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Organizing Committee
UFRO Universidad de La Frontera
University of Oxford
Università degli Studi di Napoli 'Federico II'
University of Pennsylvania and Università di Bologna
University of Pennsylvania
Carleton University
Abstract
Mathematics is crucial in developing the models employed in physics, and observations from physics can catalyze new branches of mathematics. Mirror symmetry is a classic example of this two-way interaction.
In the context of K3 surfaces, mirror symmetry is a relation between families whose Picard lattices are complementary in the even unimodular lattice of signature (2,18). Not every family of K3 surfaces has a mirror, but this construction has been very useful in studying K3 surfaces that are toric hypersurfaces, for example.
Physicists have studied K3 surfaces, elliptic fibrations, and mirror symmetry on them in connection with string theory; this provides a concrete setting that can lead to the discovery of new properties of Calabi-Yau threefolds. The study of K3 surfaces in families naturally leads to Calabi-Yau threefolds and manifolds of higher dimension. In recent years, several connections have been established between these varieties, Feynman integrals, and Φ4 theory.
The aim of this workshop is to bring together experts from different areas of mathematics and physics in order to foster a fruitful exchange of knowledge and achieve new insights.
In the context of K3 surfaces, mirror symmetry is a relation between families whose Picard lattices are complementary in the even unimodular lattice of signature (2,18). Not every family of K3 surfaces has a mirror, but this construction has been very useful in studying K3 surfaces that are toric hypersurfaces, for example.
Physicists have studied K3 surfaces, elliptic fibrations, and mirror symmetry on them in connection with string theory; this provides a concrete setting that can lead to the discovery of new properties of Calabi-Yau threefolds. The study of K3 surfaces in families naturally leads to Calabi-Yau threefolds and manifolds of higher dimension. In recent years, several connections have been established between these varieties, Feynman integrals, and Φ4 theory.
The aim of this workshop is to bring together experts from different areas of mathematics and physics in order to foster a fruitful exchange of knowledge and achieve new insights.
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Workshop Schedule
Monday, September 28, 2026
8:30 - 8:50 AM
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM
Beyond Calabi-Yau Threefolds
11th Floor Lecture Hall
Virtual Speaker
David Morrison, University of California, Santa Barbara
Session Chair
Adam Logan, Carleton University
Abstract
The Calabi-Yau theorem gives a parameter space for Ricci-flat metrics on a (differentiable) Calabi-Yau Threefold X: namely, The bundle of Kähler cones over the moduli space of complex structures on X. In physics, this parameter space is extended by taking the product with H^2(X,R)/H^2(X,Z), thereby ``complexifying'' the Kähler cones.
Mirror symmetry should exchange the roles of complex structure and complexified Kähler class in the physical theory. Since those two parts of the parameter space do not have the same form, if mirror symmetry is to hold globally and not just locally, there must be deformations of the physical theories which go beyond Calabi-Yau Threefolds, and there must be some sub-structure of the complex structure moduli space which reflects the extension of the mirror complexified Kähler space.
We will survey what is known about this question.
10:00 - 10:30 AM
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM
Why do Feynman integrals know Calabi-Yau geometry?
11th Floor Lecture Hall
Speaker
Charles Doran, University of Alberta
Session Chair
Adam Logan, Carleton University
Abstract
Feynman integrals provide a rich source of Calabi-Yau geometry, but the geometric origin of these structures is often obscure. I will describe a simple point of view that begins with two-dimensional examples and leads naturally to moduli problems for vectors and polyhedra in three dimensions. I will discuss recent results for tetrahedral moduli and some new phenomena for pentahedral geometry that point toward a broader geometric framework for the Calabi-Yau structures appearing in Feynman integrals.
