## Geometry and Physics Seminar

### 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 2 (2014)

• 13/1/2014 - Jose Gonzalez (UBC) @UBC: Topics on algebraic cobordism.

We will discuss some topics on the theory of algebraic cobordism from joint work with Kalle Karu. We will explain the computation via envelopes and the descent exact sequence. We review the relation of algebraic cobordism with Chow and K-theory. We will also discuss an extension of the cobordism rings of smooth varieties to the singular setting via a well-behaved operational bivariant theory. As an example, we describe the operational equivariant cobordism of toric varieties.

• 22/1/2014- Daniel Robbins (Amsterdam) @ UAlberta. Constraining higher derivative corrections with T-duality

From a target space perspective, T-duality relates certain pairs of string theory backgrounds with a U(1) isometry. If we perform a Kaluza-Klein reduction on the corresponding circle, T-duality then acts as a symmetry of the reduced theory, and this symmetry can be argued to constrain the higher derivative couplings, which in turn constrains the couplings of the higher dimensional theory. I will explain an unsophisticated brute force implementation of this procedure and show how it can be used to completely fix the four-derivative action of type II O-planes coupling to NS-NS sector bulk fields.

• 27/1/2014 - Ed Richmond (UBC) @UBC. The fibre bundle structure of smooth and rationally smooth Schubert varieties [PDF]

A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fi bre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showing that a Schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fi bre bundle of rationally smooth Grassmannian Schubert varieties. The proof depends on deep combinatorial results in Coxeter groups. This is joint work with William Slofstra.

• 29/1/2014 - Sara Filippini (Zurich and UAlberta) @UAlberta. Type equations and tropical curves.

We revisit the wall-crossing behaviour of solutions of a class of  thermodynamic Bethe Ansatz type integral equations, expressed as sums of ''instanton corrections''. We explain how a set of tropical curves (with signs) emerges naturally from each instanton correction, then show that the weighted sum over all such curves is in fact a tropical count. This goes through to the q-deformed setting. This construction can be regarded as a formal mirror-symmetric statement in the  framework proposed by Gaiotto, Moore and Neitzke. Joint work with J. Stoppa.

• 3/2/2014 - Ben Williams (UBC) @UBC: The Higher Chow Groups of GLn.

If X is a variety over a field, the Chow groups of X are defined in terms of closed subvarieties of X and form a kind of cohomology theory for X. Higher Chow groups, defined by Spencer Bloch in the 1980s in terms of subvarieties on X x A^m and now related to the theory of motivic cohomology, extend the theory of Chow groups, and may be nonzero even when the ordinary Chow groups vanish. I will explain how the higher Chow groups of GLn are related to a 'suspension' of the ordinary Chow groups of projective space. Time permitting, I will conjecture how this relationship might extend to 'suspensions' of the Chow groups of Grassmanians.

• 5/2/2014 - Terry Gannon (UAlberta) @UAlberta. P-curvature and modular forms.

(Vector-valued) modular forms appear all over string theory. Often, one wants to know whether the Fourier coefficients are integers, or if it is already known they must be integral, one would like to know consequences. In my talk I'll describe the most effective tool for these sorts of questions: p-curvature.

• 26/2/2014 - Daniel Halpern-Leistner (Columbia) @UBC. Instability in algebraic geometry

In order to construct moduli spaces in algebraic geometry, one typically must specify a notion of semi-stability for the objects one wishes to parameterize. To the objects that are omitted, the unstable objects, one can often associate a real number which measures "how unstable" that object is. In fact we can think of the moduli stack of all objects as stratified by locally closed substacks corresponding to objects of varying degrees of instability. The key examples of this phenomenon are the Kempf-Ness stratification of the unstable locus in GIT and the Shatz stratification of the moduli of G-bundles on a smooth projective curve. I will discuss a framework for describing stability conditions and stratifications of an arbitrary algebraic stack which provide a common generalization of these examples. Time permitting, I will discuss how some commonly studied moduli problems, such as the moduli of K-stable varieties and the moduli of Bridgeland-semistable complexes on a smooth projective variety, fit into this framework. One key construction assigns to any point in an algebraic stack a potentially large topological space parameterizing all possible `iso-trivial degenerations' of that point. When the stack is BG for a reductive G, this recovers the spherical building of G, and when the stack is X/T for a toric variety X, this recovers the support of the fan of X. NOTE: unconventional day and time: 1:30-2:30pm. Location as usual.

• 26/2/2014 - Peter Overholser (UAlberta) @UAlberta. An introduction to the Gross-Siebert program (Part I).

I will attempt to give an overview of the Gross-Siebert program, emphasizing its guiding principles and their connection to the Strominger-Yau-Zaslow conjecture.

