Claremont Center for the Mathematical Sciences

Pre-Colloquium Non-Colloquium Party
2018-09-05 16:00 - 18:00 (America/Los_Angeles) | Millikan Courtyard, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
The traditional year-opening social event for the Claremont Colleges Mathematics Community will be held in the Millikan Courtyard. Spouses, partners, and family are welcome. Professors Ali Nadim (CGU) and Blerta Shtylla (POM), co-chairs, hope to see everyone there for refreshments, and other pleasant pursuits.

Applied math organizational meeting
2018-09-10 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
We will have an organizational meeting for the applied math seminar today. Anyone who is interested in suggesting speakers and/or organizing applied math seminar is welcome to come.

Small representations of integers by integral quadratic form (Lenny Fukshansky, CMC)
2018-09-11 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Given an isotropic integral quadratic form which assumes a value t, we investigate the distribution of integer points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of quadratic forms, we produce bounds on height of points outside of some algebraic sets in a quadratic space at which the form assumes the value t. Our bounds on height are explicit in terms of heights of the form, the space, the algebraic set and the value t. Joint work with W. K. Chan.
The Fall 2018 organizational meeting for the ANTC seminar will be held at noon in the same room, preceding the talk.

An Algebra of Arcs and Knots on a Surface (Helen Wong, CMC)
2018-09-12 16:15 - 17:15 | Argue Auditorium, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
The end of the previous century saw radical changes to three-dimensional topology, which arose from two completely different approaches. One breakthrough came from Bill Thurston’s introduction of hyperbolic geometry into the field. The second one came from the Vaughn Jones’s discovery of a new “quantum” invariant for knots that brought in insight and techniques from mathematical physics and non-commutative algebra. It is widely believed that the two approaches are related.
In this talk, we will focus on a point of connection introduced over five years ago by J. Roger and T. Yang. They defined an algebra of arcs and knots on a surface that is both obviously related to the quantum invariants and motivated by hyperbolic geometry. We will give a overview of what’s known about this algebra of arcs and knots on a surface. Warning: there will be lots of pictures of arcs and knots on surfaces.

Diffusion, Social Networks, and Logic (Pavel Naumov, CMC)
2018-09-17 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
Once a new commercial product, technology, political opinion, or social norm is adopted by a few people, these few often put peer pressure on others to consider adopting it as well. Those who adopt next put even more pressure on the rest of the population. This cascading “epidemic” effect is often called diffusion in social networks. There are many natural questions that can be asked about diffusion. Which initial group of people should get “infected” by a new product to ensure its adoption by the largest possible group? Which group should be convinced that an idea is bad, in order to avoid its wide spread? How does marketing affect the diffusion? In this talk I will introduce the most commonly used mathematical model of diffusion and talk about several of my papers on logical systems that capture properties of this model.

Inversions for reduced words (Sami Assaf, USC)
2018-09-18 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
The number of inversions of a permutation is an important statistic that arises in many contexts, including as the minimum number of simple transpositions needed to express the permutation and, equivalently, as the rank function for weak Bruhat order on the symmetric group. In this talk, I’ll describe an analogous statistic on the reduced expressions for a given permutation that turns the Coxeter graph into a ranked poset with unique maximal element. This statistic simplifies greatly when shifting our paradigm from reduced expressions to balanced tableaux, and I’ll use this simplification to give an elementary proof computing the diameter of the Coxeter graph for the long permutation.
This talk is elementary and assumes no background other than passing familiarity with the symmetric group.

