Departmental Colloquium
Spring 2025
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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January 7 Special Colloquium |
Rigid-analytic spaces are geometric objects described by convergent power series over the field of p-adic numbers Q_p. Just as complex-analytic spaces provide a robust framework for analytic geometry over C, rigid-analytic spaces offer a natural setting for analytic geometry over Q_p. In this talk, I will give a gentle introduction to the theory of rigid-analytic spaces and then discuss a version of Poincaré Duality for these spaces, as conjectured by Peter Scholze in 2012.
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January 23 Special Colloquium |
Chen Wan
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The Langlands program is a web of far-reaching and influential conjectures about connections between number theory, representation theory and geometry proposed. Within this program, the relative Langlands program has emerged as one of its most important and productive branches. In this talk, I will give an overview of key problems in the relative Langlands program, with a focus on the elegant theory of relative Langlands duality, recently developed by Ben-Zvi, Sakellaridis, and Venkatesh. I will also present some of my works in this area.
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February 27 |
Amie Wilkinson
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The centralizer $Z(f)$ of a diffeomorphism $f: M \to M$ of a closed manifold $M$ is the group of all diffeomorphisms commuting with $f$; it is the collection of dynamical symmetries of $f$. The centralizer of $f$ always contains the group $\langle f \rangle$ generated by $f$ as a normal subgroup, and conjecturally the two typically coincide (that is, ``the generic diffeomorphism has only trivial symmetries''). In this talk, I will describe some results and conjectures in a project with Danijela Damjanovic, Chengyang Wu and Disheng Xu that addresses the question: what happens when $Z(f)$ is bigger than $\langle f \rangle$?
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March 6 |
Mark Iwen
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Let $M$ be a smooth submanifold of $\mathbb{R}^n$ equipped with the Euclidean (chordal) metric. This talk will consider the smallest dimension, $m$, for which there exists a bi-Lipschitz function $f:M \to \mathbb{R}^m$ with bi-Lipschitz constants close to one. We will begin by presenting a bound for the embedding dimension $m$ from below in terms of the bi-Lipschitz constants of $f$ and the reach, volume, diameter, and dimension of $M$. We will then discuss how this lower bound can be applied to show that prior upper bounds by Eftekhari and Wakin on the minimal low-distortion embedding dimension of such manifolds using random matrices achieve near-optimal dependence on dimension, reach, and volume (even when compared against nonlinear competitors). Using this as motivation, we will then discuss non-linear so-called “terminal embeddings” of sets which allow for powerful extensions of the famous Johnson-Lindenstrauss Lemma beyond what any linear map can achieve. In addition to nearly preserving all distances within a set, these extensions also nearly preserve all distances from outside the set to elements in the set, all while remaining continuous! If you haven’t seen them before, come and meet them — you won’t regret it.
This talk will draw on joint work with various subsets of Rafael Chiclana Vega, Mark Roach, Benjamin Schmidt, and Arman Tavakoli.
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March 20 |
Jesse Wolfson
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How hard is it to solve a degree 7 polynomial? How complicated is a branched cover, an algebraic group, a local system? Resolvent problems, e.g. Hilbert's 13th problem, form a rich and challenging framework for asking these questions, and one that both invites and resists ideas and methods from Hodge theory (classical and p-adic). In this talk, I'll try to give the audience a sense of this area, what we can say, and what we are still possibly very far from being able to answer. Much of this is joint work with Benson Farb and Mark Kisin.
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March 25 Note special day |
Julia Pevtsova
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Tensor triangular geometry associates a rich geometric structure, the spectrum, to a “nice” category
where one can add, subtract and multiply objects. Examples include modules over a commutative
ring and representations of a finite group. I’ll give an introduction to tensor triangular geometry via
the classical notion of support and present some examples of calculations of spectra where pictures of
different degrees of beauty will appear. The examples will fall into two families: when the multiplicative
unit has a lot of symmetries – going back to the work of D. Quillen in the 70s - and when it does not.
Based on joint work with T. Barthel, D. Benson., S. Iyengar, H. Krause and V. Serganova, A. Sherman.
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March 27 |
Dejan Slepcev
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Motivated by the task of sampling measures in high dimensions we will discuss a several gradient flows in the spaces of measures, including the Wasserstein gradient flows of Maximum Mean Discrepancy and relative entropy, the Stein Variational Gradient Descent and a new Radon-Wasserstein gradient flows. For all the flows we will consider their deterministic interacting-particle approximations. The talk will highlight some of the properties of the flows and indicate their differences. In particular we will discuss how well can the interacting particles approximate the target measures. The talk is based on joint works with Elias Hess-Childs, Anna Korba, Sangmin Park, Lihan Wang, and Lantian Xu.
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April 3 |
Mauro Maggioni
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I will discuss recent results in two research directions at the intersection of statistical learning and modeling of dynamical systems.
First, we consider systems of interacting agents or particles, which are commonly used in models throughout the sciences, and can exhibit complex, emergent large-scale dynamics, even when driven by simple interaction laws. We consider the following inference problem: given only observations of trajectories of the agents in the system, can we learn the unknown laws of interactions? We cast this as an inverse problem, discuss when this problem is well-posed, construct estimators for the interaction kernels with provably good statistical and computational properties, even in the nonparametric estimation regime when only minimal information is provided about the form of such interaction laws. We also demonstrate numerically that the estimated systems can accurately reproduce the emergent behaviors of the original systems, even when the observations are so short that no emergent behavior was witnessed in the training data. This is joint work with M. Zhong (UH), S. Tang (UCSB), and F. Lu (JHU).
In the second part of the talk, I will discuss recent applications of deep learning in the context of digital twins in cardiology, and in particular the use of operator learning architectures for predicting solutions of parametric PDEs, or functionals thereof, on a family of diffeomorphic domains, which we apply to the prediction of medically relevant electrophysiological features of heart digital twins. This is joint work with M. Yin (JHU), N. Charon (UH), R. Brody (JHU), L. Lu (Yale), D. Popescu, and N. Trayanova (JHU).
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April 8 Note special day |
Claudia Miller
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In the first part, I will describe Kähler differentials and derivations and some of the history behind the classic Lipman-Zariski Conjecture, as well as a generalized question proposed by Graf. Together with Vassiliadou, we give a partial answer to Graf’s question for a certain class of varieties. In the second portion, we turn our attention to higher order differential operators, finding explicit generators and free resolutions in some low orders for the hypersurfaces studied by Bernstein-Gel’fand-Gel’fand and Vigué. This is joint work with Diethorn, Jeffries, Packauskas, Pollitz, Rahmati, and Vassiliadou. No specific algebraic or geometric background is required.
Note, Dr. Miller will also give a Career Path Talk for students on April 8 (Tuesday) at 2pm in LCB 222 and there will be coffee with the speaker and AWM on April 9 (Wednesday). To join for coffee, meet at 2pm in the JWB lounge and then the group will walk to a coffee shop.
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April 10 |
Ben Antieau
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I will describe the results of joint work with Achim Krause and Thomas Nikolaus on a computer algorithm to compute the algebraic K-groups of simple rings like $\mathbb{Z}/4$ or $\mathbb{Z}/9$. No background in algebraic K-theory is assumed. I will then explain how theoretical breakthroughs in p-adic cohomology theories, especially the theory of prismatic cohomology, make our algorithm possible.
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April 24 |
Rustum Choksi
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TBD
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