Calendar
- 17October 17, 2021No events
- 18October 18, 2021No events
- 19October 19, 2021
CMSA Combinatorics, Physics and Probability Seminar: Ising model, total positivity, and criticality
The Ising model, introduced in 1920, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality.
The totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig’s theory of total positivity and canonical bases, and is closely related to cluster algebras and scattering amplitudes.
I will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian, obtained in our recent work with P. Pylyavskyy. Building on this connection, I will give a new boundary correlation formula for the critical Ising model.
https://harvard.zoom.us/j/94191911494?pwd=RnN3ZnIwcFYwd0QyT0MwZWVISmR5Zz09
Password: 1251442
- 20October 20, 2021
CMSA Colloquium: Categorification and applications
I will give a survey of the program of categorification for quantum groups, some of its recent development and applications to representation theory.
Zoom link: https://harvard.zoom.us/j/95767170359 (Password: cmsa)
Effective height bounds for odd-degree totally real points on some curves
1 Oxford Street, Cambridge, MA 02138 USAI will give a finite-time algorithm that, on input (g,K,S) with g > 0, K a totally real number field of odd degree, and S a finite set of places of K, outputs the finitely many g-dimensional abelian varieties A/K which are of GL_2-type over K and have good reduction outside S.
The point of this is to effectively compute the S-integral K-points on a Hilbert modular variety, and the point of that is to be able to compute all K-rational points on complete curves inside such varieties.
This gives effective height bounds for rational points on infinitely many curves and (for each curve) over infinitely many number fields. For example one gets effective height points for odd-degree totally real points on x^6 + 4y^3 = 1, by using the hypergeometric family associated to the arithmetic triangle group \Delta(3,6,6).
- 21October 21, 2021
CMSA Interdisciplinary Science Seminar: Mathematical resolution of the Liouville conformal field theory
The Liouville conformal field theory is a well-known beautiful quantum field theory in physics describing random surfaces. Only recently a mathematical approach based on a well-defined path integral to this theory has been proposed using probability by David, Kupiainen, Rhodes, Vargas.Many works since the 80’s in theoretical physics (starting with Belavin-Polyakov-Zamolodchikov) tell us that conformal field theories in dimension 2 are in general « Integrable », the correlations functions are solutions of PDEs and can in principle be computed explicitely by using algebraic tools (vertex operator algebras, representations of Virasoro algebras, the theory of conformal blocks). However, for Liouville Theory this was not done at the mathematical level by algebraic methods.I’ll explain how to combine probabilistic, analytic and geometric tools to give explicit (although complicated) expressions for all the correlation functions on all Riemann surfaces in terms of certain holomorphic functions of the moduli parameters called conformal blocks, and of the structure constant (3-point function on the sphere). This gives a concrete mathematical proof of the so-called conformal bootstrap and of Segal’s gluing axioms for this CFT. The idea is to break the path integral on a closed surface into path integrals on pairs of pants and reduce all correlation functions to the 3-point correlation function on the Riemann sphere $S^2$. This amounts in particular to prove a spectral resolution of a certain operator acting on $L^2(H^{-s}(S^1))$ where $H^{-s}(S^1)$ is the Sobolev space of order -s<0 equipped with a Gaussian measure, which is viewed as the space of fields, and to construct a certain representation of the Virasoro algebra into unbounded operators acting on this Hilbert space.
This is joint work with A. Kupiainen, R. Rhodes and V. Vargas.
Zoom ID: 950 2372 5230 (Password: cmsa)
CMSA Quantum Matter in Mathematics and Physics Seminar: Electric-magnetic duality and the Geometric Langlands duality
I will give a pedagogical review of the connection between electric-magnetic duality and the Geometric Langlands duality.*Note special time*—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):
https://www.youtube.com/channel/UCBmPO-OK1sa8T1oX_9aVhAg/playlists
https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Wwhttps://harvard.zoom.us/j/977347126
Password: cmsaArithmetic intersection and measures of maximal entropy
About 10 years ago, Xinyi Yuan and Shouwu Zhang proved that if two holomorphic maps f and g on P^N have the same sets of preperiodic points (or if the intersection of Preper(f) and Preper(g) is Zariski dense in P^N), then they must have the same measure of maximal entropy. This was new even in dimension N=1. I will describe some ingredients in their proof, while emphasizing the dynamical history behind this result. I will also sketch the proof of a theorem of Levin and Przytycki from the 1990s, in dimension N=1, that two (non-exceptional) maps have the same measure of maximal entropy if and only if they “essentially” share an iterate.
for more information, go to:
- 22October 22, 2021
CMSA Quantum Matter in Mathematics and Physics Seminar: Recent Holographic Developments on the Black Hole Information Problem
*Note special time*—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):
https://www.youtube.com/channel/UCBmPO-OK1sa8T1oX_9aVhAg/playlists
https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Wwhttps://harvard.zoom.us/j/977347126
Password: cmsa - 23October 23, 2021No events