Faculty Publications and Research

Permanent link for this collectionhttps://hdl.handle.net/2022/19037

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Hyperbolic Geometry
(2026-01-08) Matthias Weber
These notes serve as an elementary introduction to hyperbolic geometry suitable for an undergraduate course. Topics include circles, inversion, Möbius transformations, disk and upper half plane models of the hyperbolic plane, the hyperbolic distance, spherical geometry, reflection groups, Dyck's theorem, Ford circles.
Differential Geometry of Curves and Surfaces
(2026-01-07) Matthias Weber
This is a textbook for a one-semester undergraduate course in the Differential Geometry of Curves and Surfaces. Familiarity with Real Analysis is helpful. This goal is to provide a complete proof of the Gauß-Bonnet theorem. There are hyperlinks in the margin of the PDF linking to some 80 Mathematica notebooks, which are also supplied. A version with solutions to most exercises will be provided in May.
Complex Analysis
(2026-01-06) Matthias Weber
This is a textbook for a one-semester graduate course in Complex Analysis. Familiarity with Real Analysis is helpful. All proofs attempt to be organic. This goal is to provide a complete proof of the Riemann Mapping Theorem. There are two optional chapters about elliptic functions and geometric function theory.
Games Graphs Groups
(2025-10-15) Matthias Weber
Lecture Notes for a 300 level mathematics class targeted at math majors and minors. There are few prerequisites. The topics focus on mathematical games and puzzles, the table of contents tells pretty clearly what is happening: 1. Combinatorial Game Theory 2. Polyforms 3. Dissections 4. Playing with Graphs and Maps 5. Solitaire Games Some of the material is classical, some is new. There are lots of exercises, some with solutions. There are also explorations. These are longer and open-ended activities. All images have alt-texts, and the PDF is PDF/UA-2 compliant, according to VeraPDF 1.28.2 Enjoy.
Incidences
(2025-08-28) Weber, Matthias
Lecture Notes about incidence geometry suitable for an undergraduate class
Convergence of Hill's method for nonselfadjoint operators
(Society for Industrial and Applied Mathematics, 2012) Johnson, M.A.; Zumbrun, K.
By the introduction of a generalized Evans function defined by an appropriate 2- modified Fredholm determinant, we give a simple proof of convergence in location and multiplicity of Hill's method for numerical approximation of spectra of periodic-coefficient ordinary differential operators. Our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite (automatically satisfied in the scalar case). Notably, this includes a large class of non-self-adjoint operators which previously had not been treated in a simple way. The case of general coefficients depends on an interesting operator-theoretic question regarding properties of Toeplitz matrices
A khintchine decomposition for free probability
(Institute of Mathematical Statistics, 2012) Williams, J.D.
Let μ be a probability measure on the real line. In this paper we prove that there exists a decomposition $\mu = \mu_{0}\boxplus \mu_{1}\dots\boxplus \mu_{n}$such that $\mu_{0}$ is infinitely divisible, and $\mu_{i}$ is indecomposable for $i \geq 1$. Additionally, we prove that the family of all $\boxplus$-divisors of a measure $\mu$ is compact up to translation. Analogous results are also proven in the case of multiplicative convolution
Handle addition for doubly-periodic Scherk surfaces
(De Gruyter, 2012) Weber, M.; Wolf, M.
We prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus g with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The proof of the family of immersed surfaces is by induction on genus, while the proof of embeddedness is by the conjugate Plateau method.
Tetrahedral forms in monoidal categories and 3-manifold invariants
(De Gruyter, 2012) Geer, N.; Kashaev, R.; Turaev, V.
We introduce systems of objects and operators in linear monoidal categories called $\hat{\Psi}$-systems. A $\hat{\Psi}$-system satisfying several additional assumptions gives rise to a topological invariant of triples (a closed oriented 3-manifold $M$, a principal bundle over $M$, a link in $M$). This construction generalizes the quantum dilogarithmic invariant of links appearing in the original formulation of the volume conjecture. We conjecture that all quantum groups at odd roots of unity give rise to $\hat{\Psi}$-systems and we verify this conjecture in the case of the Borel subalgebra of quantum sl$_{2}$.
Positive solutions of fractional differential equations with derivative terms
(Texas State University - San Marcos, 2012) Cheng, C.; Feng, Z.; Su, Y.
