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Autumn Semester 2016

Date / Time Speaker Title Location
16 September 2016
16:00-17:15 		Prof. Dr. Ben Bakker
University of Georgia
Details

Algebraic Geometry and Moduli Seminar

Title Mercat's conjecture and higher rank Brill-Noether divisors on M_g
Speaker, Affiliation Prof. Dr. Ben Bakker, University of Georgia
Date, Time 16 September 2016, 16:00-17:15
Location HG G 43
Abstract Classical Brill--Noether theory elegantly describes linear systems on generic curves; the problem of doing the same for sections of higher rank vector bundles has been extensively studied but the picture is far from complete. In joint work with G. Farkas, we prove a conjecture of Mercat for rank 2 bundles on generic curves classifying minimal slope bundles that admit a prescribed number of sections. As in the classical case, the Brill--Noether theoretic behavior of rank 2 bundles can be packaged into a "rank two Clifford index," but while the resulting stratification of M_g is different from the gonality stratification, we nonetheless show that the divisorial strata in both cases have slope 6+12/(g+1). The proof of the main theorem proceeds by specializing to curves on K3 surfaces, and time permitting we'll discuss the relationship with wall-crossing of Bridgeland stability conditions.
Mercat's conjecture and higher rank Brill-Noether divisors on M_gread_more
HG G 43
23 September 2016
16:00-17:15 		Dr. Gergely Berczi
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Curve counting via non-reductive GIT for graded groups
Speaker, Affiliation Dr. Gergely Berczi, ETH Zürich
Date, Time 23 September 2016, 16:00-17:15
Location HG G 43
Abstract I will start with a short report on recent progress in constructing quotients by actions of non-reductive algebraic groups and extending Mumford's geometric invariant theory to a wide class of non-reductive linear algebraic groups which we call graded groups. I will then explain how certain components of the Hilbert scheme of points on smooth varieties can be described as non-reductive quotients and why this description is especially efficient to study the topology of Hilbert schemes. In particular I will explain how equivariant localisation can be used to develop iterated residue formulae for tautological integrals on geometric subsets of Hilbert schemes and I present new closed formulae counting curves on surfaces (and more generally hypersurfaces in smooth varieties) with given singularity classes. This talk is based on joint work with Frances Kirwan and Andras Szenes.
Curve counting via non-reductive GIT for graded groupsread_more
HG G 43
28 September 2016
13:30-15:00 		Dr. Alexander Buryak
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Partition function of the double ramification hierarchy and the reduced Gromov-Witten potential
Speaker, Affiliation Dr. Alexander Buryak, ETH Zürich
Date, Time 28 September 2016, 13:30-15:00
Location HG G 43
Abstract Using the intersection numbers of a given cohomological field theory with the double ramification cycles and the top Chern class of the Hodge bundle, one can construct a system of PDEs of certain type. We call this system the double ramification (DR) hierarchy. There is a natural way to associate a partition function to the DR hierarchy. Conjecturally, this partition function is related to the partition function of the cohomological field theory by a certain elementary transformation. Remarkably, the conjecture is equivalent to a system of relations between the correlators of the cohomological field theory that seems to be new. The talk is based on joint works with B. Dubrovin, J. Guere and P. Rossi.
Partition function of the double ramification hierarchy and the reduced Gromov-Witten potentialread_more
HG G 43
30 September 2016
16:00-17:15 		Dr. Hyenho Lho
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Genus two quasimap invariants for local CP2
Speaker, Affiliation Dr. Hyenho Lho, ETH Zürich
Date, Time 30 September 2016, 16:00-17:15
Location HG G 43
Abstract We calculate the genus two quasimap(stable quotient) invariants for local P^2 and verify the result predicted by physicists. It provides the first mathematical proof that quasimap(stable quotient) invariants are exactly same with the topological string amplitudes in B-model side for genus 2. If time is allowed, we discuss the way to understand holomorphic anomaly equations in terms of quasimap(stable quotient) invariants. This is joint work in progress with Rahul Pandharipande.
Genus two quasimap invariants for local CP2read_more
HG G 43
7 October 2016
16:00-17:15 		Dr. Javier Fresan
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Exponential motives
Speaker, Affiliation Dr. Javier Fresan, ETH Zürich
Date, Time 7 October 2016, 16:00-17:15
Location HG G 43
Abstract Exponential periods form a class of complex numbers containing the special values of the gamma and the Bessel functions, the Euler constant and other interesting numbers which are not expected to be periods in the usual sense of algebraic geometry. However, we can interpret them as coefficients of the comparison isomorphism between two cohomology theories associated to varieties with a potential: the de Rham cohomology of a connection with irregular singularities and the so-called “rapid decay” cohomology. I will explain how this point of view allows one to construct a Tannakian category of exponential motives and to produce Galois groups which conjecturally govern all algebraic relations among these numbers. No prior knowledge of motives will be assumed, and I will focus on examples rather than on the more abstracts aspects of the theory. This is a joint work with Peter Jossen.
Exponential motivesread_more
HG G 43
12 October 2016
13:30-15:00 		Junliang Shen
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories I
Speaker, Affiliation Junliang Shen, ETH Zürich
Date, Time 12 October 2016, 13:30-15:00
Location HG G 43
Abstract By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. In the lecture series, I will explain a mathematical approach to prove (part of) the Huang-Katz-Klemm Conjecture. Our method is to use an involution on the derived category of the CY 3-fold X, and wall-crossing techniques developed by Toda. I will also discuss the connection to the Oberdieck-Pandharipande Conjecture, which concerns the enumerative geometry of the product of a K3 surface and an elliptic curve. For a fixed primitive curve class of genus h on K3, we prove that the generating series of curve counting invariants is a quasi-Jacobi form of index h-1 and weight -10. This is compatible with, and gives strong evidence for the OP Conjecture. The talks are based on joint work with Georg Oberdieck.
Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories Iread_more
HG G 43
14 October 2016
16:00-17:15 		Ignacio Barros
Humboldt Universität zu Berlin
Details

