Aswin Rangasamy Venkatesan - Qualifying Examination
Title: Quiver representations and syzygies of Gorenstein algebras
Date:
-
Location:
POT 745
Event Series:
DISCRETE CATS SEMINAR
Aswin Rangasamy Venkatesan - Qualifying Examination
Title: Quiver representations and syzygies of Gorenstein algebras
Date:
-
Location:
POT 745
Event Series:
Discrete CATS seminar
Zoom talk
Speaker: Gábor Hetyei, UNC Charlotte
Title: Labeling regions in graphical hyperplane arrangements using the Farkas' lemma
Abstract:
We call a hyperplane arrangement graphical if each hyperplane in it is defined by an equation of the form $x_i-x_j=c$. Combining Carver's variant of the Farkas' lemma with the Flow Decomposition Theorem we show that the regions of any graphical arrangement may be bijectively labeled with a set of weighted digraphs containing directed cycles of negative weight only. Relatively bounded regions correspond to strongly connected digraphs.
The study of the resulting labelings allows us to add the omitted details in Stanley's proof on the injectivity of the Pak-Stanley labeling of the regions of the extended Shi arrangement, to generalize the ceiling diagrams in the deleted Shi and Ish arrangements studied by Armstrong and Rhoades and to introduce a new labeling of the regions in the Fuss-Catalan arrangement. We also show how Athanasiadis-Linusson labelings may be used to directly count regions in a class of arrangements properly containing the extended Shi arrangement and the Fuss-Catalan arrangement.
Date:
Location:
745 POT
Event Series:
Evan Henning, Qualifying Exam
Qualifying Exam
Speaker: Evan Henning, University of Kentucky
Title: Permutation Hopf algebras
Abstract:
Hopf algebras are a very natural algebraic structure in the study of combinatorics. This is due to the fact that many combinatorial objects admit canonical operations of combination and decomposition.
One of the most foundational combinatorial objects of mathematics is that of permutations. There are many Hopf algebras which can be put on the linear span of permutations, the most famous of which is the Malvenuto and Reutenauer Hopf algebra of permutations. This is a shuffle-like Hopf algebra that is self-dual and surjects onto the Hopf algebra of quasisymmetric functions. In this talk, we focus on the work of Mingze Zhao and Huilan Li and consider two other Hopf algebras that can be put on permutations.
These two new Hopf algebras give permutations a tensor and shuffle algebra structure. We will prove the duality between these Hopf algebras and will consider a generalization of these Hopf algebras, which can be extended to juggling.
Date:
Location:
745 POT
Event Series:
Evan Henning, Qualifying Exam
Qualifying Exam
Speaker: Evan Henning, University of Kentucky
Title: Permutation Hopf algebras
Abstract:
Hopf algebras are a very natural algebraic structure in the study of combinatorics. This is due to the fact that many combinatorial objects admit canonical operations of combination and decomposition.
One of the most foundational combinatorial objects of mathematics is that of permutations. There are many Hopf algebras which can be put on the linear span of permutations, the most famous of which is the Malvenuto and Reutenauer Hopf algebra of permutations. This is a shuffle-like Hopf algebra that is self-dual and surjects onto the Hopf algebra of quasisymmetric functions. In this talk, we focus on the work of Mingze Zhao and Huilan Li and consider two other Hopf algebras that can be put on permutations.
These two new Hopf algebras give permutations a tensor and shuffle algebra structure. We will prove the duality between these Hopf algebras and will consider a generalization of these Hopf algebras, which can be extended to juggling.
Date:
Location:
745 POT
Event Series:
Discrete CATS seminar
Speaker: Goran Omerdic, Western Kentucky University
Title: Distributive lattice models for one-rowed representations of the classical Lie algebras
Abstract:
Lie algebras are canonically used to describe the symmetries of continuous functions. Such algebras are rich in enumerative properties. We consider the classical Lie algebras, which are comprised of the "special linear," "symplectic" and "orthogonal" Lie algebras through a lens of algebraic combinatorics, using colored modular and distributive lattice models to describe said properties. Research related to special linear, symplectic and odd orthogonal Lie algebras has been fruitful, yielding families of lattice models and related coefficients to generate direct graphs. We use these methods to explore the properties of even orthogonal Lie algebras, their lattice and one-rowed tableau representations.
Date:
Location:
745 POT
Event Series:
Discrete CATS seminar
Speaker: George Nasr, Augustana University
Title: IDP for 2-Partition Maximal Symmetric Polytopes
Abstract:
The Integer Decomposition Property (IDP) for a polytope P essentially asks if the points in any scaled version of a polytope can be written as a sum of points in P itself. Despite a seemingly trite definition, asking if a polytope has IDP is among the many popular problems in discrete mathematics that has a breadth of applications, from solving other questions in discrete mathematics like understanding Ehrhart polynomials, to understanding properties of abstractly defined algebraic structures associated to polytopes, to optimization for integer programing problems whose constraints define a polytope. We provide a framework for which one can approach showing the integer decomposition property for symmetric polytopes. We utilize this framework to prove a special case which we refer to as 2-partition maximal polytopes in the case where it lies in a hyperplane of R^3. Our method involves proving a special collection of polynomials have saturated Newton polytope.
Date:
Location:
745 POT
Event Series:
Emma Pickard, UK
Qualifying Exam
Speaker: Emma Pickard, University of Kentucky
Title: Degeneracy loci and permutation groups
Abstract:
Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson.
Date:
Location:
745 POT
Event Series:
Emma Pickard, UK
Qualifying Exam
Speaker: Emma Pickard, University of Kentucky
Title: Degeneracy loci and permutation groups
Abstract:
Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson.
Date:
Location:
745 POT
Event Series:
Emma Pickard, UK
Qualifying Exam
Speaker: Emma Pickard, University of Kentucky
Title: Degeneracy loci and permutation groups
Abstract:
Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson.
Date:
Location:
745 POT
Event Series: