"Classification, Computation, and Construction,
New Methods in Geometry"
at Math Inst., Univ. of Warwick,
Mon 15th-Wed 17th Oct 2018.
Please register at
https://mrc.warwick.ac.uk/events.php
The workshop poster is available at
http://homepages.warwick.ac.uk/~masda/3CinG_workshop_poster.pdf
Confirmed speakers:
Ali Craw (Bath)
Stephen Coughlan (Bayreuth)
Tom Ducat (Heilbronn)
Michel van Garrel (Warwick)
Mark Gross (DPMMS)
Paul Hacking (UMass)
Fabian Haiden (Harvard)
Al Kasprzyk (Nottingham)
Travis Mandel (Edinburgh)
Dhruv Ranganathan (MIT)
Miles Reid (Warwick)
Martin Ulirsch (Frankfurt and Warwick)
Ziquan ZHUANG (Princeton)
Also participating
Hamid Ahmadinezhad, Christian Boehning, Gavin Brown, Tom Coates, Alessio
Corti, Alessio D'Ali, Enrico Fatighenti, Sara Filippini, Isac Heden,
Liana Heuberger, Elena Kalashnikov, Imran Qureshi, LI Chunyi, Tim
Logvinenko, Diane Maclagan, Nabijou Navid, Thomas Prince, Victor
Przyjalkowski, Greg Sankaran, Alan Thompson, ZHANG Weiyi
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Timetable (subject to corrections)
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Mon 15th Oct
10:30-11:30 in MS.05 Fabian Haiden (Harvard)
lunch break
2:00-3:00 in B3.02 Martin Ulirsch (Frankfurt)
tea or coffee break
3:30-4:30 in B3.02 Travis Mandel (Edinburgh)
5:00-6:00 in B3.02 Ali Craw (Bath)
dinner
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Tue 16th Oct
9:00-10:00 in B3.02 Al Kasprzyk (Nottingham)
coffee break
10:15-11:45 Discussion: Open Problems, Future Aims (in common room)
12:00-13:00 in B3.03 Tom Ducat (Heilbronn Inst., Bristol)
lunch break
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Tue 16th Oct Joint meeting with COW
2:00-3:00 in MS.05 Ziquan Zhuang (Princeton)
3:00-4:00 in MS.04 Paul Hacking (UMass, AMherst)
tea or coffee break
4:30-5:30 in B3.03 Dhruv Ranganathan (MIT), Logarithmic Gromov-Witten
theory with expansions
5:45 in B1.12 3CinG Steering committee meeting
Workshop dinner at 7:30 in Arden House
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Wed 17th Oct
9:00-10:00 in B3.02 Stephen Coughlan (Bayreuth)
tea or coffee break
11:00-12:00 in B3.02 Miles Reid (Warwick)
12:00-13:00 lunch break
2:00-3:00 pm in MS.03 Michel van Garrel (Warwick)
coffee break
4:30-5:30 pm in MS.03 Mark Gross (DPMMS)
final talk end Wed 17th Oct 5:30 pm
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Titles
Stephen Coughlan: On explicit constructions of canonical 3-folds
Ali Craw: Birational geometry of symplectic quotient singularities
Tom Ducat: Constructing Fano 3-folds from cluster varieties
Michel van Garrel: Genus 0 log Gromov-Witten invariants of maximal tangency
Mark Gross: Canonical scattering diagrams
Paul Hacking: Mirror symmetry for Fano varieties
Fabien Haiden: Categorical Kaehler geometry
Al Kaspzryk: Quiver mutations from the view of Fanosearch
Travis Mandel: Towards log quantum cohomology
Dhruv Ranganathan: Logarithmic Gromov—Witten theory with expansions
Miles Reid: Some explicit deformations
Martin Ulirsch: From algebraic to tropical divisors (and back again)
Ziquan ZHUANG: Birational superrigidity and K-stability
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Abstracts
Stephen Coughlan (Bayreuth), On explicit constructions of canonical 3-folds
Abstract: I will discuss methods for constructing new canonical 3-folds from old ones using birational geometry and degeneration techniques. These methods are particularly well-suited to studying questions concerning the geography of invariants of 3-folds. Many of the new constructions can be expressed as hypersurfaces inside toric varieties.
Ali Craw (Bath), Birational geometry of symplectic quotient singularities
Abstract: For a finite subgroup G of SL(2,C) and for n>1, I'll describe the movable cone of the Hilbert scheme of n points on the minimal resolution of the Kleinian singularity C^2/G. The key is to show that every projective, crepant resolution of C^{2n}/G_n (here, G_n is the wreath product of G with S_n) is a fine moduli space of \theta-stable modules over the framed preprojective algebra of G, where \theta is a choice of generic stability condition. This is joint work with Gwyn Bellamy.
Tom Ducat (Heilbronn), Constructing Fano 3-folds from cluster varieties
Abstract: Cluster varieties give rise to a large class of affine Gorenstein varieties equipped with a big torus action and a lot of symmetry. This makes them ideal candidates to use as key varieties. Concentrating on three rank 2 cluster varieties of finite type, I will explain how they can be used to construct new cases of Q-Fano 3-folds (amongst other interesting varieties). This is joint work with Stephen Coughlan (U. of Bayreuth).