11:30 AM - 12:15 PM
Hypergeometric Decompositions of K3 Surface Pencils
11th Floor Lecture Hall
Speaker
Adriana Salerno, American Mathematical Society
Session Chair
Adam Logan, Carleton University
Abstract
In this talk we will look at five pencils of projective quartic Delsarte K3 surfaces. Over finite fields, we give explicit formulas for the point counts for each family, written in terms of hypergeometric sums. Over the complex numbers, we match the periods of the corresponding family with hypergeometric differential operators and series. We also obtain a decomposition of the incomplete L-function of each pencil in terms of hypergeometric L-series and Dedekind zeta functions. This gives an explicit description of the hypergeometric motives geometrically realized by each pencil.
This is joint work with Rachel Davis, Jessamyn Dukes, Thais Gomes Ribeiro, Eli Orvis, Leah Sturman, and Ursula Whitcher.
12:30 - 2:30 PM
Lunch/Free Time
2:30 - 3:15 PM
Calabi-Yau varieties as A-hypergeometric motives
11th Floor Lecture Hall
Speaker
Ursula Whitcher, Mathematical Reviews (American Mathematical Society)
Session Chair
Adam Logan, Carleton University
Abstract
The periods of Calabi-Yau varieties realized as hypersurfaces in Gorenstein Fano toric varieties satisfy GKZ A-hypergeometric equations. We show that the same A-hypergeometric data yields point counts on Calabi-Yau varieties over finite fields, using a notion of finite field A-hypergeometric sums defined by Adolphson. This talk describes joint work with Thais Gomes Ribeiro, Tyler Kelly, Adriana Salerno, and John Voight.
3:30 - 4:00 PM
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM
Complex structures on S^6
11th Floor Lecture Hall
Speaker
Philip Engel, University of Illinois, Chicago
Session Chair
Adam Logan, Carleton University
Abstract
A recent manuscript written by Claude under the direction of Levent Alpoge constructs a 1-parameter family of compact complex threefolds diffeomorphic to S^6. I will describe the main geometric ideas behind the construction and outline the proof. If time permits, I will remark on possible future research directions.
5:00 - 6:30 PM
Reception
11th Floor Collaborative Space
Tuesday, September 29, 2026
9:00 - 9:45 AM
From Conway Moonshine to K3: Non-Invertible Symmetries
11th Floor Lecture Hall
Speaker
Sarah Harrison, Northeastern University
Session Chair
Jonathan Heckman, University of Pennsylvania
Abstract
Topological defect lines provide a way to extend the notion of symmetry in two-dimensional conformal field theory beyond ordinary group actions. I will discuss recent work on such defects in two related supersymmetric settings: the Conway moonshine module and non-linear sigma models on K3 surfaces. Though these theories are irrational with respect to their superconformal algebras, supersymmetry-preserving defects are constrained by their action on natural 24-dimensional lattices, namely the Leech lattice on the Conway side and the D-brane charge lattice on the K3 side. This leads to a proposed extension of the known Conway/K3 correspondence from symmetry groups to tensor categories of defects, with examples exhibiting Tambara–Yamagami and Fibonacci-type fusion rules.
10:00 - 10:30 AM
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM
Searching for Special Lagrangian cycles in Calabi-Yau Manifolds
11th Floor Lecture Hall
Speaker
Fabian Ruehle, Northeastern University
Session Chair
Jonathan Heckman, University of Pennsylvania
Abstract
The study of Special Lagrangian three-cycles (sLags) in Calabi-Yau threefolds is an interesting problem in Physics and Mathematics. A three-cycle L is Lagrangian if the Kahler form of the Calabi-Yau manifold vanishes when restricted to L, and it is special if the complex phase of the holomorphic top form restricted to L is constant. SLags are volume-minimizing in their homology class, and they feature prominently in the SYZ conjecture of mirror symmetry. Despite their importance, almost no concrete examples of sLags are known. We will report on work in progress where we develop an approach that combines tools from machine learning (neural networks), artificial life (genetic algorithms with quality-diversity optimization), and persistent homology (a tool from topological data analysis that assigns homological data to point clouds) to search for new sLags in Calabi-Yau manifolds.
