Department of

Mathematics


Seminar Calendar
for Commutative Ring Theory events the year of Tuesday, January 1, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, March 14, 2019

3:00 pm in 243 Altgeld Hall,Thursday, March 14, 2019

To Be Announced

Satya Mandal (University of Kansas)

Abstract: Title: Splitting property of projective modules, by Homotopy obstructions Speaker: Satya Mandal, U. of Kansas \noindent{\bf Abstract:} Follow the link: http://mandal.faculty.ku.edu/talks/abstractIllinoisMarch19.pdf Alternate version: The theory of vector bundles on compact hausdorff spaces $X$, guided the research on projective modules over noetherian commutative rings $A$. There has been a steady stream of results on projective modules over $A$, that were formulated by imitating existing results on vector bundles on $X$. The first part of this talk would be a review of this aspects of results on projective modules, leading up to some results on splitting projective $A$-modules $P$, as direct sum $P\cong Q\oplus A$. % Our main interest in this talk is to define an obstruction class $\varepsilon(P)$ in a suitable obstruction set (preferably a group), to be denoted by $\pi_0\left({\mathcal LO}(P) \right)$. Under suitable smoothness and other conditions, we prove that $$ \varepsilon(P)\quad {\rm is~trivial~if~and~only~if}~ P\cong Q\oplus A $$ Under similar conditions, we prove $\pi_0\left({\mathcal LO}(P) \right)$ has an additive structure, which is associative, commutative and has n unit (a "monoid").

Thursday, June 13, 2019

3:00 pm in 243 Altgeld Hall,Thursday, June 13, 2019

Wilf's conjecture by multiplicity

Winfried Bruns (Mathematik/Informatik Universität Osnabrück)

Abstract: Let S be a numerical semigroup. Its embedding dimension e(S) is the minimal number of generators, the Frobenius number F(S) is the largest integer not in S , and n(S) counts the elements in S that are < F(S). Wilf's conjecture states that F(S) < e(S)n(S). It has been proved in many cases, but remains a major open problem in the combinatorial theory of numerical semigroups. We will show that for fixed multiplicity m=m(S), the smallest nonzero element of S, the conjecture can be decided algorithmically by polyhedral methods using the parametrization of multiplicity m semigroups by the lattice points of the Kunz polyhedron P(m). With them we have verified the conjecture for m up to 18.

Friday, November 1, 2019

2:00 pm in 347 Altgeld Hall,Friday, November 1, 2019

Categorification and quantum symmetry

Colleen Delaney (Indiana University)

Abstract: One variation on the theme of ``quantum symmetry" is a categorical group action on a unitary modular tensor category, which can be interpreted physically as a global symmetry of a 2-dimensional topological quantum phase of matter. Much of our understanding of tensor category theory and hence topological phases comes from categorification: from generalizing theorems we have about rings to theorems about categories. For example, categorifying an easy theorem in commutative ring theory, the work of Etingof, Nikshych, and Ostrik established an equivalence between categorical G-actions on modular tensor categories (MTCs), and so-called G-crossed braided extensions of MTCs. Physicists Barkeshli, Bonderson, Cheng, and Wang then recognized that this correspondence can be understood as a tensor-categorical formulation of gauge coupling, wherein G-crossed braided extensions of MTCs give an algebraic theory of symmetry-enriched topological (SET) phases of matter. While the abstract theory of Etingof, Nikshych, and Ostrik is well understood, even constructing the de-categorified part of G-crossed braided extensions of MTCs, namely their fusion rings, is challenging problem in general. We will give a two-part talk, starting with an introduction to the algebraic theory of SET phases described above. In the second part of the talk we describe a topological phase-inspired approach to constructing the fusion rings of certain G-crossed extensions called permutation extensions and share work in progress with E. Samperton in constructing their categorifications.