## 2017 Research Symposium Poster Abstracts

**“The Weinstein Conjecture for Iterated Planar Open”**

**“Existence of totally reflexive modules in local graded rings 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

**“Computational Algebraic Geometry in Finite Fields”**

Ryann Cartor, University of Louisville

**“Solving Advection-Diffusion Problem Using A Scalable Hybrid Schwarz Domain Decomposition Algorithm”**

**“Topology of Non-k-Equal Configurations on Graphs”**

**“Analyzing Transient Solutions of the CME and CLE”**

**“A Robust Interrupted Time Series Model for Analyzing Complex Healthcare Intervention Data”**

**“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”**

**“Feature Selection for Learning Performance Models of Electrical Stimulation for Spinal Cord Injury”**

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

**“Quantitative Mostow Rigidity: Relating volume to topology for hyperbolic 3-manifolds”**

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”**

**:**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”**

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

**“Implicitly Enriched Galerkin Mapping Methods of Fourth-Order Partial Differential Equations Containing Singularities”**

**“Regularity conditions for quadratic and Hermitian forms”**

Alicia Marino, Wesleyan University

**“Non-self-adjoint subalgebras of C∗-algebras and transformation groups”**

^{∗}-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”**

**“Strong pseudoconvexity in Banach spaces”**

Sofia Ortega Castillo,CIMAT Guanajuato

**“The Augmentation Category Map Induced by Exact Lagrangian Cobordisms”**

**“Geometry and Topology of Fake Projective Spaces”**

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

**Adaptive Group Bridge for Marginal Cox Proportional Hazards Models with multiple diseases”**

**“The Boson-Fermion Correspondence”**

**“Semigroups applied to hashing: a new platform for Cayley hashes”**

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”**

**“****Spatial measures of genetic heterogeneity during carcinogenesis****“**

Katie Storey, University of Minnesota

**“Learning Functions from Time-Varying Measurement Data”**

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

**“Peri-Catalan Numbers and Free Quasigroups”**

**“The Rate of Convergence of Strong Euler Approximation for Levy-driven SDEs”**

Fanhui Xu, University of Southern California