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SUMMARY:Northfield Undergraduate Mathematics Symposium
DESCRIPTION:Each year Carleton and St. Olaf students work on a variety of
interesting research problems in mathematics, both here in Northfield and
around the country. Several of these students will be sharing the work th
ey did this past summer at the 2016 Northfield Undergraduate Mathematics S
ymposium. Please join us for as many of the talks as you can attend, as w
ell as for a pizza dinner. These talks all promise to be fascinating in t
heir own right, but each one also counts for half of the eight talks junio
r and senior math and math/stats majors need to attend to satisfy their le
cture attendance requirement. Below is a schedule for the symposium, foll
owed by the titles and abstracts for the talks.\nSchedule of Talks\n3:40 –
4:00 pm m-gapped Progressions and van der Waerden Numbers\n
Daniel Lewitz, Carleton College\n4:05 – 4:25 pm An Invo
lution Proof of a Borwein Theorem for Overpartitions\n
Conrad Parker and Melanie Stevenson, St. Olaf College\n4:30 – 4:5
0 pm A Combinatorial Model of Quantum Skew Symmetric Matrices\n
Eleanor Campbell, Carleton College\n4:55 – 5:15 pm P
artial Differential Modeling in the Kidney\n Qu
inton Neville, St. Olaf College\n5:20 – 5:55 pm Dinner (will be provided
)\n6:00 – 6:20 pm Finding Minimal Spanning Forests in a Graph\n
Abdel-Rahman Madkour and Philip Nadolny, St. Olaf Coll
ege\n6:25 – 6:45 pm The Metric s-t path Travelling Salesperson Problem a
nd the\n Randomized Christofides Algorithm\n
Shatian Wang, Carleton College\nTitles and Abst
racts\nTitle: m-gapped Progressions and van der Waerden Numbers\nSpeaker:
Daniel Lewitz, Carleton College\nAbstract: An m-gapped progression is a
generalization of an arithmetic progression in which the gaps in the prog
ression only need to belong to a set of $m$ elements, rather than all be t
he same. In the same way that arithmetic progressions pertain to the van d
er Waerden numbers, W(k; r), m-gapped progressions pertain to a function w
e call Bm(k; r). We will examine some results about the nature of the func
tion Bm(k; r), both in general and for special case when r = 2. In particu
lar, we will show how there are exact results for Bm(k; r) when k is relat
ively small. This work was done with Catherine Cooper, Trinity College; A
lex Stoll, Clemson University; and Bruce Landman, University of West Georg
ia.\nTitle: An Involution Proof of a Borwein Theorem for Overpartitions\n
Speakers: Conrad Parker and Melanie Stevenson, St. Olaf College\nAbstract
: In 1990, P. Borwein conjectured a + − − sign pattern for polynomials co
unting certain signed integer partitions. We conjecture a + − 0 pattern f
or the generating function for overpartitions into parts not divisible by
3 and give an involution-based proof of the 0 case of this conjecture usin
g pentagonal numbers and the Jacobi triple product. We also share a proof
of a generalization of this case involving quadratic nonresidues modulo a
prime.\nTitle: A Combinatorial Model of Quantum Skew Symmetric Matrices\
nSpeaker: Eleanor Campbell, Carleton College\nAbstract: The quantized co
ordinate ring of m × n Quantum matrices, or simply quantum matrices, holds
deep connections to the theory of totally nonnegative matrices, wave inte
ractions and knot theory. We examine the less understood theory of quantu
m skew-symmetric matrices Oq(Skn) over a field k. This algebra is known t
o be generated by a set of generators yi,j, 1 ≤ i < j ≤ n, which satisfy c
ertain commutativity relations dependent on some element q ∈ k. We view O
q(Skn) from a combinatorial perspective. We prove Oq(Skn) is isomorphic t
o an algebra called An over k, defined graphically. An is generated by el
ements xij, where each xij is the sum of the weights of paths from i to j
in a particular directed graph. The weights are obtained from elements of
a space with simpler commutativity relations dependent on q. Using induc
tive methods on the graph, we prove that the generators of An satisfy the
same commutativity relations of Oq(Skn), allowing for a new combinatorial
perspective that may be used to study this algebra. \nTitle: Partial Diff
