WINRS, Mid-Atlantic Region:
WINRS, Mid-Atlantic Region:
Algebra and Combinatorics Session:
• Victoria Akin (Duke University): Automorphisms of the Punctured Mapping Class Group.
We can describe the point-pushing subgroup of the mapping class group topologically as the set of maps that push a puncture around loops in the surface. However, we can characterize this topological subgroup in purely algebraic terms. Using group theoretic tools and a classic theorem of Burnside, we can recover a result of Ivanov-McCarthy establishing the triviality of Out(Mod±). To this end, we’ll demonstrate that the point-pushing subgroup is “unique” in the mapping class group.
• Sahana Balasubramanya (UNC, Greensboro): Hyperbolic structures on wreath products.
The poset of hyperbolic structures on a group G is still very far from being understood and several questions remain unanswered. In this talk, I will speak about some new results that describe hyperbolic structures on the wreath product Gwr Z, for any group G. As a consequence, I answer two open questions regarding quasi-parabolic structures: I will give an example of a group G with an uncountable chain of quasi-parabolic structures and give examples of groups that have finitely many quasi-parabolic structures.
• Talia Fernos (UNC, Greensboro): Groups and CAT(0) Cube Complexes
CAT(0) cube complexes have interesting interconnections between geometry, analysis, and algebra. In this introductory talk, we discuss a variety of aspects of groups acting on CAT(0) cube complexes. We will give examples of groups acting properly and examples who never act without fixed points. We will also discuss the existence of rank-1 and regular elements.
• Rebecca Goldin (George Mason University): On some (equivariant) combinatorics related to flag manifolds.
Schubert calculus arose from 19th Century explorations of enumerative geometry, but has evolved into the study of specific rings associated with homogeneous spaces. The topic includes, for example, examining the product structure of the cohomology ring of the Grassmannian of k-dimensional subspace of complex n-space, as well as the K-theory or equivariant K-theory of the complete flag variety. In a geometrically motivated basis for each ring, the product of any two basis elements, expanded in that basis, has coefficients that are "positive" in an appropriate sense.
Remarkably, equivariant cohomology and K-theory--which take into account a group action by a torus T (an abelian group)--have some advantages over their non-equivariant counterparts, even as they have strictly more information. I will give an overview of the structure of these rings, in terms of combinatorics and algebra only. These descriptions provide hope for future positive formulas on structure constants.
• Patricia Hersh (NC State): Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology.
Given a polytope P and generic linear functional c, one obtains a directed graph G(P,c) by taking the 1-skeleton of P and orienting each edge e(u,v) from u to v for c(u) < c(v). We will discuss the question of finding sufficient conditions on P and c so that G(P,c) will not have any directed paths which revisit a face of P after departing from it. This is equivalent to the question of finding conditions on P and c under which the simplex method for linear programming will be efficient under all choices of pivot rules. Conditions are given which provably yield a corollary of the desired nonrevisiting property. One of the proposed conditions is that G(P,c) be the Hasse diagram of a partially ordered set, which is equivalent to requiring nonrevisiting of 1-dimensional faces. This opens the door to the usage of poset-theoretic techniques. This also leads to a result for simple polytopes in which G(P,c) is the Hasse diagram of a lattice L that the order complex of each open interval in L is homotopy equivalent to a ball or a sphere, with applications to weak Bruhat order, the Tamari lattice and Cambrian lattices. We will give an overview of this story, providing background along the way.
• Bakul Sathaye (Wake Forest University): Non-hyperbolic Coxeter Groups with Menger Boundary
Many word hyperbolic groups have Gromov boundary homeomorphic to the Menger curve. However, until recently it was not known whether this is true in the broader setting of CAT(0) groups. In this talk, we discuss the first explicit examples of non-hyperbolic CAT(0) groups with Menger boundary. These groups are a family of Coxeter groups with isolated flats. This is joint work with M. Haulmark and G. C. Hruska.
Algebraic Geometry, Commutative Algebra and Number Theory Session:
• Candace Bethea (University of South Carolina): Degree computations in A^1-homotopy theory and an application to Riemann-Hurwitz over a non-perfect field.