11:30 AM - 12:15 PM
Z3 orbifold K3s, their symmetries, and how to track them
11th Floor Lecture Hall
Speaker
Katrin Wendland, Trinity College Dublin
Session Chair
Jonathan Heckman, University of Pennsylvania
Abstract
For each K3 surface, Mukai showed that the group of holomorphic symplectic automorphisms is a subgroup of the Mathieu group M24. By the Torelli Theorem, these symmetries are conveniently described by their action on the lattice of integral cohomology, but by Mukai’s and Kondo’s work also on some Niemeier lattice, depending on the details of the K3 surface.
For Kummer surfaces, whose geometry has been investigated in detail by Nikulin, we showed more than a decade ago that this leads to a concrete realization of all symmetries as elements of the Mathieu group M24. We now provide an analogue of Nikulin's studies of Kummer surfaces for Z3 orbifold K3s. We obtain a classification of all symmetries of these special K3 surfaces, and we realize the resulting symmetry group concretely as subgroups of the Mathieu groups M12 and M24.
This is joint work with Kasia Budzik, Anne Taormina, Mara Ungureanu and Ida Zadeh.
12:30 - 2:30 PM
Lunch/Free Time
2:30 - 5:00 PM
3:30 - 4:00 PM
Coffee Break
11th Floor Collaborative Space
Wednesday, September 30, 2026
9:00 - 9:45 AM
K3 Surfaces, Elliptic Fibrations, and Orthogonal Modular Forms
11th Floor Lecture Hall
Speaker
Andreas Malmendier, Utah State University
Session Chair
Dino Festi, Università degli Studi di Napoli 'Federico II'
Abstract
K3 surfaces provide a natural meeting point between lattice theory, elliptic fibrations, and modular forms, with important connections to mirror symmetry and string theory. In this talk I will describe a series of explicit constructions that lead from geometric models of K3 surfaces to their orthogonal modular forms.
I will begin with two-elementary K3 surfaces with finite automorphism group, including their canonical double-sextic and quartic models and the associated configurations of rational curves and elliptic fibrations. I will then turn to recent joint work with Adrian Clingher and Brandon Williams on K3 surfaces with transcendental lattices 2H+L(-1) where L is of type (A_n) or (D_n). For Picard ranks (10) through (17), explicit Jacobian elliptic fibrations give modular forms for the corresponding orthogonal groups, and the coefficients of the Weierstrass equations realize generators of the graded rings of orthogonal modular forms. I will explain how these constructions fit together and what they reveal about the arithmetic and mirror-symmetric structure of the K3 moduli spaces.
10:00 - 10:30 AM
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM
The Variation of Calabi-Yau Hodge Structures and Feynman Integrals
11th Floor Lecture Hall
Virtual Speaker
Albrecht Klemm, University of Bonn
Session Chair
Dino Festi, Università degli Studi di Napoli 'Federico II'
Abstract
Pure and mixed variations of Hodge structures in families of Calabi–Yau varieties are encoded by parameter-dependent period and chain integrals. These integrals satisfy homogeneous Picard–Fuchs differential ideals of Calabi–Yau type or, equivalently, flat Gauss–Manin connections and their extensions. They describe many classes of higher-loop Feynman integrals, which can be evaluated particularly efficiently as solutions of resonant GKZ-type differential ideals and their extensions, supplemented by boundary conditions at discriminant divisors with distinguished monodromy. The resulting functions generalize both the iterated integrals underlying multiple polylogarithms, which describe broad classes of one-loop integrals, and the elliptic polylogarithms relevant to two-loop integrals. We relate this geometric description to integration-by-parts methods and to the associated connections in (\epsilon)-factorized form, and discuss applications to multiloop amplitudes.