Algebraic Geometry and Moduli Seminar

Title Towards the birational classification of strata of holomorphic differentials
Speaker, Affiliation Ignacio Barros, Humboldt Universität zu Berlin
Date, Time 14 October 2016, 16:00-17:15
Location HG G 43
Abstract The Zariski closure of such spaces was recently understood and the task of giving a full birational classification is still in diapers. K3 surfaces come naturally into play when we want to construct rational curves through general points on this spaces. We use some Mukai models to prove unirationality of strata in small genus. The muse that serves us as role model is the space of odd theta characteristics, whose complete birational classification was successfully carried out by Farkas and Verra.
Towards the birational classification of strata of holomorphic differentials read_more
HG G 43
19 October 2016
13:30-15:00 		Junliang Shen
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories II
Speaker, Affiliation Junliang Shen, ETH Zürich
Date, Time 19 October 2016, 13:30-15:00
Location HG G 43
Abstract By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. In the lecture series, I will explain a mathematical approach to prove (part of) the Huang-Katz-Klemm Conjecture. Our method is to use an involution on the derived category of the CY 3-fold X, and wall-crossing techniques developed by Toda. I will also discuss the connection to the Oberdieck-Pandharipande Conjecture, which concerns the enumerative geometry of the product of a K3 surface and an elliptic curve. For a fixed primitive curve class of genus h on K3, we prove that the generating series of curve counting invariants is a quasi-Jacobi form of index h-1 and weight -10. This is compatible with, and gives strong evidence for the OP Conjecture. The talks are based on joint work with Georg Oberdieck.
Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories IIread_more
HG G 43
21 October 2016
16:00-17:15 		Johannes Schmitt
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Moduli spaces of twisted k-differentials
Speaker, Affiliation Johannes Schmitt, ETH Zürich
Date, Time 21 October 2016, 16:00-17:15
Location HG G 43
Abstract We present the definition of the moduli spaces of twisted k-differentials, which are closed substacks of \bar M_g,n constructed by Farkas and Pandharipande. On the open part M_g,n they are defined by the condition that a weighted sum of the marked points agrees, as a divisor class, with the k-th power of the canonical line bundle of the curve. We give an indication how to compute the dimension of their components and a conjectural expression of their (weighted) fundamental class in terms of a tautological cycle studied by Pixton.
Moduli spaces of twisted k-differentialsread_more
HG G 43
* 26 October 2016
14:00-15:00 		Junliang Shen
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories III
Speaker, Affiliation Junliang Shen, ETH Zürich
Date, Time 26 October 2016, 14:00-15:00
Location HG G 43
Abstract By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. In the lecture series, I will explain a mathematical approach to prove (part of) the Huang-Katz-Klemm Conjecture. Our method is to use an involution on the derived category of the CY 3-fold X, and wall-crossing techniques developed by Toda. I will also discuss the connection to the Oberdieck-Pandharipande Conjecture, which concerns the enumerative geometry of the product of a K3 surface and an elliptic curve. For a fixed primitive curve class of genus h on K3, we prove that the generating series of curve counting invariants is a quasi-Jacobi form of index h-1 and weight -10. This is compatible with, and gives strong evidence for the OP Conjecture. The talks are based on joint work with Georg Oberdieck.
Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories IIIread_more
HG G 43
28 October 2016
16:00-17:15 		Dr. Jérémy Guéré
Humboldt Universität, Berlin
Details

