Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, and others.  In this talk I'll provide an introduction to studying groups and spaces from this point of view. I'll focus on a few of my favorite "hyperbolic features" and how they manifest in many examples. This talk will include joint work with M. Hagen and A. Sisto, as well as with C. Abbott and M. Durham. 

We present new objects called quilts of alternating sign matrices with respect to two given posets. Quilts generalize several commonly used concepts in mathematics. For example, the rank function on submatrices of a matrix gives rise to a quilt with respect to two Boolean lattices. When the two posets are chains, a quilt is equivalent to an alternating sign matrix and its corresponding corner sum matrix. Such rank functions are used in the definition of Schubert varieties in both the Grassmannian and the complete flag manifold. Quilts also generalize the monotone Boolean functions counted by the Dedekind numbers, which is known to be a #P-complete problem. Quilts form a distributive lattice with many beautiful properties and contain many classical and well known sublattices, such as the lattice of matroids of a given rank and ground set. While enumerating quilts is hard in general, we prove two major enumerative results, when one of the posets is an antichain and when one of them is a chain. We also give some bounds for the number of quilts when one poset is the Boolean lattice. Several open problems will be given for future development. This talk is based on joint work with Matjaz Konvalinka in arxiv:2412.03236.

In collaboration with Han, Padurariu, and Park, we show that the number of $5$-isogenies of elliptic curves defined over $\mathbb{Q}$ with naive height bounded by $H > 0$ is asymptotic to $C_5\cdot H^{1/6} (\log H)^2$ for some explicitly computable constant $C_5 > 0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\mathbb{Q}$-isomorphism between the stack $\mathscr{X}_0(5)$ and the generalized Fermat equation $x^2 + y^2 = z^4$ with $\mathbb{G}_m$ action of weights $(4, 4, 2)$.

Pretalk: I will explain how to count isomorphism classes of elliptic curves over the rationals. On the way, I will introduce some basic stacky notions: torsors, quotient stacks, weighted projective stacks, and canonical rings.

[pre-talk at 3:00PM]

The study of orbit dynamics for the upper triangular subgroup $P \subset \mathrm{SL}(2, \mathbb R)$ holds fundamental significance in homogeneous and Teichmüller dynamics. In this talk, we shall discuss the quantitative density properties of $P$-orbits for translation surfaces near Teichmüller curves. In particular, we discuss the Teichmüller space $H(2)$ of genus two Riemann surfaces with a single zero of order two, and its corresponding absolute period coordinates, and examine the asymptotic dynamics of $P$-orbits in these spaces.

Given a random walk on a countable group, the Poisson boundary is a measure space which captures all asymptotic events of the markov chain. The Poisson boundary can sometimes be identified with a concrete geometric "boundary at infinity", but almost all previous results relied strongly on moment conditions of the random walk. I will discuss a technique which allows us to identify the Poisson boundary on any group with hyperbolic properties without moment conditions, new even in the free group case, making progress on a question of Kaimanovich and Vershik. This is joint work with Behrang Forghani, Joshua Frisch, and Giulio Tiozzo.

Growth of bacterial colonies on solid surfaces is commonplace; yet, what occurs inside a growing colony is complex even in the simplest cases. Robust colony expansion kinetics featuring linear radial growth and saturating vertical growth in diverse bacteria indicates a common developmental program, which will be elucidated in this talk using a combination of findings based on modeling and experiments.  Agent-based simulations reveal the crucial role of emergent mechanical constraints and spatiotemporal dynamics of nutrient gradients which govern observed expansion kinetics. The consequences of such emergent features will also be examined in the context of pattern formation in multi-species bacterial communities. Future directions and opportunities in theoretical modeling of such pattern formation systems will be discussed.

In this talk, we discuss the semiclassical asymptotics for Bergman kernels in exponentially weighted spaces of holomorphic functions. We will first review various approaches to the construction of asymptotic Bergman projections, for smooth weights and for real analytic weights. We shall then explore the case of Gevrey weights, which can be thought of as the interpolating case between the real analytic and smooth weights. In the case of Gevrey weights, we show that Bergman kernel can be approximated in certain Gevrey symbol class up to a Gevrey type small error, in the semiclassical limit. We will also introduce some microlocal analysis tools in the Gevrey setting, including Borel's lemma for symbols and complex stationary phase lemma. This talk is based on joint work with Hang Xu.

In this talk, I will present a full resolution of the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + {\rm div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a very natural Carleson condition (a parabolic analog of the so-called DKP-condition).

We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. We note that answer to this question was not known previously even in the  "small Carleson case", that is, when the Carleson norm of coefficients is sufficiently small.

In the elliptic case the analogous question was only fully resolved recently (2022) independently by two groups using two very different methods; one involving S. Hofmann, J. Pipher and the presenter, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges. The result is a joint work with L. Li and J. Pipher.

Consider minx≥ 0 f(x), where the objective function f: ℝ→ ℝ is smooth, and the variable is required to be nonnegative. A naive "squared variable" technique reformulates the problem to minv∈ ℝ f(v2). Note that the new problem is now unconstrained, and many algorithms, e.g., gradient descent, can be applied. In this talk, we discuss the disadvantages of this approach known for decades and the possible surprising fact of equivalence for the two problems in terms of (i) local minimizers and (ii) points satisfying the so-called second-order optimality conditions, which are keys for designing optimization algorithms. We further discuss extensions to the vector case (where the vector variable is required to have all entries nonnegative) and the matrix case (where the matrix variable is required to be a positive semidefinite) and demonstrate such an equivalence continues to hold.

In this talk, various aspects of the game of chess will be explored.  Like true philosophers, we will make random observations, pose rhetorical questions and draw strange parallels, all without claim to the truth, while touching on various topics from the nature of randomness to ways to maximize cognitive performance. Familiarity with the game is not required but will be helpful.