In this article, we are concerned with the existence of positive solutions for nonlinear fractional differential equation whose nonlinearity contains the first-order derivative, $$\displaylines{ D_{0^+}^{\alpha}u(t)+f(t,u(t),u'(t))=0,\quad t\in (0,1),\; n-1<\alpha\leq n,\cr u^{(i)}(0)=0, \quad i=0,1,2,\dots,n-2,\cr [D_{0^+}^{\beta}u(t)]_{t=1}=0, \quad 2\leq\beta\leq n-2, }$$ where $n>4 $ $(n\in\mathbb{N})$, $D_{0^+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order $\alpha$ and $f(t,u,u'):[0,1] \times [0,\infty)\times(-\infty,+\infty) \to [0,\infty)$ satisfies the Caratheodory type condition. Sufficient conditions are obtained for the existence of at least one or two positive solutions by using the nonlinear alternative of the Leray-Schauder type and Krasnosel'skii's fixed point theorem. In addition, several other sufficient conditions are established for the existence of at least triple, $n$ or $2n-1$ positive solutions. Two examples are given to illustrate our theoretical results.
Some notes on the paper “The equivalence of cone metric spaces and metric spaces”
(Springer, 2012) Asadi, M.; Rhoades, B.E.; Soleimani, H.
In this article, we shall show that the metrics defined by Feng and Mao, and Du are equivalent. We also provide some examples for one of the metrics.
On the algebraic K-theory of formal power series
(Cambridge University Press, 2012) Lindenstrauss, A.; McCarthy, R.
In this paper we extend the computation of the the typical curves of algebraic K-theory done by Lars Hesselholt and Ib Madsen to general tensor algebras. The models used allow us to determine the stages of the Taylor tower of algebraic K-theory as a functor of augmented algebras, as defined by Tom Goodwillie, when evaluated on derived tensor algebras. For $R$ a discrete ring, and $M$ a simplicial $\LARGE{\tau}\normalsize{_{R}}$-bimodule, we let $\LARGE{\tau}\normalsize{_{R}\big(M\big)}$ denote the (derived) tensor algebra of $M$ over $R$, and $\LARGE{\tau}\;\normalsize{^{\pi}_{R}}$ denote the ring of formal (derived) power series in $M$ over $R$. We define a natural transformation of functors of simplicial $R$-bimodules $\phi: \sum\tilde{K}\big(R;\; \big)\rightarrow\tilde{K}\big(\LARGE{\tau}\normalsize{_{R}\big(\; \big)\big)}$.which is closely related to Waldhausen's equivalence $\sum\tilde{K}\big(\text{Nil}\big(R; \;\big)\big)\rightarrow\tilde{K}\big(\LARGE{\tau}\normalsize{^{\pi}_{R}\big( \big)\big)}$. We show that $\phi$ induces an equivalence on any finite stage of Goodwillie's Taylor towers of the functors at any simplicial bimodule. This is used to show that there is an equivalence of functors $\sum W\big(R; \;\big)\rightarrow^{\simeq}\text{holim}_{n}\tilde{K}\big(\LARGE{\tau}\normalsize{_{R}\big(\; \big)/I^{n+1}\big)}$, and for connected bimodules, also an equivalence $\sum\tilde{K}\big(R; \;\big)\rightarrow^{\simeq}\tilde{K}\big(\LARGE{\tau}\normalsize{_{R}\big( \;\big)\big)}$.
An algebraic characterization of expanding Thurston maps
(American Institute of Mathematical Sciences, 2012) Haïssinsky, P.; Pilgrim, K.M.
Let $f:S^{2} \rightarrow S^{2}$ be a postcritically finite branched covering map without periodic branch points. We give necessary and sufficient algebraic conditions for $f$ to be homotopic, relative to its postcritical set, to an expanding map $g$.
Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary
(American Institute of Mathematical Sciences, 2012) Gie, G.-M.; Makram, H.; Tema,, R.
We consider the Navier-Stokes equations of an incompressible fluid in a three dimensional curved domain with permeable walls in the limit of small viscosity. Using a curvilinear coordinate system, adapted to the boundary, we construct a corrector function at order $\varepsilon^{j}$, $j = 0, 1$, where $\varepsilon$ is the (small) viscosity parameter. This allows us to obtain an asymptotic expansion of the Navier-Stokes solution at order $\varepsilon^{j}$, $j = 0, 1$, for $\varepsilon$ small . Using the asymptotic expansion, we prove that the Navier-Stokes solutions converge, as the viscosity parameter tends to zero, to the corresponding Euler solution in the natural energy norm. This work generalizes earlier results in [14] or [26], which discussed the case of a channel domain, while here the domain is curved.
Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type
(Springer, 2012) Cho, Y.J.; Rhoades, B.E.; Saadat, R.; Samet, B.; Shatanawi, W.
In this article, we study coupled coincidence and coupled common fixed point theorems in ordered generalized metric spaces for nonlinear contraction condition related to a pair of altering distance functions. Our results generalize and modify several comparable results in the literature.
Coupled coincidence points for two mappings in metric spaces and cone metric spaces
(Springer, 2012) Long, W.; Rhoades, B.E.; Rajović, M.
