2017 Research Symposium Poster Abstracts

“The Weinstein Conjecture for Iterated Planar Open”

Bahar Acu, University of Southern California
Abstract: The Weinstein conjecture states that, on a compact contact manifold, any Reeb vector field carries at least one periodic orbit. The conjecture was proven for all closed 3-dimensional manifolds by Taubes, but it is still open in higher dimensions. We will provide the motivation behind this conjecture along with related results in certain cases and iterate the ideas and the machinery used in dimension 3 to higher dimensions to prove the Weinstein conjecture on a special class of Weinstein fillable contact manifolds.
“Existence of totally reflexive modules in local graded rings with Hilbert series \(1+et+(e-1)t^2\)” 
Sharma Poudyal Basanti, University of Texas at Arlington
Abstract: Let \((A,m)\) be a Noetherian local graded ring with Hilbert series \(1+et+(e-1)t^2\).

It is known that the existence of exact zero divisors implies the existence of non-free totally reflexive modules. We are interested in the existence of these modules in the absence of exact zero divisors. In a recent study, Vraciu and Atkins constructed an example of a ring of codimesion 8 that does not have exact zero divisors, but has non-free totally reflexive modules. In this talk, we will give a class of rings of codimension 5 and higher admitting totally reflexive modules, but without having exact zero divisors.

“Reduced Whitehead groups of prime exponent algebras over p-adic curves”
Nivedita Bhaskhar, University of California, Los Angeles

Abstract: The question of whether every reduced norm one element of a central simple algebra A is a product of commutators was formulated in 1943 by Tannaka and Artin in terms of the triviality of the reduced Whitehead group SK1(A) = SL1(A)/[A∗ , A∗ ]. In 1991, Suslin conjectured that if the index of the central simple algebra A/K is not square free, then SK1(A) is generically non-trivial, i.e, there exists a field extension F/K such that SK1(A ⊗K F) is non-trivial. This conjecture is supported by evidence provided by the affirmative answer of Merkurjev for algebras with indices divisible by 4. 
However it is a theorem of Merkurjev/Rost that for central simple algebras of degree 4, the Whitehead group is trivial over fields of cohomological dimension 3. This is a consequence of an injection of SK1(A) into a sub-quotient of the degree 4 Galois cohomology group which led Suslin to ask whether the same was true for index ` 2 algebras for any prime ` over cohomological dimension 3 fields. In this poster, we address this question for ` torsion, degree ` 2 algebras over function fields of p-adic curves where ` is any prime not equal to p. The proof relies on the techniques of patching as developed by Harbater-Hartmann-Krashen and exploits the arithmetic of these fields to show triviality of the reduced Whitehead group.

“Computational Algebraic Geometry in Finite Fields”

