Algebra Seminar
(Mon 3:30-4:30pm, Kap 265)
Organizer: Cris Negron
Topics include (but are not limited to):
Algebra, representation theory, algebraic and arithmetic geometry, homotopy theory, and mathematical physics.
Fall 2025
| Date | Location | Speaker | Abstract |
|---|---|---|---|
| Sept 8 | Yifeng Huang (USC) | Coh zeta functions for quadratic orders Abstract: The Coh zeta function, a 1/|Aut|-weighted count of a ring's finite modules, is a basic invariant that unifies problems in enumerative algebra such as counting matrices and representations. While these zeta functions for (germs of) singular curves have revealed surprising connections to q-series, making precise predictions has remained challenging. This talk provides a unified picture for the known y2=xn singularities by re-interpreting them as quadratic orders: ramified for odd n and split for even n. This algebraic viewpoint reveals a crucial missing piece in the landscape---the inert quadratic orders---which is necessary to complete the picture (via quadratic twists), and is the focus of the new work. |
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| Sept 15 | Vesselin Dimitrov (CalTech) | A new approach to effective Diophantine approximation Abstract: I will introduce you to the method of arithmetic holonomy bounds which we currently develop in collaboration with Frank Calegari and Yunqing Tang. To stick to a simple but fundamental example, I will explain how to effectivize the Thue—Siegel theorem for the special case of high order roots from a rational number, using an analysis surprisingly close to the traditional hypergeometric method but now cast within our new framework of multivalent holonomy bounds. The two other known ways to effectivize this ‘binomial' case of Thue—Siegel are Baker’s theory of linear forms in logarithms of algebraic numbers, and Bombieri’s equivariant Thue—Siegel method. Our holonomy bounds method begins with the original line of Axel Thue and adds to it a technique of David and Gregory Chudnovsky (originally used in the 1980s in their study of the Grothendieck—Katz p-curvature conjecture about the algebraicity of the solutions of certain linear ODEs). |
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| Sept 22 | Anton Kapustin (CalTech) | Homotopy theory, Quantum Field Theory, and Quantum Lattice Systems Abstract: It is now widely accepted in the theoretical physics community that any local QFT must account for higher or generalized symmetries, and that the mathematical structure which describes them is not a group, but a higher group, i.e. a connected homotopy type. Yet the origin of homotopy-theoretic notions in QFT remains poorly understood. I will discuss this issue in the context of quantum lattice systems. They share many similarities with QFT but their theory can be put on a solid mathematical footing using operator-algebraic methods. I will show that homotopy-theoretic structures naturally appear when one considers restrictions of symmetry transformations to spatial regions. Specifically, higher groups arise through their algebraic models: crossed modules in one spatial dimension, crossed squares of groups in two dimensions, and crossed cubes in general dimensions. |
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| Sept 29 [organizer away] |
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| Oct 6 [organizer away] |
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| Oct 13 | Davide Passaro (CalTech) | A VOA Formulation of the False-Mock Conjecture for 3-Manifold Invariants Abstract: The Ẑ invariants are q-series quantum invariants of 3-manifolds that refine WRT invariants and are conjecturally a step toward a categorified 3-manifold invariant. A central open problem concerns how Ẑ invariants behave under orientation reversal of the underlying manifold. The False Mock Conjecture posits a modular relation between Ẑ(M3) and Ẑ(-M3), exchanging quantum and mock modular behavior. In this talk, I will describe a vertex operator algebra (VOA) formulation of this conjecture, where orientation reversal induces a duality between VOAs associated to M3 and -M3. This will be an introductory talk, aimed at conveying the main ideas behind the Ẑ program, the False Mock Conjecture, and their interpretation in terms of VOAs, without assuming prior familiarity with either side of the correspondence. |
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| Oct 20 | |||
| Oct 27 | |||
| Nov 3 | Sam Qunell (UCLA) | 2-categorical affine symmetries and quantum boson algebras Abstract: Representations of KLR (quiver Hecke) algebras categorify the positive part of the quantum group associated to any symmetrizable Cartan matrix. This categorical perspective makes certain symmetries more natural to study. For example, the induction and restriction functors between categories of KLR algebra modules play an important role in the theory. A closer investigation of these functors reveals surprising new symmetries. In this talk, I explain how the induction and restriction functors for KLR algebras can be used to obtain a 2-representation of the corresponding affine positive part in type A. I also describe a new categorification of a closely related algebra, the q-boson algebra, in all symmetrizable Kac-Moody types. |
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| Nov 10 | Mark Walker (U Nebraska) | On the Hodge theory of matrix factorizations Abstract: Attached to a complex homogenous polynomial with an isolated singularity, there are two ways to form a Hodge structure. The first is the classical one for the associated smooth projective hypersurface. The more exotic method uses the negative/periodic cyclic homology and topological K-theory of the associated dg category of matrix factorizations. I will describe joint work with Michael Brown in which we prove these structures are equivalent and give some computational consequences. |
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| Nov 17 | Matt Young (Utah State) | Z-hat invariants for Lie superalgebras Abstract: The goal of this talk is to explain a representation theoretic approach to physicists' so-called Z-hat invariants of 3-manifolds, as introduced by Gukov, Pei, Putrov and Vafa in the context of supersymmetric gauge theory. Specifically, we use the representation theory quantum supergroups to construct non-semisimple analogues of the modular tensor categories Reshetikhin, Turaev, Andersen and others. These categories can in turn be used to construct quantum invariants of 3-manifolds, certain limits of which recover the Z-hat invariants. Based on joint work with Francesco Costantino, Matthew Harper and Adam Robertson. |
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| Nov 24 | Jaspreet Singh (UCLA) | Smooth Calabi-Yau varieties with large index and Betti numbers. Abstract: A normal variety X is called Calabi-Yau if its canonical divisor is Q-linearly equivalent to 0. The index of X is the smallest positive integer m so that m·KX~0. We construct smooth, projective Calabi-Yau varieties in every dimension with doubly exponentially growing index, which we conjecture to be maximal in every dimension. We also construct smooth, projective Calabi-Yau varieties with extreme topological invariants; namely, their Euler characteristics and the sums of their Betti numbers grow doubly exponentially. These are conjecturally extremal in every dimension. The varieties we construct are known in small dimensions but we believe them to be new in general. This work builds off of the singular Calabi-Yau varieties found by Esser, Totaro, and Wang. |
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| Dec 1 | Aravind Asok (USC) |
| Date | Location | Speaker | Abstract |
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| Jan 19 | Holiday | ||
| Jan 26 | |||
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| Feb 9 | |||
| Feb 16 | Holiday | ||
| Feb 23 | |||
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| March 9 | |||
| March 16 | Off | ||
| March 23 | |||
| March 30 | |||
| April 6 | |||
| April 13 | |||
| April 20 | |||
| April 27 |
Mailing list: Contact me if you want to be added to the seminar mailing list.
See our friends at the Geometry seminar (M 2:00 pm) and the Combinatorics seminar (W 2:00 pm).