11:30 AM - 12:15 PM
Chern characters of coherent sheaves on smooth projective hypersurfaces
11th Floor Lecture Hall
Speaker
Tyler Kelly, Queen Mary University of London
Session Chair
Dino Festi, Università degli Studi di Napoli 'Federico II'
Abstract
Given a coherent sheaf on a smooth projective hypersurface, we will give an explicit formula for its Chern character as a Cech cocycle. If the hypersurface is even dimensional, we can describe an explicit algorithm for identifying the component of the Chern character that is the hypersurface’s primitive cohomology when viewed in the Jacobian ring, using Griffiths’ residue theorem. As an application, we will give an explicit formula for the localized Chern character of complete intersections and scrolls. This is joint work with David Favero (University of Minnesota-Twin Cities).
12:25 - 12:30 PM
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:30 - 2:30 PM
Lunch/Free Time
2:30 - 3:15 PM
An Elliptic Approach to Reid's Fantasy
11th Floor Lecture Hall
Speaker
Lara Anderson, Virginia Polytechnic Institute and State University
Session Chair
Dino Festi, Università degli Studi di Napoli 'Federico II'
Abstract
In this talk I will review recent work attempting to connect generic Calabi-Yau threefolds to those with elliptic/genus-one fibrations through simple geometric transitions. In particular, I will describe recent progress demonstrating that all known manifolds in largest known datasets of CY threefolds (the Kreuzer-Skarke threefolds constructed as toric hypersurfaces and the CICY threefolds built as complete intersections in products of projective spaces) are all connected to genus one fibrations through single geometric transitions.
3:30 - 4:00 PM
Coffee Break
11th Floor Collaborative Space
4:00 - 5:00 PM
Open Problem Session
Problem Session - 11th Floor Lecture Hall
Thursday, October 01, 2026
9:00 - 9:45 AM
Beyond Algebraic Models of Stringy Spacetime
11th Floor Lecture Hall
Speaker
Tristan Hubsch, Howard University
Session Chair
Antonella Grassi, University of Pennsylvania and Università di Bologna
Abstract
Geometry and physics intertwine throughout the study of Nature at its fundamental level, their active reciprocating feedback of key importance in string theory. Complex-algebraic methods of geometry in particular have brought about an unrivaled abundance of models, counted well into hundreds and thousands of orders of magnitude — reciprocated by the discoveries such as the wholly unexpected but profound mirror symmetry. Further and very rich possibilities have recently been made possible by generalizations that, at first glance, appear to be non-algebraic. Nevertheless, they are remarkably well-aligned within both overall mirror-symmetric and the gauged linear sigma model framework, and are amenable to almost as comprehensive quantitative analysis, and implying a foundational mirror-symmetric integration with symplectic geometry.
10:00 - 10:30 AM
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM
Mirror symmetry and Kuznetsov components.
11th Floor Lecture Hall
Speaker
Giulia Gugiatti, University of Edinburgh
Session Chair
Antonella Grassi, University of Pennsylvania and Università di Bologna
Abstract
In this talk I will discuss how the structural features of Kuznetsov components of Fano varieties are reflected in the geometry of their mirror Landau-Ginzburg models. I will focus on Fano varieties arising as cyclic covers of weighted projective spaces branched along Calabi-Yau or Fano hypersurfaces. I will discuss the case of the quartic double solid in detail. The talk is based on work in progress with Ilaria Di Dedda, Danil Kozevnikov, and Nick Sheridan.
11:30 AM - 12:15 PM
Stability Conditions and Mirror Symmetry for Generalized Complex Tori
11th Floor Lecture Hall
Speaker
Benjamin Sung, University of Southern Denmark
Session Chair
Antonella Grassi, University of Pennsylvania and Università di Bologna
Abstract
The theory of Bridgeland stability conditions assigns a complex manifold to the derived category of coherent sheaves on a smooth projective variety. The structure of this complex manifold is central for applications to algebraic geometry and mathematical physics, but describing even a connected component is often a difficult, open problem. In this talk, I will describe a conjectural framework for the stability manifolds of abelian varieties and their noncommutative analogues using generalized complex tori, extending a conjecture of Kontsevich. As applications, I will describe the stability manifold of any abelian threefold and complete the description of mirror symmetry for abelian varieties, after Golyshev-Lunts-Orlov. This is based on work in progress, joint with Fabian Haiden.