Ryann Cartor, University of Louisville

Abstract: We examine the effects on the discrete differential structure of onesided polynomial morphisms on systems of quadratic equations of Q-rank 1 over fields of characteristic two. Given a finite field GF(q) and a degree n extension, K, we determine conditions on the polynomial morphism π such that the only GF(q)-linear maps, M, inducing the symmetry D[F ◦ π](a, Mx) + D[F ◦ π](M a, x) = ΛMD[F ◦ π](a, x) for all a, x ∈ K are the maps Mx = σx for some σ ∈ K.
“Solving Advection-Diffusion Problem Using A Scalable Hybrid Schwarz Domain Decomposition Algorithm”
Lopamudra Chakravarty, Kent State University
Abstract: The solution of the linear system of algebraic equations that arise from the finite element discretization of the advection-diffusion equation is considered here. Here we study a hybrid Schwarz domain decomposition algorithm which is applied as preconditioner to solve this nonsymmetric problem. We use the GMRES method to solve the resulting preconditioned system. In each iteration step we solve a coarse finite element problem and a number of local problems. Local problems are solved in non-overlapping subdomains and ring shaped overlapping subdomains into which the original domain is subdivided. This algorithm combines the advantages of additive and multiplicative methods. Theoretical analyses show that this algorithm is scalable in the sense that the rate of convergence is independent of the mesh size and the number of subdomains. The performance of this algorithm in two dimensions is illustrated by numerical experiments.
“Topology of Non-k-Equal Configurations on Graphs”
Safia Chettih, Reed College
Abstract: Configuration spaces of n points on a graph, where no two points are equal, have homology groups that are well-known in the unordered case, while they elude a general combinatorial description. Recently, there has been new interest in configurations of n points where no k points are equal, otherwise known as non-k-equal configurations. In my poster, I will present new results which give a discretized model for non-k-equal configuration spaces on graphs, and explain the implications for the combinatorial and geometric structures interrelating configurations on graphs. 
“Analyzing Transient Solutions of the CME and CLE”
Keisha Cook, University of Alabama
Abstract: In biochemistry, species/molecules undergo randomly occurring population changes due to chemical reactions. We want to examine the probability distribution of a system. The Chemical Master Equation (CME) solution method is a set of linear, autonomous ordinary differential equations (ODEs). The Stochastic Simulation Algorithm (SSA) solution method is used to simulate the time evolution of a system of chemical reactions while taking into account the randomness of the process.
We explore the behavior and performance of these methods on a number of biological models.The following models/systems have been examined; Gene Toggle Model, Michaelis-Menten System, and Schlogl Reactions. Examining the transient solutions of each method offers a practical explanation of the system. Reactions occur at different time scales. We want to find an efficient and accurate approximation of the CME based on each respective time scale. These methods should be applicable to linear and nonlinear systems. Fast reactions are reactions that dominate the initial dynamics of a chemical reaction system. Fast reactions occur more frequently than other reactions. Slow reactions occur less frequently than other reactions. Usually the slow reactions will have a greater impact on the behavior of the system. The SSA method treats all the reactions the same. The goal is to skip over the fast reactions and only simulate the slow reactions. 
“A Robust Interrupted Time Series Model for Analyzing Complex Healthcare Intervention Data”
Maricela F Cruz, University of California, Irvine
Abstract: Current health policy calls for greater use of evidence based care delivery services to improve patient quality and safety outcomes. Care delivery is complex, with interacting and interdependent components that challenge traditional statistical analytic techniques, in particular when modeling a time series of outcomes data that might be “interrupted” by a change in a particular method of health care delivery. Interrupted time series (ITS) is a robust quasi-experimental design with the ability to infer the effectiveness of an intervention that accounts for data dependency. Current standardized methods for analyzing ITS data do not model changes in variation and correlation following the intervention and assume a pre-specified interruption time point with an instantaneous effect. This is a key limitation since it is plausible to have either anticipatory or delayed change, which can influence determination of overall effectiveness. In this paper, we describe and develop a novel ‘Robust-ITS’ model that overcomes these omissions and limitations. The Robust-ITS model formally performs inference on: (a) the change point; (b) pre- and post-intervention correlation; (c) variance of the outcome measure; and (d) pre- and post-intervention trajectory. We illustrate the proposed method by analyzing patient satisfaction data from a hospital that implemented and evaluated a new nursing care delivery model as the intervention of interest. The Robust-ITS model is implemented in a R Shiny toolbox which is freely available to the community.

“Pattern Avoidance in Restricted Growth Functions”
Samantha Dahlberg, University of British Columbia


Abstract: Wachs and White introduced four statistics on restricted growth functions or RGFs. An RGF is a sequence of positive integers w = w1w2 · · · wn such that w1 = 1 and wi ≤ 1+max(w1, w2, . . . , wi−1). We say that an RGF w avoids v if there is no subword of w which standardizes to v where standardize means we replace the ith smallest letter with i. Let Rn(v) denote the collection of length n RGFs which avoid v. In studying the generating functions for avoidance classes using Wachs and White’s statistics we find connections to other combinatorial objects. One such connection is to two-colored Motzkin paths which are counted by the Catalan numbers, Cn = 1 n+1 2n n . . The sets Rn(1212) and Rn(1221), also enumerated by Cn, have certain generating functions equivalent to the generating function for two-colored Motzkin path using the area statistic. I will present these connections in addition to other combinatorial results on RFGs.