Fall 2018 Poster Session
2018-09-19 16:15 - 17:15 | Argue Auditorium, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
CLAREMONT CENTER for MATHEMATICAL SCIENCES
Fall 2018 Poster Session
Click here for poster abstracts.
Computing Eigenmodes of the Laplace-Beltrami Operator by Using Radial Basis Functions
by Vladimir Delengov, Chiu-Yen Kao
Claremont Graduate University
Covariance-based Dissimilarity Measures Applied to Clustering Wide-sense Stationary Ergodic Processes
by Nan Rao, Qidi Peng, Ran Zhao
Claremont Graduate University
Generalized Covariation of Symmetric -stable Distributions
by Yujia Ding, Qidi Peng
Claremont Graduate University
Learning to Fail: Predicting Fracture Evolution in Brittle Materials using Recurrent Graph Convolutional Neural Networks
by Yadong Ruan, Zhengming Song, Max Schwarzer, Bryce Rogan, Diana Lee, Allon G. Percus, Viet T. Chau, Gowri Srinivasan, Hari Viswanathan, Bryan Moore
Claremont Graduate University
Tribracket Modules
by Yingqi Shi, Deanna Needell, Sam Nelson, Yingqi Shi
Claremont Mckenna College
Equiangular Tight Frames and Corresponding Lattices
by Jessie Xin, Lenny Fukshansky
Claremont Mckenna College
Quandle Coloring Quivers
by Karina Cho, Sam Nelson
Harvey Mudd College
Effective Bounds for Traces of Maass-Poincar Series
by Havi Ellers
Harvey Mudd College
Personal Beliefs and Election Forecasts
by Harry Bendekgey
Pomona College
Hemoglobin Response to Higher Order Gene Interactions
by Sylvia Akueze Nwakanma, Lillian Gonzales, Rosa Garza
Pomona College
Exploring Cell Differentiation Trajectories through Data Reduction
by Gianna Wu, Michelle Li
Pomona College

Quandle coloring quivers (Sam Nelson, CMC)
2018-09-25 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Given a finite quandle $X$, a set $S subset mathrm{Hom}(X,X)$ of quandle endomoprhisms, and an oriented knot or link $L$, we construct a quiver-valued invariant of oriented knots and links. This quiver categorifies the quandle counting invariant in the most literal sense and can be used to define many enhancements of the counting invariant. This is joint work with Harvey Mudd College student Karina Cho.

Snow Business: Scientific Computing in the Movies and Beyond (Joseph Teran, UCLA)
2018-09-26 16:15 - 17:15 | Argue Auditorium, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
New applications of scientific computing for solid and fluid mechanics problems include simulation of virtual materials in movie visual effects and virtual surgery. Both disciplines demand physically realistic dynamics for materials like water, smoke, fire, and soft tissues. New algorithms are required for each area. Teran will speak about the simulation techniques required in these fields and will share some recent results including: simulated surgical repair of biomechanical soft tissues; extreme deformation of elastic objects with contact; high resolution incompressible flow; and clothing and hair dynamics. He will also discuss a new algorithm used for simulating the dynamics of snow in Disney’s animated feature film, “Frozen.”

Reaction-Diffusion Equations under Perturbations of the Domain (Professor Jose Arrieta, Universidad Complutense de Madrid, Spain)
2018-09-27 13:15 - 14:15 | Shanahan 3465, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
We analyze the behavior of the asymptotic dynamics of dissipative reaction-diffusion equations with Neumann boundary conditions when the domain where the equation is posed undergoes certain perturbation. We will focus on the behavior of the stationary solutions, their local unstable manifolds and the attractors.
We will consider “regular” perturbations of the domain, that is, perturbations for which the spectra of the Laplace operator behaves continuously. In this case, it turns out that if all the equilibria of the unperturbed system are nondegenerate (hyperbolic), then both the equilibria and their local unstable manifolds behave continuously under the perturbation. Exploiting the gradient properties of the flow we will show that the “attractors” also behave continuously.
We may also consider some “non-regular” perturbations of the domain. In this situation, the problem needs to be studied and the technique adapted for each particular case. An interesting example of non regular perturbations is the “dumbbell domain” which consists in two domains joined by a very thin channel which degenerates to a line segment. We will describe the results obtained for this perturbation.