erential Modeling in the Kidney\nSpeaker: Quinton Neville, St. Olaf Colle
ge\nAbstract: The kidney, a diverse and complicated biological system, is
most simply charged with the production of urine. At a deeper level, the
functional unit that facilitates this process is the nephron. The nephro
n utilizes the Tubuloglomerular Feedback (TGF) System to monitor chloride
levels during the production of urine, making sure the body is not retaini
ng or losing too much. In mammals, there are two types of nephrons: short
-looped and long-looped. The key difference between the short-loop and th
e long-loop is the existence of the Thin Ascending Limb (THAL) in the long
-loop, which has differing properties for spatially varying permeability a
nd maximum transport rate of chloride. Additionally, there is biological
evidence of an association between desert mammals' ability to produce more
highly concentrated urine and a higher percentage of long-looped nephrons
in their kidneys. The long-looped nephron, however, is not well characte
rized biologically, which provides motivation to attempt to explain this a
ssociation mathematically. Thus, we developed a partial differential mode
l of a long-looped nephron, derived a characteristic ordinary differential
equation, varied the length of the model THAL, and performed a bifurcatio
n analysis of the long-looped TGF system. Our analysis indicates that the
re is a higher tendency towards oscillatory solutions in the long-looped T
GF system than in the short-looped, and increasing the length of the THAL
may create a more stable TGF system. \nTitle: Finding Minimal Spanning Fo
rests in a Graph\nSpeakers: Abdel-Rahman Madkour and Philip Nadolny, St.
Olaf College\nAbstract: In the computation of multidimensional persistent
homology, a popular tool in topological data analysis, a family of planar
graphs arises. We have studied the problem of partitioning these graphs
in a way that will be useful for parallelizing the persistent homology cal
culation. Specifically, we desire to partition an edge-weighted, undirect
ed graph G into k connected components, G1, . . . , Gk. Let wi be the wei
ght of a minimum spanning tree in component Gi. For our purposes, an idea
l partition is one that minimizes max{w1,...,wk}. This problem is known t
o be NP-hard in the case of general graphs and we are unable to find this
specific problem in the graph partitioning literature. We propose two app
roximation algorithms, one that uses a dynamic programming strategy and on
e that uses a spectral clustering approach, that produce near-optimal part
itions in practice on a family of test graphs. We present detailed descri
ptions of these algorithms and the analysis of empirical performance data.
\nTitle: The Metric s-t path Travelling Salesperson Problem and the Rando
mized Christofides Algorithm\nSpeaker: Shatian Wang, Carleton College\nAb
stract: In the well-known metric Traveling Salesman Problem (metric TSP),
a complete graph G= (V, E) is given with nonnegative metric edge costs.
The goal is to find a Hamiltonian circuit in G with minimum cost. The Chr
istofides heuristic (1976) gives a nice purely combinatorial 1.5-approxima
tion algorithm for the metric TSP. An important variant of the metric TSP
is the metric s-t path TSP, in which two fixed vertices, s and t are give
n, and the goal is to find a min-cost Hamiltonian path from s to t. The C
hristofides heuristic can be easily extended to this s-t path variant, but
only with an approximation ratio of 5/3. An, Kleinberg and Shmoys (2012)
made the first improvement to 5/3 with an LP based algorithm, the randomi
zed Christofides algorithm, and achieved an approximation ratio of 1.618.
This LP based analysis has inspired a sequence of later breakthroughs, in
cluding an 1.6 bound by Sebo (2013) and an 1.566 bound by Gottschalk and V
ygen (2016). If time permits, a more general problem, the connected T-joi
n problem, will be discussed at the end of the talk.\n
LOCATION:CMC 206
URL:https://apps.carleton.edu/curricular/math/events/?event_id=1454013
LAST-MODIFIED:20161013T113939
CREATED:20160816T111422
DTSTAMP:20200710T180919
DTSTART:20161013T154000
DURATION:PT3H20M0S
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