Calendar

Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. Starting with the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. Ein and Lazarsfeld proved that if A is a very ample line bundle, then K_X + mA satisfies property N_p for any m>=n+1+p. It has ever since been an open question if the same holds true for A ample and basepoint free. In joint work with Purnaprajna Bangere we give a positive answer to this question.

Title: TBA

Abstract: TBA

In the phase diagram of many substances, the critical points have emergent conformal symmetry and are described by conformal field theories. Traditionally, physical quantities near the critical point can be computed by perturbative field theory method, where conformal symmetry is not fully utilized. In this talk, I will explain how conformal symmetry can be used to determine certain physical quantities, without even knowing the fine details of the microscopic structure. To compute the observables precisely, one needs to develop powerful numerical techniques. In the last few years, we have invented many computational tools and algorithms, and predicted critical exponents of Helium-4 superfluid phase transition and Heisenberg magnet to very high precision.

Title: TBA

Abstract: TBA

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, March 8th.

Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

Stanley’s chromatic symmetric function is a generalization of the chromatic polynomial of a graph that encodes coloring information for graphs. One open conjecture is that non-isomorphic trees have different chromatic symmetric functions. In this talk, I will give two geometric interpretations of these functions for trees. The first interprets the chromatic symmetric function of a tree as an element in the Chow ring of the permutahedral variety opening the conjecture to algebraic geometric methods. The second describes this open conjecture in terms of the theory of valuations on generalized permutahedra. These are functions on polytopes which satisfy certain inclusion-exclusion relations with respect to subdivisions. From this perspective, we make progress on the conjecture by constructing new valuations on generalized permutahedra. We will primarily focus on this convex geometric interpretation for this talk.

Can we measure the ‘effective regularity’ of spacetime from the perturbation of quasi-normal mode (QNM) overtones? Black hole (BH) QNMs encode the resonant response of black holes under linear perturbations, their associated complex frequencies providing an invariant probe into the background spacetime geometry. In the late nineties, Nollert and Price found evidence of a BH QNM instability phenomenon, according to which perturbed QNMs of Schwarzschild spacetime migrate to new perturbed branches of different qualitative behaviour and asymptotics. Here we revisit this BH QNM instability issue by adopting a pseudospectrum approach. Specifically, we cast the QNM problem as an eigenvalue problem for a non-selfadjoint operator by adopting a hyperboloidal formulation of spacetime. Non-selfadjoint (more generally non-normal) operators suffer potentially of spectral instabilities, the notion of pseudospectrum providing a tool suitable for their study. We find evidence that perturbed Nollert & Price BH QNMs track the pseudospectrum contour lines, therefore probing the analytic structure of the resolvent, showing the following (in)stability behaviour: i) the slowest decaying (fundamental) mode is stable, whereas ii) (all) QNM overtones are ultraviolet unstable (for sufficiently high frequency). Building on recent work characterizing Burnett’s conjecture as a low-regularity problem in general relativity, we conjecture that (in the infinite-frequency limit) generic ultraviolet spacetime perturbations make BH QNMs migrate to ‘Regge QNM branches’ with a precise universal logarithmic pattern. This is a classical general relativity (effective) low-regularity phenomenon, agnostic to possible detailed (quantum) descriptions of gravity at higher-energies and potentially observationally accessible.

This seminar will be held on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/general-relativity/

In the simplest spin-glass model, due to Sherrington and Kirkpatrick, the energy function involves interaction terms between all pairs of spins. A bipartite version of this model can be obtained by splitting the spins into two groups, which we can visualize as forming two layers, and by keeping only interaction terms that go from one to the other layer. For this and other models that incorporate a finite number of types of spins, the asymptotic behavior of the free energy remains mysterious (at least from the mathematical point of view). I will present the difficulties arising there, and some partial progress.

This seminar will be held on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/probability-seminar/

This seminar will take place in SC 507 at 3:30pm.

The CMSA will host a series of three 90-minute lectures on the subject of machine learning for protein folding.

Thursday, Feb 9, 2023:  3:30–5:00 pm ET

Thursday, Feb 16, 2023:  3:30–5:00 pm ET

Thursday, March 9, 2023: 3:30–5:00 pm ET

Further details TBA.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

For a Legendrian knot $K$ in a closed contact 3-manifold, we describe a necessary and sufficient condition for contact $n$-surgery along $K$ to yield a weakly symplectically fillable contact manifold, for some integer $n>0$. When specialized to knots in the standard 3-sphere this gives an effective criterion for the existence of a fillable positive surgery, along with various obstructions. These are sufficient to determine, for example, whether such a surgery exists for all knots of up to 10 crossings. The result also has certain purely topological consequences, such as the fact that a knot admitting a lens space surgery must have slice genus equal to its 4-dimensional clasp number. We will mainly explore these topologically-flavored aspects, but will give some hints of the general proof if time allows.