An Introduction to the Sato-Tate Conjecture (Edray Goins, Pomona College)
2018-10-02 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
In 1846, Ernst Eduard Kummer conjectured a distribution of values of a cubic Gauss sum after computing a few values by hand. This was forgotten about for nearly 100 years until John von Neumann and Herman Goldstine attempted to verify the conjecture as a way to test the new ENIAC machine in 1953. They found evidence that the conjecture was false, but trusted Kummer more than they did their digital computer. The conjecture would hold until 1979, when Roger Heath-Brown and Samuel Patterson proved it to be false.
A few years earlier in 1965, Mikio Sato and John Tate independently came up with a conjecture which gave the correct distribution of these cubic Gauss sums — although it was expressed slightly differently in terms of counting points of elliptic curves over finite fields. In this talk, we give an overview of the Sato-Tate Conjecture, present an approach by Jean-Pierre Serre following his paper from 1967, then sketch the 2006 proof of the conjecture following the ideas of Laurent Clozel, Michael Harris, Nicholas Shepherd-Barron and Richard Taylor.
Here are the slides of this lecture: Edray Goins’ slides.

Modeling Mechanisms of Ovulatory (Dys)Function (Erica Graham, Bryn Mawr College)
2018-10-03 16:15 - 17:15 | Argue Auditorium, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
A normally functioning menstrual cycle requires significant crosstalk between hormones originating in ovarian and brain tissues. Reproductive hormone dysregulation may disrupt function and can lead to infertility, as occurs in the common endocrine disorder polycystic ovarian syndrome (PCOS). In this talk, I will discuss a mathematical model of the ovulatory cycle that accounts for mechanisms of ovarian testosterone production and explore insulin-mediated ovulatory dysfunction. I will also explore additional model characteristics, via bifurcations and parameter sensitivity, and their respective clinical implications.

GEMS Workshop: Knots and how to tell them apart (Professor Sam Nelson, Claremont McKenna College)
2018-10-06 10:00 - 12:00 | Shanahan 1480, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
WHAT IS GEMS:
The Gateway to Exploring Mathematics program (GEMS) is a series of workshops that helps excite the interests and curiosity of young students in mathematics and science
GEMS meets once a month on a Saturday morning from 10:00 AM to 12:00 PM
GEMS is designed to reach 8th, 9th and 10th grade students who have an interest in mathematics and science
Participants interact with excellent and award winning faculty, staff, students and alumni from each of the seven Claremont Colleges
FALL 2018 DATES:
October 6, 2018
November 3, 2018
December 8, 2018
REGISTRATION:
To register for our first event on October 6, 2018 please click on the following link:
https://tinyurl.com/GEMS2018Fall
ANY QUESTIONS:
Please contact our 2018-2019 GEMS coordinator, Elsa Harris at Elsa.Harris@cgu.edu

State Polytopes of Combinatorial Neural Codes (Rob Davis, HMC)
2018-10-09 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Combinatorial neural codes are 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn in the plane as a Venn diagram-like figure. A sufficient condition to do so is for the code to have a property called k-inductively pierced. Gross, Obatake, and Youngs recently used toric algebra to show that a code on three neurons is 1-inductively pierced if and only if the toric ideal is trivial or generated by quadratics. No result is known for additional neurons in the same generality.
In this talk, we study two infinite classes of combinatorial neural codes in detail. For each code, we explicitly compute its universal Gröbner basis. This is done for the first class by recognizing that the codewords form a Lawrence-type matrix. With the second class, this is done by showing that the matrix is totally unimodular. These computations allow one to compute the state polytopes of the corresponding toric ideals, from which all distinct initial ideals may be computed efficiently. Moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.

Applications of Cayley Digraphs to Waring's Problem and Sum-Product Formulas (Yesim Demiroglu, Harvey Mudd)
2018-10-10 04:15 - 17:15 | Argue Auditorium, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: In this talk, we first present some elementary new proofs (using Cayley digraphs and spectral graph theory) for Waring’s problem over finite fields, and explain how in the process of re-proving these results, we obtain an original result that provides an analogue of Sarkozy’s theorem in the finite field setting (showing that any subset E of a finite field Fq for which |E| > (qk)/sqrt{q – 1}must contain at least two distinct elements whose difference is a kth power). Once we have our results for finite fields, we apply some classical mathematics to extend our Waring’s problem results to the context of general (not necessarily commutative) finite rings. In the second half of our talk, we present our sum-product results related to matrix rings over finite fields, which can again be proven using Cayley digraphs and spectral graph theory in an efficient way.