Algebraic Geometry and Moduli Seminar

Title Quantum K-theory for singularities
Speaker, Affiliation Dr. Jérémy Guéré, Humboldt Universität, Berlin
Date, Time 28 October 2016, 16:00-17:15
Location HG G 43
Abstract In 2007, a quantum theory for quasi-homogeneous polynomial singularities was developed by Fan, Jarvis, and Ruan. It should be seen as the counterpart of Gromov-Witten (GW) theory for hypersurfaces in weighted projective spaces via the so-called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence. In 2003, Lee established a K-theoretic version of GW theory. However, some aspects of GW theory, such as mirror symmetry, were still missing until last year. Indeed, Givental gave in 2015 a refined version called permutation equivariant GW K-theory, and proved some mirror symmetry statements in this new context, e.g. for the quintic hypersurface in P^4. In this talk, I will describe a joint work with Valentin Tonita and Yongbin Ruan, in which we define a K-theoretic version of the quantum singularity theory and we study its permutation equivariant part. I will focus on the quintic polynomial and explain how to prove mirror symmetry and the LG/CY correspondence.
Quantum K-theory for singularitiesread_more
HG G 43
2 November 2016
13:30-15:00 		Junliang Shen
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories IV
Speaker, Affiliation Junliang Shen, ETH Zürich
Date, Time 2 November 2016, 13:30-15:00
Location HG G 43
Abstract By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. In the lecture series, I will explain a mathematical approach to prove (part of) the Huang-Katz-Klemm Conjecture. Our method is to use an involution on the derived category of the CY 3-fold X, and wall-crossing techniques developed by Toda. I will also discuss the connection to the Oberdieck-Pandharipande Conjecture, which concerns the enumerative geometry of the product of a K3 surface and an elliptic curve. For a fixed primitive curve class of genus h on K3, we prove that the generating series of curve counting invariants is a quasi-Jacobi form of index h-1 and weight -10. This is compatible with, and gives strong evidence for the OP Conjecture. The talks are based on joint work with Georg Oberdieck.
Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories IVread_more
HG G 43
11 November 2016
16:00-17:15 		Dr. Emanuel Scheidegger
Universität Freiburg
Details

