Abstract: Sergey Novikov was one of the most important mathematicians of the second half of the twentieth century. His work impacted algebraic topology, surgery theory of manifolds, foliation theory, integrable systems, and mathematical physics, as well as many other subjects. He was the recipient of numerous major prizes, including the 1970 Fields Medal and the 2005 Wolf Prize. Novikov also had a huge impact on the Maryland mathematics department, where he served as a Distinguished University Professor from 1997 until his retirement in 2017. This workshop will review some of Novikov’s many contributions, and also will feature talks about how the ideas and methods that he developed are leading to current and future research in mathematics and physics.
Abstract: In this session, the RIT teams working on the Higher-dimensionality project and the Secure Assisted Quantum Computation project will present the work they did this semester.
Abstract: I will present two different compactifications of the moduli space of surfaces fibered in log Calabi-Yau pairs, coming from a generalization of quasimap theory and from KSBA-stability. This is based on a series of joint works, with Andrea Di Lorenzo; Roberto Svaldi and Junyan Zhao
Abstract: Randomness is a way to discuss generic or typical behavior in a (class of) group(s). In this talk, I will discuss random quotients of certain classes of groups. Quotients of hyperbolic groups (groups that act geometrically on a hyperbolic space) and their generalizations have long been a powerful tool for proving strong algebraic results. I will focus on random quotients of acylindrical and hierarchically hyperbolic groups (HHGs), two generalizations of hyperbolic groups that include mapping class groups, most CAT(0) cubical groups including right-angled Artin and Coxeter groups, many 3–manifold groups, and various combinations of such groups. In this context, I will explain why a random quotient of an HHG that does not split as a direct product is again an HHG, definitively showing that the class of HHGs is quite broad. I will also describe how the result can also be applied to understand the geometry of random quotients of hyperbolic and relatively hyperbolic groups. This is joint work with Dan Berlyne, Giorgio Mangioni, Thomas Ng, and Alexander Rasmussen.
Abstract: Robustness is a fundamental concept in systems science and engineering. It is a critical consideration in all inference and decision-making problems. It has recently surfaced again in the context of machine learning (ML), reinforcement learning (RL) and artificial intelligence (AI). We describe a novel and unifying theory of robustness for ML/RL/AI emanating from our much earlier fundamental results on robust output feedback control for general systems (including nonlinear, HMM and set-valued). We briefly summarize this theory and the universal solution it provides consisting of two coupled HJB equations. These earlier results rigorously established the equivalence of three seemingly unrelated problems: the robust output feedback control problem, a partially observed differential game, and a partially observed risk sensitive stochastic control problem. We first show that the “four block” view of the above results leads naturally to a similar formulation of the robust ML problem, and to a rigorous path to analyze robustness and attack resiliency in ML. Then we describe a recent risk-sensitive approach, using an exponential criterion in deep learning, that explains the convergence of stochastic gradients despite over-parametrization. Finally, we describe our most recent results on robust and risk sensitive RL for control, using exponential rewards, that emerge from our earlier theory, with the important new extension that the models are now unknown. We show how all forms of regularized RL can be derived from our theory, including KL and Entropy regularization, relation to probabilistic graphical models, distributional robustness. The deeper reason for this unification emerges: it is the fundamental tradeoff between performance and risk measures in decision making, via rigorous duality. We close with open problems and future research directions.
Abstract: Kähler quantization provides a bridge between infinite-dimensional geometric objects in Kähler geometry and finite-dimensional data arising from spaces of holomorphic sections. In this talk, I will first review this correspondence in the ample case, where it is well understood and plays a central role in the study of canonical metrics. I will then explain how this picture can be extended beyond the ample setting, where smooth positively curved metrics are no longer available. In particular, I will describe how the Monge–Ampère energy can still be recovered from finite-dimensional approximations in the semipositive and big setting. Finally, if time permits, I will outline the idea of the proof.
Abstract: Two 20 minute talks. The first is from Chris Metzler
A machine learning based approach to phase retrieval, with applications to seeing through and around obstacles.
The second talk is from Krishna Bodla
Title: "MCTS-Guided Test-Time Scaling for Verifiable Mathematical Reasoning"
Abstract: "Large language models (LLMs) have demonstrated impressive performance on isolated mathematical problems, yet they remain brittle on multi-step, olympiad-level reasoning tasks where an early error propagates irrecoverably through the entire solution chain. We propose a Monte Carlo Tree Search (MCTS) framework that reframes formal mathematical proof search as an iterative, self-correcting process operating entirely at test time requiring no additional model training or reinforcement learning fine-tuning. Our system uses an LLM to decompose the current proof state into a focused sub-goal one step at a time, treats each decomposition as a tree node, and employs a dual reward signal combining a critic LLM (process reward) with the Lean~4 formal verifier (outcome reward) to score and backpropagate node quality. An adaptive temperature schedule encourages diverse exploration early and focused exploitation as the search deepens. We benchmark this pipeline on MiniF2F, PutnamBench, and MathOlympiadBench using five prover models at 7B--32B scale."
Abstract: In applications, it is often impossible to measure the phase of a signal, and one must recover it from some additional physical "redundancy" present in the problem. Phase retrieval problems arise in diverse fields, ranging from crystallography to quantum mechanics. Mathematically, the analysis of the problem unites many fields, including functional analysis, harmonic analysis, PDE, AI, probability, and more. I will discuss recent advances on this topic in a way suitable for a general audience.
Abstract: Hambly and Jordan (2004) introduced the series-parallel graph, a random hierarchial lattice that is easy to define: Start with the graph consisting of one edge connecting two terminal nodes. At each subsequent step of the construction, perform independent coin flips for each edge of the graph, and replace the edge by two edges in series if the coin is heads-up or two edges in parallel if tails. This results in a sequence of random graphs, which can be interpreted as a resistor network. Hambly and Jordan showed that the logarithm of the resistance grows linearly if the coins are biased to land more often heads-up. In this talk, I will discuss what happens in the critical case when fair coins are used. Starting with a new recursive distributional equation (RDE) proposed by Gurel-Gurevich, I develop a framework for analyzing RDE's based on parabolic PDE theory and use this to characterize the asymptotic behavior of the log. resistance. In the sub- or supercritical case (where the coins are biased), I discuss a tantalizing connection to the Fisher-KPP equation and front propagation.
Abstract: Using a self-generated hypoxic assay, the amoeba Dictyostelium discoideum exhibits a striking collective aerotactic behavior: when a colony is confined, cells rapidly consume the available oxygen and organize into a dense ring that propagates outward at constant speed and density. To understand this phenomenon, we introduce a simple PDE framework based on a “go-or-grow” mechanism, capturing the interplay between cell division and aerotactic response. This model gives rise to traveling wave solutions and reveals a dichotomy between pulled and pushed waves, depending on the strength of aerotaxis. To further investigate this transition between pulled and pushed waves, we develop a stochastic finite-population counterpart and analyze it through the perspective of ancestral lineages. In the large-population limit, this approach leads to an alternative PDE model. Despite the degeneracy of this latter PDE, we are able to establish existence, uniqueness, and describe the long-time behavior of solutions. In particular, we identify a transition in the asymptotic regime as the aerotactic sensitivity varies, using a recently introduced tool known as the shape defect function, which quantifies the deviation from traveling wave profiles. Finally, we consider a related experiment in Acanthamoeba. By incorporating collision effects into a mesoscopic model, we derive a density-dependent description that highlights key differences between the two systems. These modeling predictions are supported by experimental observations.