Michel van Garrel (Warwick), Genus 0 log Gromov-Witten invariants of maximal tangency
Abstract: Let X be a smooth Fano variety and let D be a smooth anticanonical divisor on it. The curve counts of the title are an algebraic version of counts of holomorphic disks with boundary in a special Lagragian torus fiber near D. The works of Gross-Siebert, Gross-Hacking-Keel and Carl-Pumperla-Siebert show the centrality of such counts to construct the mirror of (X,D). Compared to the maximal boundary case, for D smooth many more open questions remain. After sketching the definition of these counts, I will survey joint work with Graber-Ruddat and Choi-Katz-Takahashi that show how these invariants interact with mirror symmetry.
Paul Hacking (UMass), Mirror symmetry for Fano varieties
Abstract: The mirror of a Fano n-fold is a family of Calabi--Yau (n-1)-folds over the affine line with maximally unipotent monodromy at infinity. We describe this mirror correspondence in terms of birational geometry, deformation theory, and Hodge theory. This is joint work with Corti and Petracci, and builds on work of Coates, Corti, Galkin, Golyshev, and Kasprzyk.
Fabien Haiden (Harvard), Categorical Kaehler Geometry
Abstract: I will report on an ongoing joint project with Katzarkov, Kontsevich, and Pandit to develop a notion of Kaehler geometry for abelian and derived categories. There are both Archimedean and non-Archimedean variants of the theory, which is motivated by a variety of examples from geometry and representation theory, and is closely related to algebro-geometric notions of stability, e.g. Bridgeland's theory of stability conditions on triangulated categories.
Al Kaspzryk (Nottingham), Quiver mutations from the view of Fanosearch
Abstract: It is well known that to every toric del Pezzo surface X we can associate a quiver Q, and that mutations of X give mutations of Q. Can we use the quiver to determine information about X? What can we say if we first with an arbitrary quiver? This is joint work with Thomas Prince and Ketil Tveiten.
Travis Mandel (Edinburgh), Towards log quantum cohomology
Abstract: I will discuss a conjectural notion of a log quantum cohomology ring, which should extend to log varieties the usual theory of small quantum cohomology. For log CY's with maximal boundary, the degree 0 part of the conjectural construction recovers the Frobenius structure conjecture of Gross-Hacking-Keel. I will outline a proof of the Frobenius structure conjecture for cluster varieties. As an application, I will then explain how this gives the equivalence between classical and quantum periods of Fano varieties.
Dhruv Ranganathan (MIT), Logarithmic Gromov-Witten theory with expansions
Abstract: Logarithmic Gromov-Witten theory provides a virtually smooth compactification of the space of maps from curves to a normal crossings degeneration. It allows one to decompose Gromov-Witten invariants as a sum over tropical curves, each counted with a "virtual multiplicity". A major goal of this subject is to write these virtual multiplicities as a product over vertex contributions, i.e. the gluing formula. To this end, I will describe an inverse system of virtual birational, proper models of this space, each of which parameterize maps to expansions of the target satisfying a strong transversality property. The transversality property is particularly suitable to proving a gluing formula, and opens the door to virtual localization in this setting.
Miles Reid (Warwick), Some explicit deformations
Abstract: I outline a graded ring approach to constructing QQ-Gorenstein deformation families related to "singularity content" and "mutation" of polygons. For example, the weighted projective planes PP(a^2,b^2,c^2) for (a,b,c) a Markov trip are known to smooth to PP^2 by work of Manetti and Hacking. Nowadays, this interpreted in terms of mirror symmetry, but my point of view is confined to the B side. Based on experimental cases, I believe that the graded ring corresponding to the (abc)th Veronese truncation v_abc(PP(a^2,b^2,c^2)) (its 1/3-anticanonical ring) has a k-dimensional unobstructed smoothing that successively cancels the mutations of the Markov spectrum. The calculations are similar in feel to the easier cases of diptych varieties.
Martin Ulirsch (Frankfurt and Warwick), From algebraic to tropical divisors (and back again)
Abstract: Tropical geometry is a framework to formalize many of the combinatorial gadgets associated to degenerations and compactifications of algebraic varieties. Its main new insight is to think about these as objects of geometric inquiry themselves and to study their properties by what is true in algebraic geometry. This talk will give an introduction to tropical geometry guided by the analogy between algebraic and tropical curves and, in particular, their respective divisor theories.
Its main topic will be the process of tropicalization, i.e. the process of going from a divisor on an algebraic curve to a divisor on an associated tropical curve, and on the so-called realizability problem, i.e. the problem of deciding which tropical divisor are tropicalization of algebraic divisors of the same rank and multiplicity profile. In particular, I will outline a complete solution to this problem for effective canonical divisors and outline a partial solution for principal divisors.
This talk is based on joint work with Madeline Brandt, Bo Lin, Martin Moeller, Annette Werner, and Dmitry Zakharov.
Ziquan ZHUANG (Princeton), Birational superrigidity and K-stability
Abstract: Birational superrigidity and K-stability are properties of Fano varieties that have many interesting geometric implications. For instance, birational superrigidity implies non-rationality and K-stability is related to the existence of Kähler-Einstein metrics. Nonetheless, both properties are hard to verify in general. In this talk, I will first explain how to relate birational superrigidity to K-stability using alpha invariants; I will then outline a method of proving birational superrigidity that works quite well with most families of index one Fano complete intersections and thereby also verify their K-stability. This is partly based on joint work with Charlie Stibitz and Yuchen Liu.