This is joint work with Lindsey Campbell, Robert Dorward, Jonathan Gerhard, Thomas Grubb, Carlin Purcell, and Bruce Sagan.

“Cylinder Modules for Current Algebra U(sl2[t])”
Ilknur Egilmez, University of Southern California

Abstract: In this work, we study finite dimensional modules M(µ, λ) for current algebra where µ ≤ λ with λ being the highest weight. The motivation of this work comes from trace decategorification of categorified quantum groups. Trace decategorification was given as an alternate decategorification functor by Lauda and et al., and it is defined by taking the Hochschild-Mitchell homology of a category. Then, it was shown that trace of categorified quantum group U∗ is canonically isomorphic to integral idempotented version of current algebra. Studying 2-representation of U∗ gives rise to representation theory of the current algebra. By using trace and diagrammatic algebra methods we give a new approach studying representation theory of current algebra.

“Partially Magic Labelings and the Antimagic Graph Conjecture”
Maryam Farahmand, University of California, Berkeley
Abstract: The Antimagic Graph Conjecture asserts that every connected graph G = (V;E) except K2 admits an edge labeling such that each label 1; 2; : : : ; jEj is used exactly once and the sums of the labels on all edges incident to a given vertex are distinct. On the other extreme, an edge labeling is magic if the sums of the labels on all edges incident to each vertex are the same. In this poster, we approach antimagic labelings by introducing partially magic labelings, where \magic occurs” just in a sub-set of V . We generalize Stanley’s theorem about the magic graph labeling counting function to the associated counting function of partially magic labelings and prove that it is a quasi-polynomial of period at most 2. This allows us to introduce weak antimagic labelings (for which repetition is allowed), and we show that every bipartite graph satisfies a weakened version of the Antimagic Graph Conjecture.
“Feature Selection for Learning Performance Models of Electrical Stimulation for Spinal Cord Injury”
Ellen Rachel Feldman, California Institute of Technology

Abstract: Epidural spinal cord stimulation (SCS), in which implanted arrays of electrodes deliver electrical signals to spinal cord neurons, is a promising therapy for spinal cord injury (SCI). This approach enables human paraplegic patients to stand and regain partial control of leg movements, while making gains in lost autonomic function. Several parameters of the stimulation may be modified, including the choice of active electrodes, their polarities (positive, negative, or neutral), and the amplitude, frequency, and pulse width of the pulse trains applied to the active electrodes; these not only must be optimized for every patient individually, but may also vary with time. This work links computational models of epidural SCS to experimental data obtained by testing paraplegic patients’ standing performance under a range of stimulation parameters. Each set of parameters is simulated via finite element analysis to estimate the electrical activity in the spinal cord and surrounding tissues near the implant. Several types of features are then extracted from the simulation results over a range of voxel sizes. Using regression and feature selection techniques such as random forests and elastic nets, we identify the most informative electric field features (i.e. correlated with good patient motor responses) and the most important spinal cord regions to stimulate. In addition, we find that the most informative stimulating features agree with results from nerve fiber theory. Finally, we employ Gaussian process regression together with the simulation results to predict the performance of stimuli that were not tested in the patients. This procedure is applied toward suggesting additional stimulation patterns that have a sizeable probability of yielding high performance in the patients.Further applications of our work include developing algorithms to optimize stimulation configurations for SCI patients, determining optimal electrode placement, and considering novel electrode array designs. Addressing these problems may require estimating the optimal electric field for a patient; thus, we are investigating generative models to capture the joint probabilistic distribution of the features and patient responses. Stimuli could then be optimized to achieve the electrical field closest to the estimated optimum. 