12:30 - 2:30 PM
Lunch/Free Time
2:30 - 3:15 PM
Calabi-Yau Metrics and the Large Complex Structure Limit
11th Floor Lecture Hall
Speaker
Per Berglund, University of New Hampshire
Session Chair
Antonella Grassi, University of Pennsylvania and Università di Bologna
Abstract
Numerical Ricci-flat metrics on Calabi--Yau manifolds are becoming increasingly accurate. However, they often lack the interpretability required to extract theoretical insights. In this talk, we introduce a novel variant of Donaldson's algorithm using the canonical monomial basis of the ambient toric variety to compute interpretable balanced metrics even at large degrees $k$. Applying our ambient algorithm to the Dwork deformation family of Calabi--Yau hypersurfaces in $\mathbb{P}^{n+1}$, we show that the metric parameters obey novel power laws near the Large Complex Structure limit. Based on joint work with Tristan H\"ubsch, Vishnu Jejjala, Viktor Mirjani\'c and Challenger Mishra.
3:30 - 4:00 PM
Coffee Break
11th Floor Collaborative Space
4:00 - 5:00 PM
Open Problem Session
Problem Session - 11th Floor Lecture Hall
Friday, October 02, 2026
9:00 - 9:45 AM
Sunset Feynman integrals, Borcherds products, and cycles
11th Floor Lecture Hall
Speaker
Andrew Harder, Lehigh University
Session Chair
Paola Comparin, UFRO Universidad de La Frontera
Abstract
The Hessian family of K3 surfaces has been studied for many years, with much of its geometry and moduli theory developed in foundational work by Dolgachev and Keum, Dardanelli and van Geemen, Koike, and Kondō. More recently, this four-parameter family has attracted renewed interest from the physics community through its appearance in the three-loop sunset Feynman integral. I will describe a modular parametrization of the family in terms of explicit Borcherds products, giving a direct interpretation of the three-loop Feynman integral in terms of automorphic forms. I will also indicate a connection between these Borcherds products and natural higher Chow cycles on Hessian K3 surfaces. This talk is based on ongoing joint work with C. Doran, E. Pichon-Pharabod, and P. Vanhove.
10:00 - 10:45 AM
Some explicit bounds for elliptic fibrations
11th Floor Lecture Hall
Speaker
Antonella Grassi, University of Pennsylvania and Università di Bologna
Session Chair
Paola Comparin, UFRO Universidad de La Frontera
Abstract
For X a smooth elliptic K3 surface, it is known that the rank of the Mordell-Weil group can take any value between 0 and 18. In higher dimensions, the existence of a bound on rkMW(X/B) for Calabi-Yau varieties is guaranteed by recent results. Such bounds play an important role in physics. After a general introduction, I will discuss a method that paves the way to determine effective bounds on the rank of the Mordell-Weil group of higher-dimensional elliptic fibrations, and its application to threefolds of Kodaira dimension zero and Calabi-Yau fourfolds.
11:00 - 11:15 AM
Coffee Break
11th Floor Collaborative Space
11:15 AM - 12:00 PM
On a formula for the Euler numbers of threefolds with a regular K3 fibration
11th Floor Lecture Hall
Speaker
Taizan Watari, University of Tokyo
Session Chair
Paola Comparin, UFRO Universidad de La Frontera
Abstract
We will present two different derivations of a formula that governs the Euler numbers of threefolds with a regular K3 fibration.
12:30 - 2:30 PM
Lunch/Free Time
3:30 - 4:00 PM
Coffee Break
11th Floor Collaborative Space
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