• 3/3/2014 - Jim Carrell (UBC) @UBC. Surjectivity and lifting the Weyl group action to the equivariant cohomology of a Springer fibre.

A famous result of Springer says that the Weyl group of a reductive algebraic group G (over C) acts on the cohomology of the subvariety X_u of the flag variety G/B consisting of the flags fixed by a unipotent u in G. This result was unexpected since W does not act on X_u itself. Recently, Kumar - Procesi and Goresky - MacPherson showed that Springer's action lifts to the equivariant cohomology of X_u with respect to the maximal torus in C_G(u) for so called parabolic unipotents u with the proviso that the cohomology morphism j*: H*(G/B) \to H*(X_u) is surjective. In this talk we will describe the parabolic unipotents for which j* is surjective and indicate a direct proof of lifting.

• 5/3/2014 - Peter Overholser (UAlberta) @UAlberta. An introduction to the Gross-Siebert program (Part II).

Continuation of last weeks talk.

• 10/3/2014 - Andrei Caldararu (Wisconsin) @UBC. The de Rham complex from the point of view of twisted derived intersections.

I shall present results of work with Arinkin and Hablicsek which allow us to understand the Frobenius push-forward of the de Rham complex as the structure sheaf of a twisted derived intersection. Similar considerations also apply for twisted de Rham complexes, yielding results which have applications in singularity theory and in the study of matrix factorizations. Using our theorems we recover and strengthen earlier results of Deligne-Illusie, Barannikov-Kontsevich, and Sabbah. Our approach gives a new point of view on recent works of Joyce et al, generalizing results of Behrend, on understanding the holomorphic Fukaya-Floer homology.

• 12/3/2014 - James Lewis (UAlberta) @UAlberta. A Variation of the Beilinson-Hodge Conjecture.

Based on some recent joint work of J. Lewis, and others, we formulate a variation of the Beilinson-Hodge conjecture pertaining to varieties defined over the complex numbers. In this talk, we explain the motivation for this conjecture, and some evidence in support of it.

• 17/3/2014 - Jun Li (Stanford) @UBC. Categorified Donaldson-Thomas invariants for sheaves.

We will present a joint work with Young-Hoon Kiem on using family Chern-Simons charts to construct perverse sheaves on moduli of sheaves that gives a categorification of Donaldson-Thomas invariants.

• 19/3/2014 - Tyler Kelly (UPenn) @UAlberta. Towards Unifying Toric Mirror Constructions
.

I will discuss the recent work(-in-progress) on unifying various mirror constructions of various authors, such as Batyrev-Borisov and Berglund-Hübsch-Krawitz. This talk hopes to focus on questions, conjectures, and examples involved in this more generalized framework. This talk hopes not to focus on the difficulty of using a SmartBoard for seminars.

• 24/3/2014 - Emanuele Macri (Ohio State University) @UBC. Curves on Irreducible Holomorphic Symplectic Varieties.

The goal of the talk is to present derived category techniques to study holomorphic symplectic varieties. In particular, we study and answer the following questions:

(1) the Hassett-Tschinkel Conjecture on the structure of the Mori cone of curves;
(2) the Bogomolov-Tyurin-Hassett-Tschinkel-Huybrechts-Sawon Conjecture on the existence of Lagrangian fibrations;
(3) the Kawamata-Morrison Cone Conjecture.

Irreducible Holomorphic Symplectic varieties (IHS for short) are simply connected projective manifolds endowed with a unique (up to scalars) holomorphic symplectic form; K3 surfaces are the lowest dimensional example. In this talk we concentrate on IHS of K3^[n]-type, namely IHS deformation equivalent to the punctual Hilbert scheme on a K3 surface. After giving a short introduction to the basics of IHS theory, we will present recent joint work with Arend Bayer on how to prove (1), (2), and (3) for moduli spaces of sheaves on K3 surfaces, by using derived categories and Bridgeland stability. If time permits, I will also sketch how to extend these results to all IHS of K3^[n]-type, as recently proven by Bayer-Hassett-Tschinkel, Mongardi, Matsushita, Markman-Yoshioka, and Amerik-Verbitsky.

4/4/2014 - Xiaowei Wang (Rutgers, Newark) @UAlberta. Stability and compactification of moduli.In order to construct the moduli space of canonical polarized manifolds, three different stability conditions have been introduced, namely, KSBA-stability, K-stabilty and asymptotic GIT stability. In this talk, we try to explore the relations among them. In particular, any canonical polarized manifold is stable with respect to all three conditions above, however the compactifications they give are different. As a consequence, we answer a longstanding question by showing that asymptotically GIT Chow semistable varieties do not form a proper family. NOTE: Note the unusual date of the seminar.

