| Stephen Abkin |
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| Yoav Binyamin |
A Combinatorial Game of Doom and Despair |
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For this 2-player Nim game, you start with an n-tier pyramid of chips.
Player 1 goes first, and can take any chip that does not have two chips directly above it.
Upon doing so, it will also remove all chips supported by the removed chip.
Player 2 does the same with the modified pyramid and play continues until the final chip is taken,
and the last to take a chip is the winner.
We provide a theorem on the number of game states and number of game states up to ``isomorphism,'' as well as a
full criterion for game states to be isomorphic, enforcing an interesting meaning of isomorphism on Dyck paths.
Additionally, we provide new conjectures on the game's nimber values and optimal strategy using empirical evidence.
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| Jack Bollenbacher |
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| Samuel Gu |
Investigating the Strength of Eisermann's Modulo 32 Ribbon Obstruction |
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Determining if a link is ribbon is a central problem in geometric topology.
While classical algebraic obstructions, such as vanishing linking numbers lk=0, zero signature
σ=0, and the Fox-Milnor condition, are used to identify slice suspects, Eisermann recently
introduced a modern necessary condition based on a modulo 32 congruence.
This project investigates whether this condition is strictly stronger than the classical suite for multi-component links.
We conducted a computational sweep of over 30,000 links using SnapPy and SageMath, analyzing the 14-crossing and 15-crossing datasets.
We sought "imposters" that satisfy classical algebraic sliceness but fail the Modulo 32 test.
No counterexamples were found; all 357 suspects at 15 crossings satisfied the condition and were verified as ribbon.
We also developed an analytic framework for the 2-component case, providing evidence that for low-complexity links, classical criteria may be sufficient to satisfy this modern obstruction.
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| Joshua Im |
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| Marcus Lapina |
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| Uihyeon Lee |
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| Dani Scoville |
Exploring Tipping Cascades in Numerical Simulations of Antarctic Ice Shelves |
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Tipping cascades occur in networks of coupled dynamical systems where nodes on the network can undergo saddle-node
bifurcations, and where interactions between nodes can cause system-wide cascades of tipping behavior.
We consider the tipping behavior of ocean melting of ice shelves around the Antarctic Ice Sheet, which can be
described by a simple mathematical model of ocean circulation and melt.
Complex numerical models of Antarctic ice shelves have shown that shelf cavities generally exist in either high or
low melt rate steady states.
We use a network of coupled models of Antarctic ice shelf melt to explore the phase space of this dynamic network of
interacting ice shelves around Antarctica.
We interrogate controls on the coupled tipping behavior through analysis of the behavior of many randomly
generated ice shelf systems.
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| Aaryan Sharma |
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| Surya Shetty |
LLM-IDEA: An AI Agent That Knows When the Data Cannot Answer the Question |
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Large language models are increasingly used as automated scientists: they propose equations to explain data,
design experiments, and revise their models.
But a model that fits the data perfectly can still be wrong in ways the data can never reveal, because some
parameter combinations produce identical observations.
This is the classical problem of identifiability, well studied in control theory and systems biology but almost
absent from the recent literature on AI-driven discovery.
We present LLM-IDEA, an agentic framework that treats identifiability as a first-class diagnostic.
Before proposing a new experiment, the agent computes whether its current model's parameters can be determined
from the observations available, names the directions in parameter space that remain ambiguous, and, when no
experiment in the allowed class can resolve them, certifies that fact rather than continuing to search.
We validate the diagnostic on classical dynamical systems, on a custom physics world with hidden structure, and on
the benchmark used by the closest competing system, where it identifies in advance the exact cases where no further
experimentation can help.
A controlled factorial study shows that combining the diagnostic with a diverse family of proposer agents reliably
reaches discovery depths that neither achieves alone.
This is joint work with Dr. Ulisses Braga-Neto (TAMIDS Scientific Machine Learning Lab). |
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| Mckinley Xie |
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