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**Authors: **Neer Bhardwaj and Lou van den Dries

**On the Pila-Wilkie theorem**

https://arxiv.org/abs/2010.14046

**Abstract:** In this expository paper we give an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We include a full treatment of a result due to Pila and Bombieri, and of a variant of the Yomdin-Gromov theorem that are used in this proof.

**Authors: **Neer Bhardwaj and Minh Chieu Tran

**The additive groups of ℤ and ℚ with predicates for being square-free
Journal: **

*The Journal of Symbolic Logic*

https://doi.org/10.1017/jsl.2020.30

**Abstract:**We consider the four structures (

**ℤ**;SF

^{ℤ}), (

**ℤ**; <,SF

^{ℤ}), (

**ℚ**;SF

^{ℚ}), and (

**ℚ**; <,SF

^{ℚ}) where

**ℤ**is the additive group of integers, SF

^{ℤ}is the set of

*a*>

**ℤ**such that

*v*(

_{p}*a*) < 2 for every prime

*p*and corresponding

*p*-adic valuation

*v*,

_{p}**ℚ**and SF

^{ℚ}are defined likewise for rational numbers, and < denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

**Author:** Robert Dicks

**A higher weight analogue of Ogg's theorem on Weierstrass points**

https://arxiv.org/abs/2006.09520

**Abstract**: For a positive integer N, we say that ∞ is a Weierstrass point on the modular curve X0(N) if there is a non-zero cusp form of weight 2 on Γ0(N) which vanishes at ∞ to order greater than the genus of X0(N). If p is a prime with p∤N, Ogg proved that ∞ is not a Weierstrass point on X0(pN) if the genus of X0(N) is 0. We prove a similar result for even weights k≥4. We also study the space of weight k cusp forms on Γ0(N) vanishing to order greater than the dimension.

Authors: J. Connor Grady, Ching Hung Lam, James E. Tener, and Hiroshi Yamauchi

**Classification of extremal vertex operator algebras with two simple modules
Journal:**

*Journal of Mathematical Physics*

https://aip.scitation.org/doi/full/10.1063/1.5121446

**Abstract:**In recent work, Wang and Tener defined a class of “extremal” vertex operator algebras (VOAs), consisting of those with at least two simple modules and conformal dimensions as large as possible for the central charge. In this article, we show that there are exactly 15 character vectors of extremal VOAs with two simple modules. All but one of the 15 character vectors are realized by a previously known VOA. The last character vector is realized by a new VOA with central charge 33.

**Authors:** Chiun-Chuan Chen, Ting-Yang Hsiao, and Li-Chang Hung

**Discrete N-barrier maximum principle for a lattice dynamical system arising in competition models**

**Journal:** *Discrete and Continuous Dynamical Systems*

**Abstract:** We show that an analogous N-barrier maximum principle remains true for lattice systems. This extends the results from continuous equations to discrete equations. In order to overcome the difficulty induced by a discretized version of the classical diffusion in the lattice systems, we propose a more delicate construction of the N-barrier which is appropriate for the proof of the N-barrier maximum principle for lattice systems. As an application of the discrete N-barrier maximum principle, we study a coexistence problem of three species arising from biology and show that the three species cannot coexist under certain conditions.

**Authors: **József Balogh, Béla Csaba, Yifan Jing, and András Pluhár

**Title: On the discrepancies of graphs**

**Journal: ***Electronic Journal of Combinatorics*

https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i2p12

**Abstract:** In the literature, the notion of discrepancy is used in several contexts, even in the theory of graphs. Here, for a graph $G$, $\{−1, 1\}$ labels are assigned to the edges, and we consider a family $\mathcal{S}_G$ of (spanning) subgraphs of certain types, among others spanning trees, Hamiltonian cycles. As usual, we seek for bounds on the sum of the labels that hold for all elements of $\mathcal{S}_G$, for every labeling.

