldson-Thomas invariants are fundamental deformation invariants of Calabi-Yau threefolds. We describe a recent conjecture of Oberdieck and Pandharipande which predicts that the (three variable) generating function for the Donaldson-Thomas invariants of K3xE (the product of a K3 surface and an elliptic curve) is given by the reciprocal of the Igusa cusp form of weight 10. For each fixed K3 surface of genus g, the conjecture predicts that the corresponding (two variable) generating function is given by a particular meromorphic Jacobi form. We prove the conjecture for K3 surfaces of genus 0 and genus 1. Our computation uses a new technique which mixes motivic and toric methods.

• 16/3/2015 - Tom Coates (Imperial College) @UBC: Mirror Symmetry and the Classification of Fano Manifolds.[video]
• 23/3/2015 - Vladimir Chernousov (UAlberta) @UBC Algebraic groups and maximal tori. [video]

We will survey recent developments dealing with characterization of absolutely almost simple algebraic groups having the same isomorphism/isogeny classes of maximal tori over the field of definition. These questions arose in the analysis of weakly commensurable Zariski-dense subgroups. While there are definitive  results over number fields (which we will briefly review), the  theory over general fields is only emerging. We will formulate the  existing conjectures, outline their potential applications, and  report on recent progress. Joint work with A. Rapinchuk and  I. Rapinchuk.

• 30/3/2015 - An Huang (Harvard) @UBC Period integrals and their differential systems.

Period integrals are geometrical objects which can be realized as special functions, or sections of certain bundles. Their origin goes back to Euler, Gauss and Legendre in the study of complex algebraic curves. In their modern version, period integrals naturally arise in Hodge theory, and more recently in mathematical physics, and the theory of hypergeometric functions. I will give an overview of a recent program to use differential equations and D-module theory to study period integrals. Connections to hypergeometric functions of Gel'fand-Kapranov-Zelevinsky (GKZ) will also be considered. We will see that the theory is intimately related to a particular infinite dimensional representation of a reductive Lie algebra, and the topology of certain affine varieties. I will describe how the theory could help calculate period integrals, and offers new insights into the GKZ theory, and mirror symmetry for toric and flag varieties. This talk is based on joint works with S. Bloch, B. Lian, V. Srinivas, S-T. Yau, and X. Zhu.