“From Natural Images to MRIs Using TDA to analyze image data”
Maria Gommel, University of Iowa

Abstract: Topological Data Analysis (TDA) is a relatively new area of study that uses tools from algebraic topology to uncover the underlying shape of a given data set. One of the most recognized applications of TDA is the work of Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, and Afra Zomorodian. In their paper “On the Local Behavior of Spaces of Natural Images” [Int J Comput Vis (2008) 76:1-12], the authors analyze pixel patches extracted from a series of black and white natural images. Through the use of topological methods, the authors discovered that the densest set of patches form a shape topologically equivalent to a Klein bottle.
My work implements similar methodology to that of Carlsson et. al. to analyze MRI image data from patients at risk for Juvenile Huntington’s Disease (JHD). We look at the following two questions. First, if we apply the same approach as Carlsson et. al. to MRI images instead of natural images, do we get similar topological results? Second, can we find topological differences in data from the MRIs of healthy controls versus MRIs of those at risk for JHD? My poster will focus on the procedures used and progress made in answering these questions, as well as a discussion of proposed future work.
“Quantitative Mostow Rigidity: Relating volume to topology for hyperbolic 3-manifolds”
Rosemary Guzman, University of Illinois at Urbana-Champaign 
Abstract: A celebrated result of Mostow states that if M, N are two closed, connected, orientable, hyperbolic n-manifolds which are homotopy equivalent in dimensions $n \ge 3$, then M, N are equivalent up to isometry. 

This unique geometric-topological relationship has been the framework for many important results in the field, including notable results providing lower bounds on the volume of M, and results relating volume to homology (Culler-Shalen). 

Here, we will focus on the case where the fundamental group of M has a property, “k-free,” for $k \ge 5$, and discuss current work toward an improvement on the volume bound from the current known bound of 3.44 which holds for $k \ge 4$. This is joint work with Peter Shalen.

“Time-homogeneous parabolic Wick-Anderson model in one space dimension: regularity of solution”
Hyun-Jung Kim, University of Southern California
Abstract: : We consider the stochastic parabolic Anderson model driven by time-independent random potential with Wick product on a bounded interval (a, b) or the whole line R: ut(t, x) =uxx(t, x) + u(t, x) W˙ (x) u(0, x) =u0(x). (1)

Our aim is to define an Lp(Ω) bounded (chaos) solution by means of chaos expansion in a probability space (Ω, F, P) and to establish the optimal space-time regularity of the solution to (1) under a minimal assumption on the initial condition u0.Two main noticeable features of our model in the paper are 1) time-independent white noise W˙ (x), 2) special multiplication called Wick product between u and W. Even though time-dependent models have been well-studied thanks to the Itˆo theory, time-independent models like our model still need more improvement. Instead of using the usual point-wise product, the model (1) is interpreted in the renormalized sense by using the Wick product.

This allows us to reduce the stochastic model (1) into countably many deterministic propagators of (1). With the help of the hypercontractivity property of the Ornstein-Uhlenbeck operator and the Kolmogorov’s continuity theorem, we achieve the same optimal space-time regularity results either on a bounded interval (a, b) or the whole line R.

“Height Pairing for Cycles and Determinant Line Bundle”
Yordanka Aleksandrova Kovacheva, The University of Chicago

Abstract: In this poster I want to present my work on height pairing of cycles modulo relations and the corresponding determinant line bundle and point possible directions for future research. More specifically, I consider the map CHp (X) × CHq(X) → P ic(S) of Chow groups of a variety X over a base S. Here p+q = d+1, where d is the relative dimension of the morphism X → S. I treat the Chow groups CHp (X) as categories with the obvious objects and morphisms arising from the Zp (X, 1) term in Bloch’s complex modulo the image of Tame symbols of K2-chains. This pairing coincides with the Knudsen-Mumford determinant line bundle using the structure sheaves of the cycles on X.

