Abstracts

Week of March 2, 2025

March 3, 2025
Geometry, Symmetry and Physics Tetrahedral Symbol and Relative Langlands Duality 4:30pm -
KT 801
In the quantum theory of angular momentum, the Racah–Wigner coefficient, often known as the 6j symbol, is a numerical invariant assigned to a tetrahedron with half-integer edge-lengths. The 6 edge-lengths may be viewed as representations of SU(2) satisfying certain multiplicity-one conditions. One important property of the 6j symbol is its hidden symmetry outside the tetrahedral ones, originally discovered by Regge.
 
In this talk, we explore a generalized construction, dubbed the tetrahedral symbol, in the context of rank-1 semisimple groups over local fields, and explain how the extra symmetries may be explained by relative Langlands duality. Joint work with Akshay Venkatesh.
March 4, 2025
Geometry & Topology Drilling veering triangulations and applications to pseudo-Anosov flows 4:00pm -
KT 207

By a result of Agol and Guéritaud, a transitive pseudo-Anosov flow F on a closed three-manifold N, along with a certain collection O of its closed orbits, determines a veering triangulation V of N - O that encodes F. Together with Henry Segerman, we devised and implemented an algorithm that takes as an input V and a closed orbit c of F that is not in O, and returns a veering triangulation of N - O - c that also encodes F. I will discuss the main ideas behind the algorithm, its key challenges, and its role in collecting experimental evidence for certain conjectures about pseudo-Anosov flows.
Applied Mathematics Forbidden 0-1 Patterns and the Pach-Tardos Conjecture 4:00pm -
LOM 215

This talk will survey the extremal theory of pattern-avoiding 0-1 matrices, and some of their applications in geometry, combinatorics, and algorithms.  If P is a 0-1 matrix, Ex(P,n) is the maximum number of 1s in an n x n 0-1 matrix that does not contain any submatrix that dominates P.  Every 0-1 pattern P can be regarded as the incidence matrix of a bipartite graph, in which the two sides of the bipartition are ordered.  Thus, this definition can be seen as a generalization of the Turan extremal function (for subgraph avoidance).

Pattern-avoiding 0-1 matrices have been studied since the late 1980s, and yet the precise relationship between 0-1 matrices and Turan theory is still poorly understood.  For many years the foremost open problem has been to characterize the extremal functions of acyclic patterns (those whose graphs correspond to forests).  In 2005 Pach and Tardos conjectured that Ex(P,n) = O(n polylog(n)), for any acyclic P.  We give a simple refutation of the Pach-Tardos conjecture by giving a class of acyclic patterns for which Ex(P,n) > n 2^{sqrt{log n}}.
March 6, 2025
Analysis Fisher information for the space-homogeneous Boltzmann equation 4:00pm -

https://yale.zoom.us/j/95303636613
This talk is concerned with the space-homogeneous Boltzmann equation. We will see that the Fisher information of solutions decreases with time for physically relevant potentials, including the case of hard-spheres and soft and very soft potentials. An important consequence of this new a priori estimate is the well-posedness of the equation and the regularity of solutions.
Quantum Topology and Field Theory Transparent SL_n-skeins 4:30pm -
KT 801

For a Lie group G, the G-skein module of a 3-dimensional manifold M is a fundamental object in Witten’s interpretation of quantum knot invariants in the framework of a topological quantum field theory. It depends on a parameter q and, when this parameter q is a root of unity, the G-skein module contains elements with a surprising “transparency” property, in the sense that they can be traversed by any other skein without changing the resulting total skein. I will describe some (and conjecturally all) of these transparent elements in the case of the special linear group SL_n. The construction is based on the very classical theory of symmetric polynomials in n variables.  
March 7, 2025
Friday Morning Seminar TBA 10:00am -
KT 801

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!
Lang Lecture Multiple Dirichlet series and Nichols algebras 3:00pm -
KT 205

Multiple Dirichlet series are meromorphic functions in  several complex variables developed in analytic number theory to study moments of L-functions. Nichols algebras are noncommutative algebras developed to study Hopf algebras and quantum groups. In joint work in progress with Ian Whitehead, we give a uniform construction of multiple Dirichlet series in the function field setting, unifying many previous constructions and producing new examples. I will give an introduction to these two areas and explain how our work relates them.