Agent-Based and Continuous Models of Locust Hopper Bands: The Role of Intermittent Motion, Alignment, Attraction and Repulsion (Andrew J. Bernoff, HMC)
2018-10-15 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
Locust swarms pose a major threat to agriculture, notably in northern Africa and the Middle East. In the early stages of aggregation, locusts form hopper bands. These are coordinated groups that march in columnar structures that are often kilometers long and may contain millions of individuals. We propose a model for the formation of locust hopper bands that incorporates intermittent motion, alignment with neighbors, and social attraction, all behaviors that have been validated in experiments. Using a particle-in-cell computational method, we simulate swarms of up to a million individuals, which is several orders of magnitude larger than what has previously appeared in the locust modeling literature. We observe hopper bands in this model forming as a fingering instability. Our model also allows homogenization to yield a system of partial integro-differential evolution equations. We identify a bifurcation from a uniform marching state to columnar structures, suggestive of the formation of hopper bands.

The Bateman—Horn Conjecture, Part I: heuristic derivation (Stephan Garcia, Pomona)
2018-10-16 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
The Bateman—Horn Conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the Green—Tao theorem, and a variety of famous conjectures, such as the Twin Prime Conjecture. In this expository talk, we start from basic principles and provide a heuristic argument in favor of the conjecture. This talk should be accessible to undergraduates with a background in modular arithmetic.

Great Expectations (Matthew Junge, Duke Univ.)
2018-10-17 16:15 - 17:15 | Argue Auditorium, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
The mean of a random quantity is supposed to confirm our expectations. What happens when it defies them? We will look at a few famous expected values; some old, some new, all great.

Isometric Circle Actions (Catherline Searle, Wichita State)
2018-10-24 16:15 - 17:15 | Argue Auditorium, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
I will begin by describing a number of important examples of isometric actions of circles in Euclidean space and their restrictions to subspaces of Euclidean space. The goal of the talk will be to see how isometric actions of circles and tori can be used to “recognize” the space on which they are acting.

Minimal Gaussian Partitions, Clustering Hardness and Voting (Steven Heilman, USC)
2018-10-29 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
A single soap bubble has a spherical shape since it minimizes its surface area subject to a fixed enclosed volume of air. When two soap bubbles collide, they form a “double-bubble” composed of three spherical caps. The double-bubble minimizes total surface area among all sets enclosing two fixed volumes. This was proven mathematically in a landmark result by Hutchings-Morgan-Ritore-Ros and Reichardt using the calculus of variations in the early 2000s. The analogous case of three or more Euclidean sets is considered difficult if not impossible. However, if we replace Lebesgue measure in these problems with the Gaussian measure, then recent work of myself (for 3 sets) and of Milman-Neeman (for any number of sets) can actually solve these problems. We also use the calculus of variations. We will discuss applications of this Gaussian “multi-bubble” problem to optimal clustering of data and to designing elections that are resilient to hacking.

Uniform asymptotic growth of symbolic powers (Robert Walker, University of Michigan)
2018-10-30 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980’s with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a brief introduction to AG, to two notions of taking powers of ideals (regular vs symbolic) in Noetherian commutative rings, and to the ideal containment problem that I study in my thesis. Part two of my talk will focus on stating the main results of my thesis in a user-ready form, giving a “comical” example or two of how to use them. At the risk of sounding like Paul Rudd in Ant-Man, I hope this talk with be awesome.

Saving Bats from Fungal Diseases with Linear Algebra (Nina Fefferman, U of Tennessee-Knoxville)
2018-10-31 16:15 - 17:15 | Argue Auditorium, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: Bats in North America have been dying off due to the invasion of a fungal disease (White Nose Syndrome). In this talk, I’ll present a very simple linear algebraic model to predict the magnitude of the die-offs. By comparing these models to some data about actual bat survival, my collaborator and I also hypothesized that the disease might be causing rapid evolution in the bat populations and this could give some populations better hope of surviving. I’ll go through these models and show how the predictions they make are different from models that don’t include bat evolution. I’ll also talk a little about some of the intervention strategies that have been proposed to help bat populations survive, and use these models to show how some of them might accidentally hurt rather than help, if we don’t figure out whether the main impacts of disease are evolutionary or immunological first.