“Designing Optimal Combination Therapies to Minimize Drug Resistance in Solid Tumors via Evolutionary Stochastic Modeling”

Danika Gray Lindsay, University of Minnesota

Abstract: Many experimental studies have shown that a key factor in driving the emergence and evolution of drug resistance in cancer is tumor hypoxia, or the existence of low-oxygen regions within a tumor. These regions of low oxygen concentration lead to the formation of localized environmental niches where drug-resistant cell populations can evolve and survive. Hypoxia-activated prodrugs (HAPs) are compounds designed to penetrate to hypoxic regions of a tumor first before releasing an active drug. Several of these HAPs are currently in clinical trials. However, preliminary results have not shown a survival benefit in some of these trials. We hypothesize that the efficacy of treatments involving these prodrugs depends heavily on identifying the correct treatment schedule, and that mathematical modeling can be used to help design potential therapeutic strategies combining HAPs with standard therapies to achieve long-term tumor control or eradication.

We develop this framework in the specific context of non-small cell lung cancer (NSCLC), which is commonly treated with a drug known as erlotinib. We design an evolutionary stochastic mathematical model in which the population of cancer cells is described using a multi-type, non-homogeneous, continuous time birth-death process. We analyze this model and use it to predict treatment outcomes for a tumor undergoing a variety of combination therapies with erlotinib and evofosfamide, a HAP currently in clinical trials. Our model predicts that tumor eradication may actually be possible using these drugs together, whereas monotherapy with either of these drugs alone always leads to the inevitable development of drug resistance and disease progression. Furthermore, we develop a model to describe the combination toxicity constraints on the patient and use these constraints to optimize treatment strategies over the space of tolerated schedules in order to identify specific combination schedules that lead to optimal tumor control. These results have the potential to significantly improve treatment outcomes for patients with NSCLC.

“Constructing Symplectic Varieties Using Geometric Invariant Theory”
Nicolette M Jimenez, United States Military Academy

