| Stephen Abkin |
Sign-Switched Antithetic Coupling for Signed Gradient-Particle Methods |
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Particle methods approximate solutions of differential equations using moving particles.
Each particle carries a positive or negative weight, indicating whether it adds to or subtracts from the reconstructed solution.
Random movements represent diffusion, but they also introduce sampling error.
Averaging several simulations reduces this error at the cost of more computation.
We study how to obtain accurate averages with fewer simulations for one-dimensional equations describing transport and diffusion.
We run simulations in pairs and match their particles by position.
Particles with weights of the same sign receive opposite random movements, while those with opposite signs receive the same movement.
This helps the random fluctuations in the two solutions cancel when averaged.
Each simulation on its own retains the same distribution of results as an independent run.
With particle positions and partners fixed, we prove that this rule minimizes the random variation contributed by each matched pair at
the final diffusion step.
We also calculate the reduction exactly.
In tests on Burgers' equation, our method reaches a common accuracy target in about 40% less time than matching only particles with
weights of the same sign.
Further tests show how the benefit varies with the starting solution, observation time, and equation.
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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 |
Algorithmic Bounds for Bond-Edge Optimization under Strict DNA Self-Assembly Constraints |
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Within the flexible tile model of DNA self-assembly, where vertices represent branched junction molecules and
edges represent bonds, we study the problem of minimizing the number of bond-edge types needed to assemble a
target graph without also forming any nonisomorphic graph of the same or smaller order.
We develop an algorithmic framework for bounding this minimum using 2-switch operations.
Efficiently computable criteria, including triangle-count arguments, certify when edge pairs must receive distinct bond-edge types.
These constraints define a conflict graph whose chromatic number gives a lower bound, and resolving ambiguous cases through
direct isomorphism testing strengthens it.
The resulting algorithms make it practical to analyze graphs that lack a complete theoretical characterization.
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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 |
On the existence of hyperelliptic curves over Q(T) with certain Jacobian ranks< |
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Let y2 = f(x,T) be a hyperelliptic curve of genus g ≥ 1, defined over Q(T).
We prove the existence of infinitely many imaginary hyperelliptic curves with a fixed genus g having a certain
rank for 5≤ r ≤ 4g+2, and a similar result for real hyperelliptic curves with a fixed genus g having a
certain rank for 6 ≤ r ≤ 4g+4.
We begin by constructing such curves and prove the rank using two methods.
First, we apply the generalized Nagao conjecture, which relates the first moment and the rank of the Jacobian
variety JX(Q(T)), and that the conjecture holds for our curves,
making the result unconditional.
Furthermore, we explicitly construct rational points in the Mordell-Weil group and use Shioda-Tate to prove that the
rank is equal to r.
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| Marcus Lapina |
Connections Between Hypergraph and Vector Parking Functions |
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Parking functions are well-studied objects in the field of combinatorics.
There are many ways to generalize these objects.
We work with two ways to generalize them: hypergraph parking functions and vector parking functions.
Our main goal is to classify which types of hypergraphs, or more specifically multihypergraphs, have the property that
their set of hypergraph parking functions are the same as the set of vector parking.
This is a generalization of a theorem of Gaydarov and Hopkins from 2016.
This classification is motivated by chip-firing and the problem of counting superstable configurations on a
hypergraph/multihypergraph which have a one-to-one correspondence with hypergraph parking functions.
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| Dominic Motekaitis |
Applying the approximate SU(3) and SU(4) to study atomic nuclei |
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Recent experiments in measuring exotic nuclear states such as Carbon-12 and Beryllium-8, especially their alpha-clustered states, have
necessitated a physics-informed framework to analyze experimental data.
Alpha clustering offers an elegant explanation to nuclear stability beyond magic-number considerations from the nuclear shell model, as
well as has historically explained stellar astrophysical processes (the famous Hoyle state).
However, alpha clustering is challenging to describe in terms of first principles methods.
To address this, we can exploit the emergent Elliott SU(3) symmetry as well as the Wigner SU(4) spin/isospin symmetry,
under which the alpha particle has a particularly simple structure.
By leveraging the SU(3) symmetry of the nuclear shell model, we can explain trends in the relative strengths of alpha-particle transfer
reactions from group theory alone.
This work discusses the application of tools from group theory to studying exotic nuclei and understanding the dynamics of novel nuclear
reactions.
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| Charlie Pyle |
Modal Analysis of a Chevrolet Silverado Truck Using Implicit and Explicit Finite Element Simulations in LS-DYNA. |
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Modal analysis characterizes structural vibrations critical for vehicle dynamics and passenger ride comfort.
We study the correlation between linear implicit eigenvalue theory and nonlinear explicit transient dynamics using a
full-scale finite element model of a Chevrolet Silverado pickup truck in LS-DYNA.
An implicit simulation is used to calculate the first 100 eigenmodes capturing rigid-body translations, unconstrained
mechanism motions, and structural vibrations.
We explicitly simulate a drop test with sensors placed around the truck and use Fast Fourier Transforms to extract
vibrational peaks and reconstruct 69 structural modes.
These simulations have applications to passenger comfort, fault detection, and vehicle health monitoring.
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| Uihyeon Lee |
Asymptotic Behavior of N-gons Under Cyclic Snap Operation |
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For any N-gon with N>3, we define the 'snap' operation, which takes a vertex vi
to the midpoint of the diagonal of its adjacent vertices.
We show that if we cyclically repeat this operation infinitely many times, the vertices of the N-gon
will converge to one point defined by a convex combination its vertices.
Moreover, we show that the area of a convex quadrilateral under this process halves at each iteration. |
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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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| Mark Shiliaev |
On SI-groups and subgroups of the first Grigorchuk group. |
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A group is considered SI if every chief factor of that group is abelian.
This property was introduced by G. Kurosh and K. Chernikov in 1947 as a weaker and more general version of solvability.
However, there are no known examples of an SI group that has a non SI subgroup and does not contain a
free group on two generators.
One previously proposed candidate to be such a group was the first Grigorchuk group.
The first Grigorchuk group is SI because it is a branch group, which is not a property that its subgroups possess.
We show that all subgroups of the Grigorchuk group are still SI.
In the process we also show that the SI property lifts from a normal subgroup to a group if the index of that subgroup is a
power of 2.
We also show that if every finitely generated subgroup of some group G is SI, then every subgroup of G is SI.
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| Mckinley Xie |
Multiple Distance Ramsey Bounds for Graphs in Euclidean Spaces |
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Given some finite set of real numbers A, consider the infinite graph on Rn such that
there is an edge between a pair of vertices if the Euclidean distance between them lies in A.
Then, given some arbitrary finite graph H, what is the number of colors necessary to color Rn
such that it contains no monochromatic copy of H?
We will provide a lower bound to this question through the use of partition rank and flattening rank methods.
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