One goal in homotopy theory is to understand homotopy classes of self-maps of spheres. In classical topology, the degree map gives us an isomorphism of [S^n, S^n] with the integers. I will talk about two ways to compute degrees in A^1 homotopy theory: Cazanave's computation of naive homotopy classes of rational functions from P^1 to itself and Kass and Wickelgren's local to global motivic degree results. I will end with an application of these degree computations to an enriched Riemann-Hurwitz formula over a non-perfect field, the case of a perfect field done by Marc Levine in 2017.
• Emily Bergman (University of Delaware): Secondary Constructions of PN and APN Functions.
Highly non-linear functions are important in cryptology as they provide security against differential cryptanalysis. Differential uniformity is a measure of a function's linearity, with highly non-linear functions having low differential uniformity. There is a natural lower bound for the differential uniformity of any function, and functions over finite fields that are optimal with respect to this bound are known as perfect nonlinear (PN) for odd characteristic, and almost perfect nonlinear (APN) for even characteristic. We shall discuss the concept of differential uniformity, and PN and APN functions. In particular, our goal is to develop secondary constructions for functions with low differential uniformity; in other words we want to develop methods by which a given PN or APN function can be altered to obtain a new function with low differential uniformity, preferably optimal. We will discuss previous research, including connections with mutually orthogonal systems, along with our initial attempts to develop a secondary construction.
• Ela Celikbas (West Virginia University): Embeddings of canonical modules and resolutions of connected sums.
It is well-known that, for a Cohen-Macaulay local ring S with a canonical module \omega_S, if S is generically Gorenstein, then \omega_S can be identified with an ideal of S, that is, \omega_S embeds into S.
In this talk, we are concerned with a specific embedding of a canonical module of R/I_{m,n}$ to itself, where I_{m,n} is an ideal generated by all square-free monomials of degree m in a polynomial ring R with n variables. We discuss how to construct such an embedding using a minimal generating set of Hom_R(R/I_{m,n}, R/I_{m,n}). Using this embedding, we give a resolution of connected sums of certain Artinian k-algebras where k is a field. (This talk is based on a recent joint work with Jai Laxmi and Jerzy Weyman.)
• Alicia Lamarche (University of South Carolina): Exceptional collections of toric varieties associated to root systems.
Given a root system R, one can construct a toric variety X(R) by taking the maximal cones of X(R) to be the Weyl chambers of R. The automorphisms of R act on X(R); and a natural question arises: can one decompose the derived category of coherent sheaves on X(R) in a manner that is respected by Aut(R)? Recently, Castravet and Tevelev constructed full exceptional collections for D^b(X(R)) when R is of type A_n. In this talk, we'll discuss progress towards answering this question in the case where R is of type D_n, with emphasis on the 'base' case of D_4.
• Rebecca R.G. (George Mason University): Singularities of commutative rings of all characteristics via big Cohen-Macaulay modules.
In characteristic p>0, many of the existing results on the singularities of commutative rings were proved using tight closure, a technique developed by Mel Hochster and Craig Huneke. In rings of characteristic 0 or mixed characteristic, similar results have been achieved by reduction to characteristic p. We discuss a characteristic-free approach to the study of singularities of commutative rings that relies on particular modules called big Cohen-Macaulay modules.
• Ashley Wheeler (Mount Holyoke College): Matroids and the bracket algebra.
The bracket ring is the K-subalgebra of K[X] generated by the size r minors of X (an r x n matrix; K = field). It is well-known as the homogeneous coordinate ring for the Grassmann variety. But we can also think of the columns of X as naming coordinates of n points in (r-1)-dimensional projective space. Thinking this way, it turns out a Plücker coordinate indexed by a set of columns of X vanishes, if and only if the corresponding r points are dependent (i.e. coplanar, collinear, etc.). The question is, given a prescribed configuration of n points in projective (r-1)-space, what is the matroid defined by the corresponding dependencies?
Analysis, PDE and Probability Session:
• Constanze Liaw (University of Delaware): Spectral Theory of Finite Rank Perturbations.