Abstract: I will provide strategies on how one constructs new projective varieties using classical ones. In the case when n = 2, I will show that the Hamiltonian reduction of b×Cn is a complete intersection by using a well-known technique in symplectic geometry. I will construct four projective varieties using geometric invariant theory (GIT). I will also construct an affine variety via the affine quotient using the four irreducible components. Finally, I will discuss how one should construct a hyperkahler metric on these types of varieties.
“Implicitly Enriched Galerkin Mapping Methods of Fourth-Order Partial Differential Equations Containing Singularities”
Sinae Kim, University of North Carolina at Charlotte 
Abstract: We introduce two enrichment methods (explicit and implicit) in the framework of IGA (Isogeometric Analysis) to solve fourth-order equations containing singularities, since the standard IGA and conventional FEM do not yield reasonable solutions to the problems. We demonstrate that both enrichment methods yield good approximate solutions: explicit enrichment methods give large matrix condition number and face singular integration, while implicit enrichment methods overcome the limitations. 
“Regularity conditions for quadratic and Hermitian forms”
Alicia Marino, 
Wesleyan University
Abstract: A main question in the arithmetic theory of quadratic forms is the representation problem: given an integral quadratic form f, for which integers a does there exist a solution to f(x) = a? Attempts to answer this question have led to the study of many different types of quadraticforms. We call a positive definite quadratic form regular if whenever there is a solution over the p-adic integers for every prime p then there exists a solution over the rational integers. We can strengthen this notion of regularity to strict regularity by demanding that the solutions are primitive, i.e. the coordinates of the solutions are coprime. These notions of regularity can be extended to the study of integral Hermitian forms. I will present my thesis work on integral quadratic forms which satisfy a higher dimensional analogue of the strict regularity condition, and also present a recent result, joint with J. Liu, regarding the finiteness of strictly regular ternary integral Hermitian forms.
“Non-self-adjoint subalgebras of C∗-algebras and transformation groups”
Kathryn A McCormick, University of Iowa
Abstract: An analysis is presented here of operator algebras constructed as cross-sectional algebras of certain holomorphic matrix bundles. The focus is on the boundary representations of the containing C-algebras. More specifically, let R be finitely connected, bordered Riemann surface with associated deck transformation group G, viewed as acting on the unit disk, D. Then the action of G on D extends to a free and proper action of G on a certain open subset De of D which has quotient space R; i.e., De/G ‘ R. A representation ρ of G in P U(n, C) is then used to build a Mn(C)- bundle, M, over R in the standard fashion. The operator algebra in question, denoted A(R, ρ), is the collection of all continuous cross sections of M over R that are holomorphic on R. A(R, ρ) generates the C-algebra of all continuous cross sections of M, written C(R, ρ). We calculate the boundary representations of C(R, ρ) for A(R, ρ), and show that the C-envelope of A(R, ρ) is the space of all continuous cross sections of M restricted to the boundary ∂R. Our analysis is conducted using the fundamental groupoids of R and ∂R. We discuss, as well, how our analysis leads to isomorphism invariants for algebras of the form A(R, ρ).
“Progress on the 1/3 − 2/3 Conjecture”
Emily J Olson, Michigan State University
Abstract: In a partially ordered set P, let a pair of elements (x, y) be called α-balanced if the proportion of linear extensions that has x before y is between α and 1 − α. The 1/3 − 2/3 Conjecture states that every finite poset which is not a chain has some 1/3-balanced pair. While the conjecture remains unsolved, we present progress in certain types of posets, including products of two chains, Boolean and set partition lattices, and posets corresponding to Young diagrams.

“Strong pseudoconvexity in Banach spaces”
Sofia Ortega Castillo,
CIMAT Guanajuato

Abstract: I will present a survey on pseudoconvexity in Cn in terms of plurisubharmonic exhaustion functions, plurisubharmonic C2 defining functions as well as local conditions. With that in mind, I will present a characterization of strong pseudoconvexity in terms of an exhaustion fuction, and a notion of uniform pseudoconvexity inspired by uniform convexity. Moreover, I will present examples of Banach spaces with uniformly pseudoconvex unit ball, e. g. 2-uniformly PLconvex spaces. This will help motivate a suitable definition of strong pseudoconvexity in the infinite-dimensional and non-smooth boundary context, and to further provide counterexamples to strict pseudoconvexity in this general case. Since 2-uniformly PL-convex spaces are a main example in my work, I will also present a number of characterizations of r-uniform PL-convexity.
“The Augmentation Category Map Induced by Exact Lagrangian Cobordisms”
Yu Pan, Duke University
Abstract: To a Legendrian knot, one can associate an A∞ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials. 

“Geometry and Topology of Fake Projective Spaces”
Priyanka Rajan, University of California, Riverside
Abstract: A fake real projective space is a manifold homotopy equivalent to real projective space, but not diffeomorphic to it. Equivalently, it is the orbit space of a free involution on a (homotopy) sphere.

In this thesis, we show that some of the fake RP6s , constructed by Hirsch and Milnor in 1963, and the analogous fake RP14s admit metrics that simutaneously have almost nonnegative sectional curvature and positive Ricci curvature. These spaces are obtained by taking the Z2 quotients of the embedded images of the standard spheres of codimension one in some of Milnor’s exotic 7spheres and the analogous Shimada’s exotic 15spheres. This part of my thesis is joint work with F. Wilhelm.

