TBA
- Series
- School of Mathematics Colloquium
- Time
- Thursday, April 17, 2025 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Paul Bourgade – NYU
Work of Levin and Przytycki shows that if two non-special rational
functions f and g of degree $> 1 $over $\mathbb{C}$ share the same set of
preperiodic points, there are $m$, $n$, and $r$ such that $f^m g^n = f^r$.
In other words, $f$ and $g$ nearly commute. One might ask if there are
other sorts of relations non-special rational functions $f$ and $g$ over $\mathbb{C}$
might satisfy when they do not share the same set of preperiodic
points. We will present a recent proof of Beaumont that shows that
they may not, that if f and g do not share the same set of preperiodic
points, then they generate a free semi-group under composition. The
proof builds on work of Bell, Huang, Peng, and the speaker, and uses a
ping-pong lemma similar to the one used by Tits in his proof of the
Tits alternative for finitely generated linear groups.
TBD
Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.
TBA
Please Note: There will be a pre-seminar from 10:55 to 11:15 in Skiles 005.
A Springer fiber is the set of complete flags in Cn which are fixed by a given nilpotent matrix. It is a fundamental object of study in geometric representation theory and algebraic combinatorics. The irreducible components of a Springer fiber are indexed by combinatorial objects called standard Young tableaux. It is an open problem to describe geometric properties of these components (such as their singular loci and cohomology classes) in terms of the combinatorics of tableaux. We initiate a new approach to this problem by characterizing which irreducible components are equal to Richardson varieties, which are comparatively much better understood. Another motivation comes from Lusztig's recent study of the cell decomposition of the totally nonnegative part of a Springer fiber into totally positive Richardson cells. This is joint work in progress with Martha Precup.
Please Note: Zoom link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
Nonlinear functionals of Gaussian fields are ubiquitous in probability theory and PDEs. In work in progress with Robert Chang (Rhodes College), we introduce a family of random curves in the plane which encode the random values of certain nonlinear functionals of fractional Brownian motions on a circle with positive Hurst index s -1/2. For a special Cameron-Martin shift, the low variance limit of the fractional Brownian motion induces a LLN and CLT for the associated random curves that is nearly identical to the global behavior of Plancherel measures on large Young diagrams. The limit shape is independent of s and is that of Vershik-Kerov-Logan-Shepp. The global Gaussian fluctuations depend on s and, if we continue s to negative values, coincides with the process in Kerov's CLT for s = - 1/2. Although it might be possible to give a direct explanation for this coincidence by regularization, in this talk we give an indirect dynamical explanation by combining (i) results of Eliashberg and Dubrovin for a specific Hamiltonian QFT and (ii) the fact that in Hamiltonian systems, at short time scales, the quantum evolution of pure Gaussian wavepacket initial data agrees statistically with the classical evolution of mixed Gaussian random initial data.