Algebra Seminar

Algebra Seminar

Title: From Hilbert's 12th problem to complex equiangular lines

Abstract: We describe a connection between two superficially disparate open problems: Hilbert's 12th problem in number theory and Zauner's conjecture in quantum mechanics and design theory. Hilbert asked for a theory giving explicit generators of the abelian Galois extensions of a number field; such an "explicit class field theory" is known only for the rational numbers and imaginary quadratic fields. Zauner conjectured that a configuration of d^2 pairwise equiangular complex lines exists in d-dimensional Hilbert space (and additionally that it may be chosen to satisfy certain symmetry properties); such configurations are known only in a finite set of dimensions d.

 
We prove a conditional result toward Zauner's conjecture, refining insights of Appleby, Flammia, McConnell, and Yard gleaned from the numerical data on complex equiangular lines. We prove that, if there exists a set of real units in a certain abelian extension of a real quadratic field (depending on d) satisfying certain congruence conditions and algebraic properties, a set of d^2 equiangular lines may be constructed, when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s=0 and is closely connected to the Stark conjectures over real quadratic fields. We will work through the example d=5 in detail to help illustrate our results and conjectures.
Date:
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Location:
745 POT
Event Series:

Algebra Seminar

Title: Degenerations of cohomology rings
 
Abstract: An associative algebra is encoded by its structure constants, describing how to multiply elements in a distinguished basis and expand in that basis. Such algebras are rigid in the sense that you can't generally maintain associativity while modifying some of the structure constants. Motivated by analogues of Horn's problem on eigenvalues of sums of Hermitian matrices, Belkale and Kumar (2006) nonetheless obtained important new associative algebras from the cohomology of generalized flag varieties by setting various structure constants equal to zero. The existence of this degeneration was originally established via geometric invariant theory; another proof was supplied by Graham and Evens (2013) using relative Lie algebra cohomology. We give an elementary proof. This leads us to an additional degeneration, which we interpret geometrically. (Joint work with Dominic Searles.)
Date:
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Location:
745 POT
Event Series:

Algebra Seminar

TItle: Lech's inequality and a conjecture of Stuckrad-Vogel

 

Abstract: Let $(R, m)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module of dimension $d$. We prove that the set ${l(M/IM)/e(I,M)}$, when $I$ runs through all $m$-primary ideals, is bounded below by $1/d!e(R)$. Moreover, when the completion of $M$ is equidimensional, this set is bounded above by a finite constant depending only on $M$. This extends a classical inequality of Lech and answers a question of Stuckrad-Vogel. Our main tool is to use Vasconcelos's homological degree. The talk is based on joint work with Patricia Klein, Pham Hung Quy, Ilya Smirnov, and Yongwei Yao. 

Date:
-
Location:
745 POT
Event Series:

Algebra Seminar

Title: Adversarial Network Coding



Abstract: In the context of Network Coding, one or more sources of information attempt to transmit messages to multiple receivers through a network of intermediate nodes. In order to maximize the throughput, the nodes are allowed to recombine the received packets before forwarding them towards the sinks.



In this talk, we present a mathematical model for adversarial network transmissions, studying the scenario where one or multiple (possibly coordinated) adversaries can maliciously corrupt some of the transmitted messages, according to certain restrictions. For example, the adversaries may be constrained to operate on a vulnerable region of the network. 



If noisy channels (traditionally studied in Information Theory) are described within a theory of "probability", adversarial channels are described within a theory of "possibility". Accordingly, in this talk we take a discrete combinatorial approach in defining and studying network adversaries and channels.



We propose various notions of capacity region of an adversarial network, and illustrate a general technique that allows to port upper bounds for the capacities of point-to-point channels to the networking context. We then present some new upper bounds on the capacity regions of an adversarial network, and describe some new capacity-achieving communication schemes.



The new results in this talk are joint work with Frank R. Kschischang (University of Toronto).

Date:
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Event Series:

Algebra Seminar

Title: OI-algebras, strongly stable ideals, and cellular resolutions.


Abstract: A common occurrence, in commutative algebra and elsewhere, is a family of ideals $I_n \subseteq k[x_1,\ldots,x_n]$ in a family of polynomial rings, satisfying that $f(I_n) \subseteq I_{n+1}$ for any $f$ in a certain family of ring homomorphisms. In the context where $f$ is any order-preserving function of the indices of the variables, the theory of OI-algbras gives a categorical re-framing of this situation. In this framework one can study OI-ideals using resolutions by free OI-modules. After introducing and motivating the subject, we will exhibit a family of OI-ideals (coming from strongly-stable ideals) that have explicit free resolutions supported on OI-simplicial complexes. This is based on work in progress with Uwe Nagel.

 

Date:
-
Location:
745 POT
Event Series:

Algebra Seminar

Title: Symbolic powers via test ideals

Abstract: An important problem in commutative algebra is studying the relationship between symbolic and ordinary ideals. One striking result in this direction was found by Ein-Lazarsfeld-Smith, who showed that for regular rings in characteristic 0, the dn-th symbolic power of any ideal is contained in the n-th ordinary power of that ideal, where d is the dimension of the ring. Their method proved to be quite powerful, and was adapted to the positive characteristic setting by Hara and the mixed characteristic setting by Ma and Schwede. However, all of this work was done in the regular setting. This is because the above method relies on the so-called subadditivity property of test ideals, which only holds for regular rings.

In this talk, we will discuss an approach to extending Ein-Lazarsfeld-Smith's result to the non-regular setting by using a new subadditivity formula for test ideals. Recent joint work with Carvajal-Rojas, Page, and Tucker shows that this approach works for a large class of rings, including all Segre products of polynomial rings. Time permitting, we will discuss how applying this approach to any toric variety reduces to solving a certain combinatorial problem. 

Date:
-
Location:
745 POT
Event Series:

Algebra Seminar

Title: Local Cohomology of Thickenings
 
Abstract: Let $R$ be a standard graded polynomial ring that is finitely generated over a field, and let $I$ be a homogenous prime ideal of $R$. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of $R/I^t$, as $t$ goes to infinity, which led to the development of an asymptotic invariant by Dao and Montaño. I will discuss their results and give concrete examples of the calculation of this new invariant in the case of determinantal rings. 
Date:
-
Location:
POT 745
Event Series:
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