• 7/4/2014 - Mathieu Huruguen (PIMS/UBC) @UBC. Special reductive groups over an arbitrary field.

A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In a modern language, it can be shown that the special groups are those of essential dimension zero. In 1958 Grothendieck classified special groups in the case where the base field k is algebraically closed. In this talk I will explain some recent progress towards the classification of special reductive groups over an arbitrary field. In particular, I will give the classification of special semisimple groups, special reductive groups of inner type and special quasisplit reductive groups over an arbitrary field k. NOTE: Note the unusual time of the seminar: 1:50-2:50pm. This is in order to avoid a clash with the PIMS-CRM Fields lecture at 3pm.

• ### Term 1 (2013)

• 9/9/2013 - Hsian Hua Tseng (Ohio State University) @ UAlberta: Counting disks in toric varieties

• For a toric manifold X and a Lagrangian torus fiber L in X, Fukaya-Oh-Ohta-Ono defined open Gromov-Witten invariants which are virtual enumerations of holomorphic disks in X with boundary conditions in L. Qualitative and quantitative properties of these open Gromov-Witten invariants play important roles in the symplectic geometry and mirror symmetry of X. Open Gromov-Witten invariants are difficult to compute because standard methods in Gromov-Witten theory (such as virtual localization) are not applicable. In this talk we explain a geometric method that leads to a complete calculation of these open Gromov-Witten invariants for compact semi-Fano toric manifolds. This is joint work with K. Chan, S.-C. Lau, N. C. Leung.

• 23/9/2013 - Sabin Cautis (UBC) @ UBC: Categorical Heisenberg actions on Hilbert schemes of points [PDF]
• We define actions of certain Heisenberg algebras on the Hilbert schemes of points on ALE spaces. This lifts constructions of Nakajima and Grojnowski from cohomology to K-theory and derived categories of coherent sheaves. This action can be used to define Lie algebra actions (using categorical vertex operators) and subsequently braid group actions and knot invariants.

• 25/9/2013 - David Favero (UAlberta) @ UAlberta: Homological projective duality via variation of geometric invariant theory quotients (I)
• I will briefly Homological Projective Duality, an analog of projective duality in the setting of derived categories due to Kuznetsov. I will explain the relationship between this duality, GLSMs, and variation of geometric invariant theory quotients as part of joint work with M. Ballard, D. Deliu, U. Isik, and L. Katzarkov.

• 30/9/2013 - David Favero (UAlberta) @ UAlberta: Homological projective duality via variation of geometric invariant theory quotients (II)
• Continuation of talk on Sep. 25.

• 2/10/2013 - Simon Wood (IPMU) @ UAlberta: On the extended W-algebra of type sl_2 at positive rational level
• The extended W-algebra of type sl_2 at positive rational level is a vertex operator algebra that is of great interest in logarithmic conformal field theory. In this talk I will give an overview of how it is constructed as a subvertex operator algebra of a lattice vertex operator algebra by means of so called screening operators. I will also explain how the screening operator formalism allows one to prove c_2 cofiniteness, compute relations in Zhu's algebra and classify all simple modules of the extended W-algebra of type sl_2 at positive rational level.

• 7/10/2013 - Martijn Kool (UBC/PIMS) @ UBC: Curves on surfaces [PDF]
• The Hilbert scheme of curves in class \beta on a smooth projective surface S carries a natural virtual cycle. In many cases this cycle is zero (often when S has a holomorphic 2-form and \beta is not sub-canonical). However, in these cases one can often remove part of the obstruction bundle and obtain a non-trivial reduced virtual cycle. Both cycles have interesting applications. (1) Both are related to Pandharipande-Thomas' stable pair invariants on the total space of the canonical bundle over S. (2) The reduced virtual cycle is related to Severi degrees and classical curve counting on S. (3) The non-reduced virtual cycle is related to the Seiberg-Witten invariants of S (by work of Duerr-Kabanov-Okonek and Chang-Kiem).

• 28/10/2013 - Amin Gholampour (Maryland) @ UBC: Stable pair theory of K3 fibrations [PDF]
• I will talk about my joint work with A. Sheshmani and Y. Toda. We study the stable pair theory of K3 fibrations over curves with possibly nodal fibers. We express the stable pair invariants of the fiberwise irreducible classes in terms of the famous Kawai-Yoshioka formula for the Euler characteristics of moduli space of stable pairs on K3 surfaces and Noether-Lefschetz numbers of the fibration. In the case that the K3 fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants of the fiberwise curve classes in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.

