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Qaasim Shafi: Quasimaps and accordions

22nd September, 2021

Quasimaps provide an alternate curve counting system to Gromov-Witten theory, which are related by wall-crossing formulae. Relative (or logarithmic) Gromov-Witten theory has proved useful for constructions in mirror symmetry, as well as for determining ordinary Gromov-Witten invariants via the degeneration formula. Different versions of this theory rely on various technologies, including expansions (or accordions) as well as logarithmic structures. I will discuss how to use a hybrid of these approaches to produce a proper moduli space parametrizing quasimaps relative a smooth divisor in any genus.

Andrew Macpherson: Why are correspondences ubiquitous?

21st September, 2021

Many of the algebraic structures we construct from geometric data are represented 'motivically', that is, their structure constants are obtained by pushing and pulling 'coefficients' (e.g. functions, sheaves) along diagrams like X ← W → Y. In this quasi-survey talk, I will explain how many of the convenient properties of the algebraic categories we like to work in (e.g. vector spaces, dg-categories) are already present in categories of correspondences themselves. This explains their frequent appearance in the study of universal homology theories.

Patience Ablett: Gorenstein curves in codimension 4

21st September, 2021

While Gorenstein codimension 3 varieties are well understood from Buchsbaum-Eisenbud's structure theorem, the picture is less clear for codimension 4. In this talk we describe some constructions of stable curves corresponding to the possible Betti tables for Artin Gorenstein algebras of regularity and codimension four, as outlined in a paper of Schenck, Stillman and Yuan. These constructions use techniques from liaison theory and the Tom and Jerry formats of Brown and Reid.

Liu Shengxuan: Stability condition on Calabi-Yau threefold of complete intersection of quadratic and quartic hypersurfaces

21st September, 2021

In this talk, I will first introduce the background of Bridgeland stability condition. Then I will mention some existence result of Bridgeland stability. Next I will prove the Bogomolov-Gieseker type inequality of X(2,4), Calabi-Yau threefold of complete intersection of quadratic and quartic hypersufaces, by proving the Clifford type inequality of the curve X(2,2,2,4). Then this will provide the existence of Bridgeland stability condition of X(2,4).

Alan Thompson: The mirror Clemens-Schmid sequence

20th September, 2021

I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a 'mirror P=W' conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting.

Paul Hacking: Mirror symmetry for ℚ-Fano 3-folds

20th September, 2021

This is a report on work in progress with my student Cristian Rodriguez. The mirror of a ℚ-Fano 3-fold with b2 = 1 is a rigid K3 fibration over ℙ1 such that Hodge bundle is degree 1 and some power of the monodromy at infinity is maximally unipotent. Although prior work focused on the maximally unipotent case (without base change), perhaps a classification of such Picard-Fuchs equations is possible.

In the smooth case these fibrations were described explicitly by Przyjalkowski, and Doran-Harder-Novoseltsev-Thompson showed that they are given by etale covers of the (1-dimensional) moduli of rank 19 K3 surfaces. In the case of a single 1/2(1,1,1) singularity they are given by rigid rational curves on the (2-dimensional) moduli of rank 18 K3 surfaces, and examples suggest they are Teichmuller curves in A2 (via the Shioda-Inose correspondence relating rank 18 K3s and abelian surfaces), as studied by McMullen.

Duco van Straten: A strange Calabi-Yau degeneration

20th September, 2021

If a Calabi-Yau threefold varies in a one-parameter family and aquires some double points, a small resolution will produce a rigid space. The local monodromy at such a 'conifold transition' is of infinite order. In the talk I report on some work done with S. Cynk (Krakow), which shows similar transitions to rigid Calabi-Yaus are possible with monodromy of finite order, in sharp distinction to what can happen for K3 surfaces.

Xinyi Yuan: A uniform Bogomolov type of theorem for curves over global fields

15th September, 2021

In the recent breakthrough on the uniform Mordell-Lang problem by Dimitrov-Gao-Habegger and Kuhne, their key result is a uniform Bogomolov type of theorem for curves over number fields. In this talk, we introduce a refinement and generalization of the uniform Bogomolov conjecture over global fields, as a consequence of bigness of some adelic line bundles in the setting of Arakelov geometry. The treatment is based on the new theory of adelic line bundles of Yuan-Zhang and the admissible pairing over curves of Zhang.