Talk #1: The Dual Boundary Complex of the Moduli Space of Cyclic Compactifications

• Speaker: Toby Anderson, Harvey Mudd College
• Abstract: Moduli spaces provide a useful method for studying families of mathematical objects. We study certain moduli spaces of algebraic curves, which are generalizations of familiar lines and conics. This talk focuses on the dual boundary complex of the moduli space of genus-zero cyclic curves. This complex is itself a moduli space of graphs and can be investigated with combinatorial methods. Remarkably, the combinatorics of this complex provides insight into the geometry and topology of the original moduli space. This talk will cover the combinatorial methods used to calculate the Euler characteristic of this space as well as insights into how we might calculate its homotopy type.
• Recording: https://youtu.be/SU4D1tLJPLA

Talk #2: A meromorphic continuation of the Shintani-Barnes cocycle

• Speaker: Kairi Black, Duke University
• Abstract: The class number formula (CNF) relates the value at s = 1 of the Dedekind zeta function of a number field F to algebraic invariants of the field. Also associated to F is the sheaf-cohomological Shintani-Barnes cocycle. It is known that the Shintani-Barnes cocycle specialises to the values of partial zeta functions at s = 2, 3, 4, .... Motivated by the Stark conjectures - a generalization of the CNF - we study a meromorphic continuation of the Shintani-Barnes cocycle.

Talk #3: An Almost Linear Time Algorithm Testing Whether the Markoff Graph mod p is Connected

• Speaker: Colby Brown, University of California, Davis
• Abstract: The Markoff graphs modulo p were proven by Chen (2024) to be connected for all but finitely many primes, and Baragar (1991) conjectured that they are connected for all primes. In this talk, we showcase an algorithmic realization of the process introduced by Bourgain, Gamburd, and Sarnak [arXiv:1607.01530] to test whether the Markoff graph mod p is connected for arbitrary primes. Our algorithm runs in o(p^(1 + eplison)) time for every epsilon > 0. We demonstrate this algorithm by confirming that the Markoff graph mod p is connected for all primes less than one million.
• Recording: https://youtu.be/beJLcA0ErYY

Talk #4: Combinatorics of weighted DAG polytropes

• Speaker: Kamillo Ferry, Technische Universität Berlin
• Abstract: We study the specific family of polytropes from weighted directed acyclic graphs (DAGs) motivated by max-linear Bayesian networks. Max-linear Bayesian networks are a type of structural equation model that has been introduced by Gissibl and Klüppelberg in 2018. They are suited to model cause and effect relations between large observed values of several variables. We characterise how the combinatorics of polytropes are determined by associated tropical hyperplane arrangements and the structural identifiability of edges and their parameters given a specific choice of parameters.
• Recording: https://youtu.be/TP_ay7pMoFg

Talk #5: Ultrasolid Homotopical Algebra

• Speaker: Sofía Marlasca Aparicio, University of Oxford
• Abstract: We introduce ultrasolid modules, a new category for the purpose of formal geometry first proposed by Dustin Clausen. Ultrasolid modules over a field k build on the condensed mathematics of Clausen and Scholze, generalising the already existing theory of solid modules over Q or Fp . One can build some basic results in ultrasolid commutative algebra, and study the spectral and derived variants: E∞ ultrasolid k-algebras and animated ultrasolid k-algebras, as well as the cotangent complex in each of these settings. Finally, we apply this to deformation problems to generalise the Lurie-Schlessinger criterion in equal characteristic.
• Recording: https://youtu.be/QhuOijdBsNI

Talk #6: Cyclic covers: Hodge theory and categorical Torelli theorems

• Speaker: Hannah Dell, University of Edinburgh
• Abstract: Since any Fano variety can be recovered from its derived category up to isomorphism, we ask whether less information determines the variety - this is called a categorical Torelli question. In this talk we consider an n-fold cover X → Y ramified in a divisor Z. The cyclic group of order n acts on X. We study how a certain subcategory of Db(X) (the Kuznetsov component) behaves under this group action. We combine this with techniques from topological K-theory and Hodge theory to prove that this subcategory determines X for two new classes of Fano threefolds which arise as double covers of (weighted) projective spaces. This is joint work with Augustinas Jacovskis and Franco Rota (arXiv:2310.13651).
• Recording: https://youtu.be/JWhMr3MmTvg

Talk #7: Ideals of differential operators

• Speaker: Isaac Goldberg, Cornell University
• Abstract: I will exposit a non-commutative classification result of Yuri Berest and Oleg Chalykh, which classies the (left) ideals of the ring of differential operators on a smooth, affine curve. The set of such ideals breaks up into a disjoint union of varieties, and which variety an ideal lives in is a discrete invariant, analogous to the codimension of an ideal in the commutative setting (where the corresponding moduli space is a Hilbert scheme).
• Recording: https://youtu.be/NKx4TheUCDY