GEMS Workshop: Exploring the fascinating world of prime numbers, Part I with Professor Adolfo Rumbos, from Pomona College.
2018-11-03 10:00 - 12:00 | Shanahan 1480, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
TOPIC: Exploring the fascinating world of prime numbers, Part I
The study of patterns in the sequence of prime numbers has fascinated mathematicians for centuries. Are there formulas that generate prime numbers? Are there patterns in the distribution of prime numbers and the distribution of gaps between consecutive primes? In this series of two workshops, beginning with the proof of the infinitude of the primes and modular arithmetic, we explore some facts about prime numbers, solve some puzzles related to primes, and survey a few of the questions that are still unsolved.
WHAT IS GEMS:
The Gateway to Exploring Mathematics program (GEMS) is a series of workshops that helps excite the interests and curiosity of young students in mathematics and science
GEMS meets once a month on a Saturday morning from 10:00 AM to 12:00 PM
GEMS is designed to reach 8th, 9th and 10th grade students who have an interest in mathematics and science
Participants interact with excellent and award winning faculty, staff, students and alumni from each of the seven Claremont Colleges
FALL 2018 DATES:
October 6, 2018
November 3, 2018
December 8, 2018
REGISTRATION:
To register for our next event on November 3, 2018 please click on the following link:
https://tinyurl.com/GEMS2018Fall2
ANY QUESTIONS:
Please contact our 2018-2019 GEMS coordinator, Elsa Harris at Elsa.Harris@cgu.edu

CFTP: the algorithm ERGM deserves, but not the one it needs right now (Matt Moores, University of Wollongong)
2018-11-05 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
The exchange algorithm enables Bayesian posterior inference for models with intractable likelihoods, such as Ising, Potts, or exponential random graph models (ERGM). Crucially, this algorithm relies on an auxiliary Markov chain to obtain an unbiased sample from the generative distribution of the model. It was originally proposed to use coupling from the past (CFTP) for this purpose, but this requires the Markov chain to be uniformly ergodic. In the case of the Ising model, coupling time increases super-exponentially for parameter values larger than the critical point. Alternatives to CFTP, such as perfect slice sampling or bounding chains for Swendsen-Wang, have been proposed for the Ising model. However, there are currently no suitable alternatives for ERGM, which also features a phase transition that can cause problems with convergence. This talk will review some recent work on simulation algorithms for ERGM and discuss how this problem might be addressed.

This is joint work with Kerrie Mengersen and Chris Drovandi (QUT, Australia), Antonietta Mira (USI Lugano, Switzerland), and Alberto Caimo (Dublin Inst. Tech., Ireland).

Turning probability into polynomials (Mark Huber, CMC)
2018-11-06 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Moment generating functions (Laplace transforms) are a means for transforming probability problems into problems involving polynomials. Here I will concentrate on the binomial distribution, and use the mgf to link this distributions probabilities directly to the binomial theorem. The mgf is also a key ingredient in Chernoff bounds, which give upper bounds on the tail probabilities of binomial distributions (aka partial sums of the binomial theorem). By employing the method of smoothing and tilting, it is possible to attain bounds on the tails that go down faster than the traditional approximation heuristic that uses the Central Limit Theorem.

The Legacy of Rudolph Kalman (Andrew Stuart, Caltech)
2018-11-07 16:15 - 17:15 | Argue Auditorium, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: In 1960 Rudolph Kalman published what is arguably the first paper to develop a systematic, principled approach to the use of data to improve the predictive capability of mathematical models. As our ability to gather data grows at an enormous rate, the importance of this work continues to grow too. The lecture will describe this paper, and developments that have stemmed from it, revolutionizing fields such space-craft control, weather prediction, oceanography, oil recovery, medical imaging and artificial intelligence. Some mathematical details will be also provided, but limited to simple concepts such as optimization and iteration; the talk is designed to be broadly accessible to anyone with an interest in quantitative science.