This article is concerned with coupled coincidence points and common fixed points for two mappings in metric spaces and cone metric spaces. We first establish a coupled coincidence point theorem for two mappings and a common fixed point theorem for two $w$-compatible mappings in metric spaces. Then, by using a scalarization method, we extend our main theorems to cone metric spaces. Our results generalize and complement several earlier results in the literature. Especially, our main results complement a very recent result due to Abbas et al.
A second order algebraic knot concordance group
(Mathematical Sciences Publishers, 2012) Powell, M.
Let C be the topological knot concordance group of knots $S^{1} \subset S^{3}$ under connected sum modulo slice knots. Cochran, Orr and Teichner defined a filtration: \[C \supset F_{(0)} \supset F_{(0.5)} \supset F_{(1)} \supset F_{(1.5)} \supset F_{(2)} \supset\cdots\] The quotient $C/F_{(0.5)}$ is isomorphic to Levine’s algebraic concordance group; $F_{(0.5)}$ is the algebraically slice knots. The quotient $C/F_{(1.5)}$ contains all metabelian concordance obstructions. Using chain complexes with a Poincaré duality structure, we define an abelian group $AC_{2}$, our second order algebraic knot concordance group. We define a group homomorphism $C \rightarrow AC_{2}$ which factors through $C/F_{(1.5)}$, and we can extract the two stage Cochran–Orr–Teichner obstruction theory from our single stage obstruction group $AC_{2}$. Moreover there is a surjective homomorphism $AC_{2} \rightarrow C/F_{(0.5)}$, and we show that the kernel of this homomorphism is nontrivial.
Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with Onsager mobility
(American Institute of Physics, 2012) Liu, H.; Sengul, T.; Wang, S.
The main objectives of this article are two-fold. First, we study the effect of the nonlinear Onsager mobility on the phase transition and on the well-posedness of the Cahn-Hilliard equation modeling a binary system. It is shown in particular that the dynamic transition is essentially independent of the nonlinearity of the Onsager mobility. However, the nonlinearity of the mobility does cause substantial technical difficulty for the well-posedness and for carrying out the dynamic transition analysis. For this reason, as a second objective, we introduce a systematic approach to deal with phase transition problems modeled by quasilinear partial differential equations, following the ideas of the dynamic transitiontheory developed in Ma and Wang [Phase Transition Dynamics in Nonlinear Sciences (Springer) (to be published); Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises Vol. 53 (World Scientific, Hackensack, NJ, 2005)].
On the Taylor tower of relative $K$-theory
(Mathematical Sciences Publishers, 2012) Lindenstrauss, A.; McCarthy, R.
For $R$ a discrete ring, $M$ a simplicial $R$–bimodule, and $X$ a simplicial set, we construct the Goodwillie Taylor tower of the reduced $K$–theory of parametrized endomorphisms $\widetilde{K}\bigg(R; \widetilde{M}\big[X\big]\bigg)$ as a functor of $X$. Resolving general $R$–bimodules by bimodules of the form $\widetilde{M}\big[X\big]$, this also determines the Goodwillie Taylor tower of $\widetilde{K}\bigg(R; M\bigg)$ as a functor of $M$. The towers converge when $X$ or $M$ is connected. This also gives the Goodwillie Taylor tower of $\widetilde{K}\big(R⋉M\big)\simeq\widetilde{K}\big(R;B.M\big)$ as a functor of $M$. For a functor with smash product $F$ and an $F$–bimodule $P$, we construct an invariant $W\big(F;P\big)$ which is an analog of $TR\big(F\big)$ with coefficients. We study the structure of this invariant and its finite-stage approximations $W_{n}\big(F;P\big)$ and conclude that the functor sending $X\mapsto W_{n}\big(R;\widetilde{M}\big[X\big]\big)$ is the n–th stage of the Goodwillie calculus Taylor tower of the functor which sends $X\mapsto\widetilde{K}\big(R;\widetilde{M}\big[X\big]\big)$. Thus the functor $X\mapsto W\big(R;\widetilde{M}\big[X\big]\big)$ is the full Taylor tower, which converges to $\widetilde{K}\big(R;\widetilde{M}\big[X\big]\big)$ for connected $\text{X}$.
New bounds on cap sets
(American Mathematical Society, 2012) Katz, N.H.; Bateman, M.
We provide an improvement over Meshulam's bound on cap sets in $F^{N}_{3}$. We show that there exist universal $\epsilon > 0$ and $ C > 0$ so that any cap set in $F^{N}_{3}$ has size at most $C\frac{3^{N}}{N^{1+e}}$. We do this by obtaining quite strong information about the additive combinatorial properties of the large spectrum.

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