Hirsch and Milnor also constructed fake RP5s using invariant subspheres of codimension two. Octonionically, this construction yields closed 13manifolds, that are homotopy equivalent to RP13s. The analog to their proof that fake RP5s are not diffeomorphic to standard RP5 breaks down; since in contrast to dimension 6, there is an exotic 14sphere. We show that some of the Hirsch-Milnor RP13s are not diffeomorphic to standard RP13s. Here we obtain a complete diffeomorphism classification of the Hirsch-Milnor RP13s. This part of my thesis is joint work with C. He.

“Group Key Establishment with Physical Unclonable Functions”
Angela Robinson, Florida Atlantic University

Abstract: Physical unclonable functions (PUFs) are effective tools for secret key generation which do not rely on any computational hardness assumptions. In the literature their use for 2-party key establishment has been explored. In this paper we initiate the exploration of PUFs as a resource for group key establishment.
First, we adjust an existing security framework for group key establishment in such a way that the use of PUFs can be modeled conveniently. Hereafter, we present a 4-round solution for group key establishment whose security relies on the availability of PUFs with appropriate guarantees. For authentication purposes, we assume an existentially unforgeable signature scheme to be in place. The final key and session identifier derivation can be realized without a random oracle by using a family of collision-resistant pseudorandom functions.
Adaptive Group Bridge for Marginal Cox Proportional Hazards Models with multiple diseases”
Natasha Sahr, Medical College of Wisconsin
Abstract: Variable selection methods in linear regression such as lasso, SCAD, and group lasso have been applied to the univariate Cox proportional hazards (PH) model; however, variable selection in multivariate failure time regression analysis is a challenging and relatively unexplored research area. In this context, we propose an adaptive group bridge penalty to select variables for marginal Cox PH models with multivariate failure time data. The proposed method not only selects group variables, but also individual variables within a group. The adaptive group bridge method for marginal Cox PH models with multivariate failure time data was compared to the group bridge penalty method, and backwards, forwards, and stepwise selection. The simulation studies show that the adaptive group bridge method has superior performance compared to the other methods in terms of variable selection accuracy.
“The Boson-Fermion Correspondence”
Nicolle E Sandoval Gonzalez, University of Southern California
Abstract: The boson-fermion correspondence is ubiquitous throughout mathematical physics. In essence, it reveals how the action of the Clifford algebra on fermionic Fock space can be recovered from the action of the Weyl algebra on bosonic Fock space. Given the combinatorial nature of both spaces and their well-known representations of the Weyl and Clifford algebras, this correspondence gives immediate insight into the rich interplay between representation theory, combinatorics, and quantum physics.
My research focuses on categorifying this relationship; that is, to lift the relations between the Clifford and Heisenberg algebras and reconstruct the correspondence at the categorical level in terms of infinite chain complexes and chain homotopies. Given how little is known about the categorification of Clifford algebras, this result could have immediate consequences for the study of their higher representation theory.
“Semigroups applied to hashing: a new platform for Cayley hashes”
Bianca Sosnovski, Queensborough Community College 
Abstract: Cayley hash functions are based on the idea of using a pair of elements in a (semi)group, $A$ and $B$, to hash the 0 and 1 bit, respectively. A bit string is associated to a string of $A$’s and $B$’s and the hash value is computed by multiplying the sequence of $A$’s and $B$’s in the (semi)group.

We present a new semigroup platform for a Cayley hash function. Our proposed hash function uses a pair of two linear functions in one variable over $\mathbb{F}_p$ under composition operation. The semigroup is generated by the functions $f(x) = 2x+1$ and $g(x) = 3x+1$ modulo a prime $p > 3$. The result is an efficient hash function whose outputs are of size $2\log p$. We give explicit lower bound on the length of collisions for the proposed hash function.

This is joint work with Vladimir Shpilrain.​

“The Smallest Nontrivial Height of Totally p-adic Numbers of Degree 2 or Degree 3”
Emerald T Stacy, Oregon State University
Abstract: Given an algebraic number α, the height of α gives a measure of how arithmetically complicated α is. We say an algebraic number is totally p-adic if its minimal polynomial splits completely over Qp. Given a degree d and a prime p, there exists a smallest nontrivial height of a totally p-adic algebraic number of degree d, which we will call τd,p. This poster will show the results I have so far: a solution for d = 2 and d = 3.