**Authors: **Yifan Jing and Bojan Mohar

**Title: The genus of complete 3-uniform hypergraphs**

**Journal: ***Journal of Combinatorial Theory Series B*

https://www.sciencedirect.com/science/article/pii/S0095895619300796

**Abstract:** In 1968, Ringel and Youngs confirmed the last open case of the Heawood Conjecture by determining the genus of every complete graph $K_n$. In this paper, we determine both the orientable and the non-orientable genus of $K_n^{(3)}$ when n is even, generalizing Ringel–Youngs Theorems to hypergraphs. Moreover, it is shown that the number of non-isomorphic minimum genus embeddings of $K_n^{(3)}$ is at least $2^{\frac{1}{4}n^2\log n(1-o(1))}$.

**Authors: **Gennian Ge,** **Yifan Jing, Zixiang Xu, and Tao Zhang

**Title: Color isomorphic even cycles and a related Ramsey problem**

**Journal: ***SIAM Journal on Discrete Mathematics*

https://epubs.siam.org/doi/abs/10.1137/20M1329652

**Abstract:** Given a graph $H$ and an integer $k\geq 2$, let $f_{k}(n,H)$ be the smallest number of colors $c$ such that there exists a proper edge coloring of $K_{n}$ with $c$ colors containing no $k$ vertex-disjoint color-isomorphic copies of $H$. Using algebraic properties of polynomials over finite fields, we construct an explicit proper edge coloring of $K_{n}$ and show that $f_{k}(n, C_{4})=\Theta(n)$ when $k\geq 3$. We also consider a related generalized Ramsey problem. For given graphs $G$ and $H,$ let $r(G,H,q)$ be the minimum number of edge colors (not necessarily proper) of $G$, such that the edges of every copy of $H\subseteq G$ together receive at least $q$ distinct colors. We obtain some general lower bounds for $r(K_{n,n},K_{s,t},q)$ with a broad range of $q$.

**Authors: **Vaibhav Karve and Anil N. Hirani

**The complete set of minimal simple graphs that support unsatisfiable 2-CNFs**

**Journal: ***Discrete Applied Mathematics*

https://doi.org/10.1016/j.dam.2019.12.017

**Abstract:** A propositional logic sentence in conjunctive normal form that has clauses of length at most two (a 2-CNF) can be associated with a multigraph in which the vertices correspond to the variables and edges to clauses. We show that every 2-CNF that has been reduced under the application of certain tautologies, is equisatisfiable to a 2-CNF whose associated multigraph is, in fact, a simple graph. Our main result is a complete characterization of graphs that can support unsatisfiable 2-CNF sentences. We show that a simple graph can support an unsatisfiable reduced 2-CNF sentence if and only if it contains any one of four specific small graphs as a topological minor. Equivalently, all reduced 2-CNF sentences supported on a given simple graph are satisfiable if and only if all subdivisions of those four graphs are forbidden as subgraphs of the original graph.

**Authors:** Yu Wu, Gabriel Shindes, Vaibhav Karve, Derrek Yager, Daniel B. Work, Arnab Khakraborty, and Richard B. Sowers

**Congestion barcodes: Exploring the topology of urban congestion using persistent homology**

https://arxiv.org/abs/1707.08557

**Abstract:** This work presents a new method to quantify connectivity in transportation networks. Inspired by the field of topological data analysis, we propose a novel approach to explore the robustness of road network connectivity in the presence of congestion on the roadway. The robustness of the pattern is summarized in a congestion barcode, which can be constructed directly from traffic datasets commonly used for navigation. As an initial demonstration, we illustrate the main technique on a publicly available traffic dataset in a neighborhood in New York City.

