# Auburn Algebra Seminar

## Fall 2023

Schedules from past semestersSeminars will be held in Parker 358 on Tuesdays from 2:30 to 3:20.

### Schedule:

**August 22:**Michael Brown (Auburn)

**Title:**Orlov's Landau-Ginzburg/Calabi-Yau correspondence

**Abstract:**I will discuss a landmark theorem of Orlov that relates the derived category of sheaves on a projective complete intersection to the singularity category of its affine cone. I will also discuss work in progress with Prashanth Sridhar, where we aim to generalize Orlov's theorem to the setting of differential graded algebras.

**August 29:**Hal Schenck (Auburn)

**Title:**Kuramoto Oscillators: algebraic and topological aspects

**Abstract:**We investigate algebraic and topological signatures of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks). We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. Furthermore, we describe a construction of networks on an arbitrary number of vertices having linearly stable states that are not twisted stable states. Joint work with M. Stillman (Cornell) and H. Harrington (Oxford).

**September 5:**

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**September 12:**

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**September 19:**

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**September 26:**

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**October 3:**Keller VandeBogert (Notre Dame University)

**Title:**Stable Sheaf Cohomology via A-infinity Koszul Duality

**Abstract:**In arbitrary characteristic, it is known that the cohomology of line bundles on flag varieties eventually behaves like a fixed polynomial functor, depending only on the line bundle. Computing this "stable" polynomial functor remains a highly nontrivial open question with relatively few cases being known explicitly. In this talk, I will illustrate how the computation of stable cohomology may be significantly simplified by instead bundling sheaf theoretic data together as an algebra, then taking advantage of the A-infinity analogue of Koszul duality. This is joint work with Claudiu Raicu.

**October 10:**

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**October 16 (Monday):**Jonathan Montaño (Arizona State University)

**Title:**K-polynomials of multiplicity-free varieties

**Abstract:**Multiplicity-free varieties are those whose multidegrees are equal to zero or one. This special condition implies important arithmetic properties of its coordinate ring and its Gröbner degenerations. In this project, we show that the support of (twisted) K-polynomials of multiplicity-free varieties form a generalized polymatroid. We apply this to show that the Möbius support of a base polymatroid is a generalized polymatroid, and to settle a particular case of a conjecture of Monical, Tokcan, and Yong. This is joint work with Castillo, Cid-Ruiz, and Mohammadi.

**October 24:**Luke Oeding (Auburn)

**Title:**Maximizing Geometric Measures of Entanglement

**Abstract:**How do we measure entanglement from a geometric point of view? For small quantum systems we can do this with algebraic and geometric information. I’ll talk about 3 qutrit systems ( 3x3x3 tensors) and recent work with Hamza Jaffali and Frederic Holweck where we use geometry to reduce the 27-dimensional space we’re working with to a 3-dimensional space that still contains all the relevant information. We then maximize the absolute value of several invariants, including the hyperdeterminant and find new candidates for maximal entanglement. I’ll give an overview of the geometry, as well as some of the standard optimization techniques we used.

**October 31:**Claudia Miller (Syracuse University)

**Title:**Cyclic Adams operations

**Abstract:**Using an idea of Atiyah from 1966, developed further in various settings by Benson, Haution, and Köck, we develop Adams operations on the Grothendieck groups of perfect complexes with support and of matrix factorizations using cyclic group actions on tensors powers. The main aim of this talk is to discuss what Adams operations are, the two traditional ways to construct them, including both lambda operations and Atiyah's innovative approach using representations of the symmetric group to understand various power operations in algebra (given a well-behaved monoidal structure). A second aim is to address a particular setting, time permitting: For complexes, Gillet and Soulé developed these using the first of these constructions and the Dold-Kan correspondence and used them to solve Serre's Vanishing Conjecture in mixed characteristic (also proved independently by P. Roberts using localized Chern characters). Their approach cannot be used in the setting of matrix factorizations, so we use Atiyah's approach, avoiding simplicial theory altogether. We may briefly mention an application to a conjecture of Dao and Kurano on the vanishing of Hochster's theta pairing for pairs of modules over an isolated hypersurface singularity in the remaining open case of mixed characteristic. This is joint work with Michael Brown, Peder Thompson, and Mark Walker.

**November 7:**Greg Smith (Queen's University)

**Title:**Cohomology of toric vector bundles

**Abstract:**A toric vector bundle is a vector bundle on a toric variety equipped with a torus action that is compatible with canonical action on the underlying variety. Klyachko proves that toric vector bundles are classified by finite-dimensional vector spaces with a suitable family of filtrations. Building on this equivalence of categories, we construct a complex of modules over the Cox ring which simultaneously encodes the cohomology of a toric vector bundle and many of its twists by line bundles. Beyond the improved computational efficiency, this approach leads to new insights into virtual resolutions and vanishing theorems. This talk is based on joint work with Michael Perlman.

**November 14:**Prashanth Sridhar (Auburn)

**Title:**Orlov’s Landau-Ginzburg/Calabi-Yau correspondence for dg-algebras

**Abstract:**This is a sequel to Michael’s talk on August 22, 2023. I’ll discuss joint work with Michael K. Brown generalizing Orlov’s theorem to dg-algebras.

**November 28:**Huajun Huang (Auburn)

**Title:**Yamamoto-Nayak's Theorem and Extensions

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