Spatial measures of genetic heterogeneity during carcinogenesis
Katie Storey, University of Minnesota

Abstract: Carcinogenesis, the transformation from healthy tissue to invasive cancer, is characterized by the progressive accumulation of genetic alterations in a small group of founder cells. In joint work with collaborators Jasmine Foo and Kevin Leder at the University of Minnesota, and Marc Ryser at Duke University, we explore the temporal dynamics of spatial heterogeneity during the process of carcinogenesis from healthy tissue. We utilize a spatial stochastic model of mutation accumulation and clonal expansion in a structured tissue to describe this process. Under a two-step tumorigenesis model, we develop and analyze a spatial analog of the Simpson Index, a traditionally non-spatial measure of diversity commonly used in ecology. Our spatial analog measures the length-scale of genetic heterogeneity during the carcinogenesis process. The evolutionary framework contributes to a growing literature focused on developing a better understanding of the spatial population dynamics of cancer initiation and progression.     
“Learning Functions from Time-Varying Measurement Data”
Giang Tran, University of Texas at Austin

Abstract: Learning the governing equations for time-varying measurement data is of great interest across different scientific fields. When such data is moreover highly corrupted, for example, due to the recording mechanism failing over unknown intervals of time, recovering the governing equations becomes quite challenging. In this work, we show that if the data exhibits chaotic behavior, it is possible to recover the underlying governing nonlinear differential equations even if a large percentage of the data is corrupted by outliers, by solving an l1 minimization problem which assumes a parsimonious representation of the system and exploits the joint sparsity in the variable representing the corrupted data.

“Pattern Formation – on the modeling of multi- constituent inhibitory systems”
Chong Wang, George Washington University

Abstract: Skin pigmentation, animal coats and block copolymers, which can be considered as multi-constituent inhibitory systems, are all around us. Theoretical analysis and numerical simulation of multi-constituent inhibitory systems will be provided here. An inhibitory system is studied as a nonlocal  geometric variational problem. The free energy of the system is the sum of two terms: the total size of the interfaces separating the constituents, and a longer ranging interaction energy that inhibits micro-domains from unlimited growth. We establish that in different parameter ranges there are corresponding assemblies of certain patterns that existas the stationary sets of the free energy functional. Numerically, a diffusive interface model is proposed and different self-assembly processes, which form various patterns, are vividly showed here.
“Peri-Catalan Numbers and Free Quasigroups”
Stefanie Wang, Iowa State University
Abstract: A quasigroup is an algebra equipped with nonassociative binary operations of multiplication, left division, and right division. Rooted binary trees with n leaves represent the number of ways to bracket a word with n arguments involving a single nonassociative binary operation. While Catalan numbers count rooted binary trees with n leaves, we are interested in the peri-Catalan numbers that count the number of quasigroup words with n arguments. We will provide an introduction to quasigroups and the motivation behind the study of peri-Catalan numbers.

“The Rate of Convergence of Strong Euler Approximation for Levy-driven SDEs”
Fanhui Xu, University of Southern California

Abstract: A SDE driven by an α-stable (Levy) process with its coefficient being Lipschitz, the drift being β-Holder continuous and α in [1,2) is considered. In particular, the existence and uniqueness of the strong solution to this SDE with non-Lipschitz drift is proved by applying Euler approximation and deriving the rate of convergence in Lp sense when β<1, β+α/2>1. When β=1, i.e. in the Lipschitz case, the rate of convergence is provided as well. It turns out that it is better than the standard estimation on SDEs driven by a Wiener process. “Ito-Tanaka trick” is adequately used in the case of Holder drift, properties of the solution to a backward Kolmogorov equation play an important role here.