**Authors:** Vivek Kaushik and Daniele Ritelli

**Evaluation of harmonic sums with integrals
Journal:**

*AMS Quarterly of Applied Mathematics*

**Abstract:**We consider the sums and with being a positive integer. We evaluate these sums with multiple integration, a modern technique. First, we start with three different double integrals that have been previously used in the literature to show which implies Euler's identity Then, we generalize each integral in order to find the considered sums. The dimensional analogue of the first integral is the density function of the quotient of independent, nonnegative Cauchy random variables. In seeking this function, we encounter a special logarithmic integral that we can directly relate to The dimensional analogue of the second integral, upon a change of variables, is the volume of a convex polytope, which can be expressed as a probability involving certain pairwise sums of independent uniform random variables. We use combinatorial arguments to find the volume, which in turn gives new closed formulas for and The dimensional analogue of the last integral, upon another change of variables, is an integral of the joint density function of Cauchy random variables over a hyperbolic polytope. This integral can be expressed as a probability involving certain pairwise products of these random variables, and it is equal to the probability from the second generalization. Thus, we specifically highlight the similarities in the combinatorial arguments between the second and third generalizations.

**Authors: **Marissa Miller

**Stable subgroups of the genus two handlebody group**

https://arxiv.org/abs/2009.05067

**Abstract:** We show that a nitely generated subgroup of the genus two handlebody group is stable if and only if the orbit map to the disk graph is a quasi-isometric embedding. To this end, we prove that the genus two handlebody group is a hierarchically hyperbolic group, and that the maximal hyperbolic space in the hierarchy is quasi-isometric to the disk graph of a genus two handlebody by appealing to a construction of Hamenstdt-Hensel. We then utilize the characterization of stable subgroups of hierarchically hyperbolic groups provided by Abbott-Behrstock-Durham. We also provide a counterexample for the higher genus analogue of the main theorem.

**Author:** M. A. Tursi

**A separable universal homogeneous Banach lattice**

https://arxiv.org/abs/2008.06658

**Abstract:** We prove the existence of a separable approximately ultra-homogeneous Banach lattice \mathfrak{BL} that is isometrically universal for separable Banach lattices. This is done by showing that the class of Banach lattices has the Amalgamation Property, and thus finitely generated Banach lattices form a metric Fraïssé class. Some additional results about the structural properties of \mathfrak{BL} are also proven.

**Authors: **Leonid V. Kovalev and Xuerui Yang

**Algebraic structure of the range of a trigonometric polynomial**

**Journal: ***Bulletin of the Australian Mathematical Society*

https://www.cambridge.org/core/journals/bulletin-of-the-australian-mathematical-society/article/algebraic-structure-of-the-range-of-a-trigonometric-polynomial/91E169A34C4CD273B266C023E2943A8C

**Abstract:** The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with certain symmetry.

**Authors: **Leonid V. Kovalev and Xuerui Yang

**Near-isometric duality of Hardy norms with applications to harmonic mappings**

**Journal: ***Journal of Mathematical Analysis and Applications*

https://www.sciencedirect.com/science/article/abs/pii/S0022247X2030202X?via%3Dihub

**Abstract: **Hardy spaces in the complex plane and in higher dimensions have natural finite-dimensional subspaces formed by polynomials or by linear maps. We use the restriction of Hardy norms to such subspaces to describe the set of possible derivatives of harmonic self-maps of a ball, providing a version of the Schwarz lemma for harmonic maps. These restricted Hardy norms display unexpected near-isometric duality between the exponents 1 and 4, which we use to give an explicit form of harmonic Schwarz lemma.

**Authors: **Leonid V. Kovalev and Xuerui Yang

**Extreme values of the derivative of Blaschke products and hypergeometric polynomials**

https://arxiv.org/abs/2007.09760

**Abstract:** A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.

**Authors: **Leonid V. Kovalev and Xuerui Yang

**Fourier series of circle embeddings **

**Journal: ***Computational Methods and Function Theory*

https://link.springer.com/article/10.1007/s40315-019-00263-2

**Abstract: **We study the Fourier series of circle homeomorphisms and circle embeddings, with an emphasis on the Blaschke product approximation and the vanishing of Fourier coefficients. The analytic properties of the Fourier series are related to the geometry of the circle embeddings, and have implications for the curvature of minimal surfaces.