Crossing the Threshold: The Role of Demographic Stochasticity in the Evolution of Cooperation (Tom LoFaro, Gustavus Adolphus College)
2018-11-08 16:15 - 17:15 | Shanahan 3465, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
When Charles Darwin began writing “On the Origin of Species” he knew that explaining cooperative behavior in the context of “survival of the fittest” was problematic. In fact, this apparent contradiction puzzled ecologists for many years after. In this talk we will discuss a mathematical model of the evolution of cooperation developed by Doebeli, Blarer, and Ackermann that incorporates ideas from game theory into a standard population genetics model. We will show that if the model is viewed deterministically then cooperative behavior cannot spread from rarity. However, if birth rates are stochastic then cooperative behavior might spread. We will explore why this is so and describe conditions that increase the probability that cooperative behavior will become established.

Digital sequences for frequency hopping CDMA systems (Lenny Fukshansky, CMC)
2018-11-12 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
Frequency hopping is a method of transmitting signals by rapidly switching between many frequency channels, following some sequence of frequencies known to the transmitter and the receiver. This technique is used in the CDMA (code division multiple access) systems, and has many civilian and military applications. For successful transmission minimizing signal interference, we want to use sets of digital frequency sequences with minimal Hamming cross-correlation, which measures frequency overlaps with time shifts between two different sequences. We discuss a construction of a new family of one-coincidence sequences like this coming from some basic arithmetic of finite fields, which have some nice properties. This is joint work with Adib Shaar, and this talk is dedicated to his memory.

Cayley digraphs of matrix rings over finite fields (Yesim Demiroglu, HMC)
2018-11-13 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
In this talk we use the unit-graphs and the special unit-digraphs on matrix rings to show that every n x n nonzero matrix over F_q can be written as a sum of two SL_n-matrices when n>1. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and prove that if X is a subset of Mat_2 (F_q) with size |X| > (2 q^3 sqrt{q})/(q – 1), then X contains at least two distinct matrices whose difference has determinant $alpha$ for any $alpha in F_q^*$. Using this result we also prove a sum-product type result: if $A,B,C,D subseteq F_q$ satisfy $sqrt[4]{|A||B||C||D|}= Omega (q^{0.75})$ as q tends to infinity, then $(A – B)(C – D)$ equals all of $F_q$. In particular, if A is a subset of F_q with cardinality $|A| > frac{3}{2} q^{3/4}$, then the subset $(A – A) (A – A)$ equals all of $F_q$. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units. This talk should be accessible to undergraduates with some background in linear algebra.

Coupled Mechanochemical Multiscale Model to Study the Growth Regulation and Morphogenesis during Tissue Development (Weitao Chen, UCR)
2018-11-14 16:15 - 17:15 | Argue Auditorium, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Growth regulation and pattern formation are two main problems in developmental biol-
ogy. How cells know when to stop growing at certain tissue size with specic shape is an important
question in both developmental biology and regenerative medicine, and it is still an unsolved mystery
in many systems. During the growth, tissues and organs always exhibit self-government to some
extent. Cells stop proliferation precisely when the intended size of the tissue or organ is achieved.
Meanwhile, dierential cell shapes in space are integrated to give rise to well-organized overall struc-
ture. Uncontrolled growth of the cells in tissues or organs will lead to abnormal development or
fatal diseases such as cancer. Therefore, developing an extensible predictive mathematical model
for exploring the mechanisms involved in the tissue development is signicant for understanding
the fundamental principles in developmental biology, with a broad range of applications from tissue
engineering to biomanufacturing and biotech industry. Experimental data suggests that mechanical
properties of cells and chemical signals in both intracellular and extracellular domains play critical
roles in size control and shape formation. Here we develop a multiscale, mechochemical coupled
model of tissue growth control. This rst-of-class modeling approach provides sub-cellular details
to both mechanical properties and chemical signaling during tissue growth. This model is applied
to test competing hypotheses in the eld to resolve the highly debated question of how tissues reach
their nal size, as well as how the tissue shape is determined simultaneously.
