A planar map is a combinatorial object built by gluing together triangles into a discrete surface with the sphere topology. Random planar maps sit at the intersection of many fields of mathematics – they can be studied enumeratively or bijectively, and their scaling limits have deep connections to conformal field theory and bosonic string theory. In this talk, I will discuss how certain models of random planar maps can be encoded using pairs of random trees, and how it helps us understand the geometry of random surfaces. No background beyond basic probability will be needed.

We show that for a reductive algebraic group \(G\) there exists an integer \(r(G)\), such that any finite set of elements in \(G\) of size more than \(r(G)\) that generates a Zariski-dense subgroup must be redundant i.e. we can remove some elements and still generate a Zariski-dense subgroup. We use this to deduce the analogous result for compact Lie groups. Thus, for example, if you have \(1000\) rotations that generate a dense subgroup of \({\rm SO}(3)\), some of them must be redundant. For non-compact Lie groups (e.g \({\rm SL}_2(\mathbb{C})\)) this fails: there are arbitrarily large irredundant topologically generating sets. The proof is mostly arithmetic: we ensure generators live in a number field in order to reduce the problem to finite groups via strong approximation and other results of this sort.

Epigenetic cell memory, the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multi-cellular organisms. It was observed in a simulation study by our collaborators that the stochastic dynamics and time-scale differences between establishment and erasure processes in chromatin modifications (such as histone modifications and DNA methylation) can have a critical effect on epigenetic cell memory. In this work, we provide a mathematical framework to rigorously validate and extend beyond these computational findings. Viewing our stochastic model of a chromatin modification circuit as a singularly perturbed, finite state, continuous time Markov chain, we extend beyond existing theory in order to characterize the leading terms in the series expansions of stationary distributions and mean first passage times. In particular, we provide an algorithm to determine the orders of the poles of mean first passage times, character
 ize the
 limiting stationary distribution and the limiting mean first passage times in terms of a reduced Markov chain. We also determine how changing erasure rates affects system behavior. The theoretical tools developed in this paper not only allow us to set a rigorous mathematical basis for the computational findings of our prior work, highlighting the effect of chromatin modification dynamics on epigenetic cell memory, but they can also be applied to other singularly perturbed Markov chains beyond the applications in this paper, especially those associated with biochemical reaction networks.

This talk is based on joint work with Simone Bruno, Felipe A. Campos, Domitilla Del Vecchio, and Ruth J. Williams.

We investigate quantum compact groups which support quantum metric space structure.  In our core example, we define an analog of the Hamming metric on the quantum permutation group $S_n^+$.  The construction of our quantum metric relies on the work of Biane and Voiculescu.  We also obtain an associated quantum 1-Wasserstein distance on the tracial state space of $C(S_n^+)$.  This is joint work with David Jekel and Anshu.

Siteswap notation provides a powerful way for jugglers to communicate and even discover new patterns. As they say, great power comes with great responsibility. In this talk we wield this tool as irresponsibly as possible, developing dual patterns, juggling with anti-balls, and more!

For a reductive $p$-adic group $G$, Kazhdan-Lusztig prove an isomorphism of the the extended affine Hecke algebra and the $G^\vee$-equivariant $K$-group of the Steinberg variety of the Langlands dual group $G^\vee$. It has a profound application of proving an important case of the local Langlands correspondence which is known as the Deligne-Langlands conjecture. For $G$ being a reductive group over an equal-characteristic local field, Bezrukavnikov upgrades Kazhdan-Lusztig's isomorphism to an equivalence of monoidal categories and proves the tamely ramified local geometric Langlands correspondence. In this talk, we discuss an ongoing project with João Lourenço on proving a tamely ramified local geometric Langlands correspondence for reductive $p$-adic groups. Time permitting, I will mention an interesting variant of Bezrukavnikov's equivalence in Ben-Zvi-Sakellaridis-Venkatesh's framework of the relative Langlands program based on a joint work in preparation with Milton Lin and Toan Pham.

[pre-talk at 1:20pm]

Liouville quantum gravity is a canonical model for random surfaces conjectured to be the scaling limit of various planar maps. Given that curvature is a central concept in Riemannian geometry, it is natural to ask whether this can be extended to LQG surfaces. Here, we define the Gaussian curvature for LQG surfaces (despite their low regularity) and study the relations with its discrete counterparts. We conjecture that this definition of Gaussian curvature is the scaling limit of the discrete curvature, we prove that the discrete curvature on the $\epsilon$-CRT map with a Poisson vertex set integrated with a smooth test function is of order $\epsilon^{o(1)},$ and show the convergence of the total discrete curvature on a CRT map cell when scaled by $\epsilon^{1/4}.$ Joint work with E. Gwynne.

In this talk we will introduce a PDE way to construct hypersurfaces which are critical for anisotropic integrands. Namely, we study energy concentration for rescalings of an anisotropic version of Allen-Cahn.

Besides a Gamma-convergence result, we will sketch a proof of the fact that energy of stable critical points (of the rescaled Allen-Cahn) concentrates along an integer rectifiable varifold, a weak notion of hypersurface, using stability (or finite Morse index) to compensate for the lack of monotonicity formulas.

Among the technical ingredients, we will see a generalization of Modica's bound and a diffuse version of the stability inequality for hypersurfaces.

This is joint work with Antonio De Rosa (Bocconi University).

We provide counterexamples showing that uniform laws of large numbers do not hold for subdifferentials under modest assumptions. Our results apply to random Lipschitz, convex, and convex-composite functions with randomness confined to the inner smooth map. Consequently, they resolve in the negative the questions posed by Shapiro and Xu [J. Math. Anal. Appl., 325(2), 2007] and highlight the obstacles nonsmoothness poses to uniform results.

In geometry, the quantum K theory of Grassmannians is a ring with a product deforming the usual K theory product. In (mathematical) physics, it is the coordinate ring of an affine variety given by the Bethe Ansatz equations. I will discuss a dictionary between these two perspectives, with emphasis on geometric interpretations. In particular, the graphical calculus from a 5-vertex lattice model yields Pieri-type rules, to quantum K multiply Schubert classes by Hirzebruch lambda y classes of tautological bundles. One may also construct eigenvectors of the previous quantum multiplication operators, called Bethe vectors, which quantize the usual classes of torus fixed points. I will discuss how the existence of these Bethe vectors leads to a theory of quantum equivariant localization for Grassmannians. This is joint work with V. Gorbounov and C. Korff, following earlier work with W. Gu, E. Sharpe, and H. Zou.