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This seminar covers a wide range of topics in Lie theory including (but not limited to) groups of Lie type, Lie algebras, Coxeter groups and their Hecke algebras, algebraic groups, and quantum groups, with emphasis on combinatorial and representation theoretic properties. Further information about upcoming seminars and people in the math department can be found at the CU Math Department Home Page.
A fundamental problem in topology is to determine when two spaces are distinct. For surfaces, classical invariants suffice, but for three-dimensional manifolds the question is considerably harder. The Witten-Reshetikhin-Turaev invariants, originating in quantum field theory, provide one family of tools. For torus bundles, i.e. 3-manifolds built by cutting a torus, applying a mapping class in SL(2,Z), and regluing, the skein algebra embeds into a non-commutative torus at roots of unity, yielding a finite-dimensional representation that makes these invariants concretely computable. Exploiting this structure, we develop a classical dynamic programming algorithm and a quantum algorithm that encodes the computation into logarithmically many qubits, achieving an exponential reduction in space. We further identify a natural coefficient-counting problem that is #P-complete yet admits efficient quantum approximation.
The first half of the talk will be expository, requiring only linear algebra and modular arithmetic. The second half will outline the algorithmic results.
How to tell shapes apart using quantum computers
Tue, Feb. 24 2:30pm (MATH 3…
Richard Green (CU)
X
The E8 root system is a highly symmetric configuration of vectors, corresponding to the 240 spheres that touch a given one in the densest possible packing of spheres in 8-dimensional Euclidean space. Despite this eightness, some combinatorial features of the root system are much easier to understand in terms of 9-dimensional coordinates. We will use this point of view to gain insight into a certain tightly structured partially ordered set, by using the combinatorics of perfect matchings and Fano planes.
This is based on work in progress with Tianyuan Xu (University of Richmond).
Gessel and Zhuang introduced the concept of shuffle-compatibility of statistics on permutations of subsets of \N to describe statistics whose multiset of values on the shuffles of two disjoint permutations is determined exactly by the size and statistic values of the two permutations being shuffled. Shuffle-compatibility implies the existence of an algebraic structure on the equivalence classes induced by the statistic. For example, the descent set statistic is shuffle-compatible, and the algebra on the equivalence classes it induces is isomorphic to the Hopf algebra of quasisymmetric functions.
In this talk, we generalize shuffle-compatibility to objects such as words, parking functions, and set partitions using the Hopf monoids associated with these objects. We cover a variety of well-known statistics on each object. We define various algebraic structures on the equivalence classes formed by these statistics that are, in many cases, quotients of (or isomorphic to) well-known combinatorial Hopf algebras such as FQSym, WQSym, PQSym, and NCSym.
Generalizing shuffle-compatibility using Hopf monoids
Tue, Mar. 24 2:30pm (MATH 3…
Juan Villareal
X
In this talk, I will present several scattering amplitude formulas arising in quantum field theory and explain how they give rise to new infinite-dimensional algebraic structures and associated symmetries. This work is based on ongoing collaboration with undergraduate students from the REU 2025.
How to tell shapes apart using quantum computers