Abstracts

Week of September 8, 2024

September 9, 2024
Group Actions, Geometry and Dynamics Entropy bounds for one-parameter diagonal flows on SL(d,R)/SL(d,Z) using linear functionals 4:15pm -
TBA
We give a method to bound the entropy of measures on SL(d,R)/SL(d,Z) which are invariant under one parameter diagonal subgroups, in terms of entropy contributions from regions of the cusp corresponding to different parabolic groups. These bounds depend on an auxiliary linear functional on the Lie algebra of the Cartan group, which can then be optimized depending on the bound one is trying to obtain. We discuss how this method is utilized to obtain new sharp bounds for cusp entropies.
 
Geometry, Symmetry and Physics Rational Cherednik Algebras and Torus Knot Invariants 4:30pm -
KT 801

The Khovanov–Rozansky homology categorifies the classical Jones and HOMFLY-PT polynomials. In this talk, we will explore how the Khovanov-Rozansky homology of the (m, n)-torus knot can be derived from the finite-dimensional representation of the rational Cherednik algebra at slope m/n, equipped with the Hodge filtration. This result confirms a conjecture by Gorsky, Oblomkov, Rasmussen, and Shende. Our approach involves the geometry of Hilbert schemes of points and character D-modules. Numerous examples will be provided to introduce and clarify the main concepts. 
September 10, 2024
Geometry & Topology Solving the word problem in the mapping class group in quasi-linear time 4:00pm -
KT 207

Mapping class groups of surfaces are of fundamental importance in dynamics, geometric group theory, and
low-dimensional topology.  The word problem for groups in general, the definition of the mapping class group, its finite generation by twists, and the solution to its word problem were all set out by Dehn [1911, 1922, 1938].  Some of this material was rediscovered by Lickorish [1960’s] and then by Thurston [1970-80’s] – they gave
important applications of the mapping class group to the topology and geometry of three-manifolds.  In the past fifty years, various mathematicians (including Penner, Mosher, Hamidi-Tehrani, Dylan Thurston, Dynnikov) have given solutions to the word problem in the mapping class group, using a variety of techniques.  All of these algorithms are quadratic-time.

We give an algorithm requiring only $O(n \log^3(n))$ time.  We do this by combining Dynnikov’s approach to curves on surfaces, M"oller’s version of the half-GCD algorithm, and a delicate error analysis in interval arithmetic.

This is joint work with Mark Bell.
Geometric Representation Theory Seminar Learning seminar on Bun_G 4:00pm -
KT 801
September 13, 2024
Friday Morning Seminar Friday Morning Seminar 10:00am -
KBT 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!