We study the well/ill-posedness of the Boltzmann equation with dispersive methods. We take the constant collision kernel case as the 1st example. We construct a family of special solutions, which are neither near equilibrium nor self-similar, and prove the ill-posedness in H^s Sobolev space for s1, despite the fact that the equation is scale invariant at s=1/2. Combining with the previous Chen-Denlinger-Pavlovic result regarding well-posedness, we have found the exact well/ill-posedness threshold.

The commensurator of a Riemannian manifold M encodes symmetries between all the finite covers of M, and lifts to a subgroup of isometries of the universal cover of M. In case M is an (irreducible) finite volume locally symmetric space, the commensurator is thus a subgroup of a semisimple Lie group G. Margulis proved that if the commensurator is dense in G, then M is arithmetic. Greenberg-Shalom asked if the same is true for infinite volume M? I will report on recent progress on this question when M regularly covers a finite volume hyperbolic manifold. This is joint work with D. Fisher and M. Mj.

Riemannian sigma model is a supersymmetric quantum mechanics which is highly interesting from the mathematical point of view; it has deep connections with Hodge theory, Morse theory, and Fredholm theory. There is an elegant construction of the theory known by physicists, namely superspace construction. I'll discuss how to mathematically understand the construction, using the language of supermanifolds. + Although I ended up using quite a different language, Tudor Dimofte's lecture 10 of BW&TD - Twisted QFT is a good source for this topic. This is where I learned this construction, and it might be interesting to watch.

In this talk, I will present $C^\infty$ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to point-wise bounds on their mass, energy and entropy densities. Joint work with Luis Silvestre.

Let E be an elliptic curve over Q, and let p be a prime of good ordinary reduction for E. Following the pioneering work of Skinner (and independently Wei Zhang), there is a growing number of results in the direction of a p-converse to a theorem of Gross-Zagier and Kolyvagin, showing that if the p-adic Selmer group of E is 1-dimensional, then a Heegner point on E has infinite order. In this talk, I'll report on an analogue of Skinner's result in the rank 2 case, in which Heegner points are replaced by certain "generalised Kato classes" introduced by Darmon-Rotger. For E without CM, such an analogue was obtained in an earlier work with M.-L. Hsieh, and in this talk I'll focus on the CM case, whose proof uses a different set of ideas.

Fano varieties are smooth projective varieties whose anticanonical class is ample. The positivity condition has far-reaching geometric implications, e.g., a Fano variety over complex numbers is simply connected, which has an analogue on the algebro-geometric side: any Fano variety is covered by rational curves, and in fact rationally connected, i.e. there are rational curves connecting any two of its points. In a series of papers, De Jong and Starr introduce and investigate possible candidates for the notion of higher rationally connectedness, inspired by the natural analogue in topology, and define that a projective variety X is 2-Fano if it is Fano and the second Chern character ch2(T_X) is positive (intersects positively with every surface in X). In a similar way, one defines n-Fano varieties for any n ≥ 2; for instance, P^n is n-Fano. In this talk, I will give evidence for the analogy with higher connectedness and present certain classification results. Despite the attention, there are few known examples of higher Fano varieties, and the sparsity even prompted a conjecture that every toric 2-Fano variety is a projective space. I will then explain partial results we obtained in this direction. This work is based on a series of projects with Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Kelly Jabbusch, Enrica Mazzon, Libby Tailor, Nivedita Viswanathan.

For elliptic curves with ordinary reductions at p, a conjecture on the vanishing order and the leading coefficient of the Bertolini--Darmon--Prasanna p-adic L-function at the trivial character was formulated by Agboola and Castella. I will talk about a generalization of their work to elliptic curves with supersingular reduction at p, which is a joint project with Castella, Hsu, Kundu, Lee and Leng for the Collaborative Research Workshop in Oregon 2022.

We study symmetries of families of circle maps that arise in connection to the theorem of Poncelet for pencils of quadrics. We find diffeomorphisms that have a one-parameter family of continuous symmetries. We introduce the concept of the eccentricity of a pencil and use it to show that families of Poncelet maps can be simultaneously reduced in a suitable canonical covering space. We show that this eccentricity is the modulus of some Jacobi elliptic functions we use to construct such covering spaces. Later, we show that the theory allows universal parameters that solve the inverse problem: Given a rotation number, find explicitly the element of a pencil whose Poncelet map has that rotation number. We provide a proof of the general Poncelet theorem using the constructed canonical covering space. (joint work with James Meiss, U of Colorado at Boulder.)

This talk deals with Markov chains on the ferromagnetic Potts model on random regular graphs. We analyse how the performance of Markov chains relates to the emergence of metastable states, which in turn correspond to stationary points of a functional called the Bethe free entropy. The proofs are based on coupling and spatial mixing arguments. Based on joint work with Andreas Galanis, Leslie Ann Goldberg, Jean Bernoulli Ravelomanana, Daniel Stefankovic and Eric Vigoda (arXiv:2202.05777).

Title: Abstract: We’ll discuss some of the history and intrigue surrounding Sarnak’s conjecture, a problem formulated in 2010 which has thus far evaded a general solution. There will be a brief discussion of the dynamical context for the problem, and a sketch of the proof for the “almost everywhere” version of the statement. We will cover some general number-theoretic background for the problem and motivation, as well as some related purely number-theoretic conjectures. We will then get into some examples of systems for which the conjecture is known to hold and the best known results towards proving the general case.