Algebraic Geometry and Moduli Seminar

Title Analytic continuation of hypergeometric functions in the resonant case and modularity at the conifold
Speaker, Affiliation Dr. Emanuel Scheidegger, Universität Freiburg
Date, Time 11 November 2016, 16:00-17:15
Location HG G 43
Abstract Hypergeometric functions of order 3 and 4 appear as hemisphere partition functions of the A-model of the quartic K3 surface and the quintic threefold and as periods of the B-model of the mirror quartic and the mirror quintic, respectively. We discuss their analytic continuation to the singular or conifold point and its conjectured relation to the L-series of the quintic at the conifold point determined by Schoen given in terms of certain modular forms.
Analytic continuation of hypergeometric functions in the resonant case and modularity at the conifoldread_more
HG G 43
23 November 2016
13:30-15:00 		Prof. Dr. Sabir Gusein-Zade
Moscow State University
Details

Algebraic Geometry and Moduli Seminar

Title Orbifold Milnor lattice, orbifold Seifert and intersection forms
Speaker, Affiliation Prof. Dr. Sabir Gusein-Zade, Moscow State University
Date, Time 23 November 2016, 13:30-15:00
Location HG G 43
Abstract For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group (an admissible one), H.Fan, T.Jarvis, and Y.Ruan defined the so-called quantum cohomology group. This group is defined in terms of the vanishing cohomology groups of Milnor fibres of restrictions of the function to fixed point sets of elements of the group. The quantum cohomology group is considered as the main object of the so called quantum singularity theory or FJRW-theory. Fan, Jarvis, and Ruan studied some structures on the quantum cohomology group which generalize similar structures in the usual singularity theory. An important role in singularity theory is played by such concepts as the (integral) Milnor lattice, the monodromy operator, the Seifert form and the intersection form. Analogues of these concepts have not yet been considered in the FJRW-theory. We define an orbifold version of the monodromy operator on the quantum (co)homology group and a lattice which is invariant with respect to the orbifold monodromy operator and is considered as an orbifold version of the Milnor lattice. The action of the orbifold monodromy operator on it can be considered as an analogue of the integral monodromy operator. Moreover, we define orbifold versions of the Seifert form and of the intersection form. The talk is based on a joint work with W. Ebeling.
Orbifold Milnor lattice, orbifold Seifert and intersection formsread_more
HG G 43
25 November 2016
16:00-17:15 		Prof. Dr. Sabir Gusein-Zade
Moscow State University
Details

