Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Calabi-Yau Manifolds & Mirror Symmetry Seminar

CYMS Seminar - Andrew Harder (Lehigh University)

Thursday, March 12th, 2020

Time: 2:40-4:00 p.m Place: Jeffery Hall 319

Speaker: Andrew Harder (Lehigh University)

Title: Calabi--Yau threefolds fibered K3 surfaces and their mirrors.

Abstract: Mirror symmetry predicts that, given a family of Calabi--Yau varieties, there is a mirror dual family of Calabi--Yau varieties, so that the algebraic aspects of the first are reflected by the symplectic aspects of the other, and vice versa. However, given a family of Calabi--Yau threefolds, it is not usually clear how its mirror family can be constructed. We address this problem for smooth Calabi--Yau threefolds that are built by smoothing degenerate Calabi--Yau threefolds made up of unions of pairs of quasi-Fano manifolds. This leads to a classification of a certain class of Calabi--Yau threefolds, and a surprising relationship to Ishkovskih's famous classification of smooth Fano threefolds of Picard rank 1. This is based on joint work with C. Doran, A. Novoseltsev, and A. Thompson.

CYMS Seminar - Jia-Wei Guo (National Taiwan University)

Thursday, February 6th, 2020

Time: 3:00 p.m Place: Jeffery Hall 319

Speaker: Jia-Wei Guo (National Taiwan University)

Title: Class number relations arising from intersections of Shimura curves and Humbert surfaces.

Abstract: By considering the intersections of Shimura curves and Humbert surfaces on the Siegel modular threefold, we obtain new class number relations. The result is a higher-dimensional analogue of the classical Hurwitz-Kronecker class number relation. This is a joint work with Yifan Yang.

CYMS Seminar - Oswaldo Sevilla

Thursday, November 14th, 2019

Time: 2:30 p.m Place: Jeffery Hall 319

Speaker: Oswaldo Sevilla (Fields Institute and Centro de Investigacion en Matematicas A.C.)

Title: Calabi Yau Threefolds arising from certain root lattices.

Abstract: I'll show my work on the construction of Calabi Yau threefolds that are constructed from the C_3 and C_4 root systems, using a construction by H. Verrill (Root lattices and pencils o f varieties, 1996) based on a paper of V. Batyrev (Dual Polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties,1994).

CYMS Seminar - Fenglong You (University of Alberta)

Thursday, October 17th, 2019

Time: 2:30 p.m Place: Jeffery Hall 319

Speaker: Fenglong You (University of Alberta)

Title: Relative Gromov--Witten theory and mirror symmetry

Abstract: Gromov--Witten theory is considered as the first modern approach in enumerative geometry. Absolute Gromov--Witten invariants provide virtual counts of curves in smooth projective varieties/orbifolds. It is known to have many nice structural properties, such as quantum cohomology, WDVV equation, Givental's formalism, mirror theorem, CohFT etc.. Relative Gromov--Witten invariants study the virtual counts of curves in varieties with tangency conditions along a divisor. In this talk, I will give an overview of some recent developments on parallel structures of relative Gromov--Witten theory. If time permits, I will also talk about some applications such as SYZ mirror symmetry and Doran--Harder--Thompson conjecture.

CYMS Seminar - Richard Gottesman (Queen's University)

Thursday, September 5th, 2019

Time: 2:30 p.m Place: Jeffery Hall 319

Speaker: Richard Gottesman (Queen's University)

Title: Vector-Valued Modular Forms and Modular Linear Differential Equations

Abstract: The sequence of denominators of the Fourier coefficients of a modular form on a congruence subgroup is always bounded. It has been conjectured that the converse is also true. We will consider this problem in the context of vector-valued modular forms and explain a strategy for proving such an unbounded denominator result. A key point is the importance of understanding the solutions of the modular linear differential equation at all of the cusps).

CYMS Seminar - Noriko Yui (Queen's University)

Thursday, October 18th, 2018

Time: 11:30 a.m - 12:20 p.m Place: Jeffery Hall 422

Speaker: Noriko Yui (Queen's University)

Title: Four-dimensional Galois representations arising from certain Calabi--Yau threefolds Part II

Abstract: We consider the (irreducible) four-dimensional Galois representations arising from certain Calabi--Yau threefolds over ${\bf{Q}}$ with all the Hodge numbers of the third cohomology groups equal to $1$. There are many examples of (families) of such Calabi--Yau threefolds. The modularity/automorphy of such Calabi--Yau threefolds will be the main topic of discussion. There are two venues to be considered. In one venue, we ought to count the number of rational points over finite fields of these Calabi--Yau threefolds to concoct their L-series. In the other venue, we ought to construct some modular varieties, in this case, conjecturally, Siegel modular forms of weight $3$ and genus $2$ on some paramodular subgroups of $Sp(4,{\bf{Z}})$, and then compute their L-series. Such modular forms may be constructed using Borcherds forms. The ultimate aim is to establish a Langlands correspondence between the two L-series, thereby establishing the modularity/automorphy of such Calabi--Yau threefolds.

This is a joint work with Yifan Yang (National Taiwan University).