• 30/10/2013 - Francois Gay-Balmaz (ENS) @ UAlberta: Integrable PDEs on semisimple Lie algebras
• Motivated partly by previous work on the zero curvature representation (ZCR) of completely integrable chiral models and partly by the underlying Hamiltonian structures of ideal complex fluids, we derive systems of partial differential equations, called G-strands, that admit a quadratic zero curvature representation for an arbitrary real semisimple Lie algebra. Using the root space decomposition, the G-strand equations can be formulated explicitly for the compact real form and the normal real form of any semisimple Lie algebra. We present several particular examples, including the exceptional group G_2. We also determine the general form of Hamilton's principles and Hamiltonians for these systems, and analyze the linear stability of their equilibrium solutions.

• 4/11/2013 - Paul Johnson (Colorado State) @ UBC: Topology and combinatorics of Hilbert schemes of points on orbifolds [PDF]
• The Hilbert scheme of n points on C^2 is a smooth manifold of dimension 2n. The topology and geometry of Hilbert schemes have important connections to physics, representation theory, and combinatorics. Hilbert schemes of points on C^2/G, for G a finite group, are also smooth, and their topology is encoded in the combinatorics of partitions. When G is a subgroup of SL_2, the topology and combinatorics of the situation are well understood, but much less is known for general G. After outlining the well-understood situation, I will discuss some conjectures in the general case, and a combinatorial proof that their homology stabilizes.

• 6/11/2013 - Alan Thompson (San Diego) @ UAlberta: Families of lattice polarized K3 surfaces with monodromy
• The concept of lattice polarization for a K3 surface was first introduced by Nikulin. I will discuss ways in which his definition can be extended to families of K3 surfaces over a (not necessarily simply connected) base curve, with the aim of gaining control over the action of monodromy upon the Néron-Severi lattice of a general fibre. I will then present an application of this to the study of Calabi-Yau threefolds that admit fibrations by Kummer surfaces.

• 18/11/2013 - Dragos Oprea (San Diego) @ UBC: The Chern classes of the Verline bundle
• The Verlinde bundles over the moduli space M_g of smooth curves have as fibers spaces of generalized theta functions i.e., spaces of global sections of determinant line bundles over moduli of parabolic bundles. I will discuss a formula for the Chern classes of the Verlinde bundles, as well as extensions over the compactification \overline M_g.

• 27/11/2013 - Stefan Mendez-Diez (UAlberta) @ UAlberta: Geometrization of N-Extended 1-Dimensional Supersymmetry Algebras
• The problem of classifying off-shell representations of the $N$-extended one-dimensional super Poincar\'{e} algebra is closely related to the study of a class of decorated graphs known as Adinkras. We will discuss how these combinatorial objects possess a form of emergent supergeometry: Adinkras are equivalent to very special super Riemann surfaces with divisors. The method of proof critically involves Grothendieck's theory of "dessins d'enfants'', work of Cimasoni-Reshetikhin expressing spin structures on Riemann surfaces via dimer models, and an observation of Donagi-Witten on parabolic structure from ramified coverings of super Riemann surfaces.

• 27/11/2013 - Michel van Garrel (Fields) @ UAlberta: Integrality of relative BPS state counts of toric Del Pezzo surfaces
• This is joint work with Tony Wong and Gjergji Zaimi. Relative BPS state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert and conjectured to be integers. For toric Del Pezzo surfaces, a proof of this conjecture will be presented.

• 2/12/2013 - Atsushi Kanazawa (UBC) @ UBC: BCOV theory and Calabi-Yau 3-folds with two large complex structure limits.
• In their famous paper in 1994, Bershadsky, Cecotti, Ooguri and Vafa derived a set of equations called the BCOV holomorphic anomaly equations. The BCOV theory presents a generalization of the classical g=0 mirror symmetry (Hodge theory) and is capable of computing higher genus Gromov-Witten invariants. The key ingredient is the special Kaehler geometry of the moduli space of Calabi-Yau 3-folds. In this talk, I will explain the basic idea of the BCOV theory and show some interesting computations.

• 9/12/2013 - Ziv Ran (Riverside) @ UBC: Lagrangian-like submanifolds in Sympletic-like Poisson manifolds
• The deformation theory of holomorphic symplectic manifolds and their Lagrangian submanifolds is well known to be well behaved, thanks to the influence of Hodge theory. We will report on recent work extending some of these results to certain Poisson manifolds and their Lagrangian-like submanifolds, using mixed Hodge theory.

• 16/12/2013 -  Peter Overholser (Fields) @ UAlberta: A tropical descendent Landau-Ginzburg potential.

I will review Gross's tropical description of mirror symmetry for P2 and explain how its structures can be expanded to yield new tropical invariants.
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