Algebraic Geometry and Moduli Seminar

Title Power structures on Grothendieck ring of varieties and Macdonald type equations for generalized orbifold Euler characteristics
Speaker, Affiliation Prof. Dr. Sabir Gusein-Zade, Moscow State University
Date, Time 25 November 2016, 16:00-17:15
Location HG G 43
Abstract One has a simple formula for the generating series of the Euler characteristics of the symmetric powers of a space: it is equal to the series $(1-t)^{-1}=1+t+t^2+\ldots$ not depending on the space in the exponent equal to the Euler characteristic of the space itself (I. Macdonald). A Macdonald type equation is a formula for the generating series of the values of an invariant for symmetric powers of spaces or varieties (or for their analogues) which gives this series as a series not depending on the space in the exponent equal to the value of the invariant for the space itself. If the invariant takes values in a ring different from the ring of integers, an expression of this sort requires an interpretation. It is given by the so called power structure over the ring. The most important example is a geometric power structure over the Grothendieck ring of complex quasiprojective varieties. One has a number of extensions of the concept of the Euler characteristic to the equivariant setting (i.e. for spaces with actions of a finite group) with values in the Burnside ring of the group and also to the generalized one with values in the Grothendieck ring of complex quasiprojective varieties. Also there exist the concept of orbifold Euler characteristic and its extensions. We shall discuss power structures over some (Grothendieck) rings and Macdonald type equations formulated in terms of them. The talk is based on joint works with I. Luengo and A. Melle-Hernandez.
Power structures on Grothendieck ring of varieties and Macdonald type equations for generalized orbifold Euler characteristicsread_more
HG G 43
7 December 2016
13:30-15:00 		Dr. Ran Tessler
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Open intersection theory for P^1: the stationary sector I
Speaker, Affiliation Dr. Ran Tessler, ETH Zürich
Date, Time 7 December 2016, 13:30-15:00
Location HG G 43
Abstract I will describe a construction of genus 0 open GW and descendant theory for (P^1,RP^1), with an eye towards higher genus and other target pairs. If time permits the non stationary sector will be addressed. A joint work with Sasha Buryak, Rahul Pandharipande, and Amitai Zernik.
Open intersection theory for P^1: the stationary sector Iread_more
HG G 43
9 December 2016
16:00-17:15 		Dr. Kaloyan Slavov
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title An application of random plane slicing to counting F_q-points on hypersurfaces
Speaker, Affiliation Dr. Kaloyan Slavov, ETH Zürich
Date, Time 9 December 2016, 16:00-17:15
Location HG G 43
Abstract Let X be an absolutely irreducible hypersurface of degree d in A^n, defined over a finite field F_q. The Lang--Weil bound gives an interval that contains #X(F_q). We exhibit an explicit interval, which does not contain #X(F_q), and which overlaps with the Lang--Weil interval for a certain range of q. The proof is elementary and uses a probabilistic counting argument. The talk will be accessible to a broad audience.
An application of random plane slicing to counting F_q-points on hypersurfacesread_more
HG G 43
14 December 2016
14:00-15:30 		Prof. Dr. John Ottem
Univ. of Oslo
Details

Algebraic Geometry and Moduli Seminar

Title Nef higher codimension cycles
Speaker, Affiliation Prof. Dr. John Ottem, Univ. of Oslo
Date, Time 14 December 2016, 14:00-15:30
Location HG G 43
Abstract While the cones effective and nef divisors are fundamental objects in algebraic geometry, the corresponding cones for cycles of higher codimension have received much less attention and remain more mysterious. We survey some recent developments in this area, and in particular discuss the example of nef 2-cycles on the variety of lines on a cubic fourfold.
Nef higher codimension cyclesread_more
HG G 43
21 December 2016
13:30-15:00 		Dr. Ran Tessler
ETH Zurich
Details

Algebraic Geometry and Moduli Seminar

Title Open intersection theory for P^1: the stationary sector II
Speaker, Affiliation Dr. Ran Tessler, ETH Zurich
Date, Time 21 December 2016, 13:30-15:00
Location HG G 43
Abstract I will describe a construction of genus 0 open GW and descendant theory for (P^1,RP^1), with an eye towards higher genus and other target pairs. If time permits the non stationary sector will be addressed. A joint work with Sasha Buryak, Rahul Pandharipande, and Amitai Zernik.
Open intersection theory for P^1: the stationary sector IIread_more
HG G 43
13 January 2017
16:00-17:15 		Prof. Dr. Bumsig Kim
KIAS (Seoul)
Details

Algebraic Geometry and Moduli Seminar

Title Quasimap wall-crossings and Mirror Symmetry
Speaker, Affiliation Prof. Dr. Bumsig Kim, KIAS (Seoul)
Date, Time 13 January 2017, 16:00-17:15
Location HG G 43
Abstract We present a wall-crossing formula for the virtual classes of ε-stable quasimaps to GIT quotients and sketch its proof for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing formula relating the genus g descendant Gromov-Witten potential and the genus g epsilon-quasimap descendant potential is established. For the quintic threefold, our results may be interpreted as giving a rigorous and geometric interpretation of the holomorphic limit of the BCOV B-model partition function of the mirror family. This is a joint work with I. Ciocan-Fontanine.
Quasimap wall-crossings and Mirror Symmetryread_more
HG G 43

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