The Kato-Rosenblum theorem and Aronszajn-Donoghue theory provide us
with reasonably good understanding of the subtle theory of rank one perturbations. We will discuss these statements.
When considering higher rank perturbations, the situation is different. While the Kato-Rosenblum theorem still ensures the stability of the absolutely continuous part of the spectrum, the singular parts behavior may be more complex.
Nonetheless, some positive results prevail in the finite rank setting.
• Katrina Morgan (University of North Carolina at Chapel Hill): Wave Decay on Asymptotically Flat Spacetimes.
The standard wave operator can be viewed as a generalization of the Laplacian on a flat Lorentzian spacetime with signature (3,1) (here we assume 3 spatial dimensions). The wave operator on curved Lorentzian spacetimes (e.g. spacetimes arising in General Relativity) is again given by the generalized Laplacian. A standard result called Sharp Huygens’ Principle establishes that solutions to the flat wave equation decay all the way to 0 in finite time at any point in space. The decay of solutions to the wave equation on a class of curved spacetimes which tend toward flat at a rate of |x|^{-1} was studied in Tataru 2013. In this case, Tataru established a t^{-3} decay rate. The current work examines the rate of wave decay in between the known cases: when the spacetime is curved but becomes flat at a faster rate of |x|^{-k} for k = 2, 3, 4, …. The techniques used in this ongoing work, including resolvent and local energy estimates, will be discussed. No familiarity with these tools will be assumed.
• Jessica Kelly (Christopher Newport University): A Lyapunov-type inequality for differential equations involving fractional derivatives.
Lyapunov's famous inequality provides a lower bound for the distance between two consecutive zeros of the nontrivial solution of the differential equation y''+qy=0 under Dirichlet boundary conditions. Since Lyapunov's result in 1907, many applications have been discovered. Additionally, there are several results containing Lyapunov--type inequalities, wherein aspects such as the differential equation and boundary conditions are modified.
In this talk, we will prove a Lyapunov--type inequality by considering the differential equation D^\alpha y+qy=0 for alpha greater than 2 and less than or equal 3, where D^\alpha is the Riemann-Liouville fractional derivative and yis a nontrivial solution satisfying the three--point boundary conditions y(a)=y(b)=y(c)=0 and y(a)=y''(d)=y(b)=0, for suitably ordered values of a,b,c,d.
• Andrea Carracedo Rodriguez (Virginia Tech): Interpolatory Model Reduction of Parametric Bilinear Systems
Subsystem interpolation for model order reduction of linear dynamical systems has been extended to the parametric linear case and to the bilinear non-parametric case. Here we extend such framework to parametric bilinear systems.
•Benjamin Russo (Farmingdale State College SUNY): Fractional derivatives and the Segal-Bargmann space.
In this talk we will introduce a generalized Segal-Bargmann space which uses the Mittag-Leffler function as its reproducing kernel. This space has been featured in the development of a finite difference method. However, it has yet to be investigated as a space of entire functions. We will discuss some preliminary results in comparison to the Segal-Bargmann space and potential applications. This is joint work with Joel Rosenfeld and Warren Dixon.
• Katie Quertermous (James Madison University): C*-algebras Generated by Composition and Toeplitz Operators.
If \varphi is an analytic self-map of the unit disk D, then the composition operator C_{\varphi} : f \mapsto f \circ \varphi is a bounded operator on the Hardy space H^2(D). We are primarily interested in composition operators induced by linear-fractional self-maps of D. In this talk, we investigate the C*-algebras generated by composition operators induced by linear-fractional maps and either the unilateral shift T_z or the ideal K of compact operators and explore how descriptions of the structures of these C*-algebras, modulo the ideal K, can be applied to obtain information about the spectra of algebraic combinations of composition and Toeplitz operators.
Applied Mathematics and Biomathematics Session:
• Lauren Childs (Virginia Tech): Stochastic simulation model of within-mosquito malaria parasite diversity generation.
Plasmodium falciparum, the malaria parasite causing the most severe disease in humans, undergoes an asexual stage within the human host, and a sexual stage within the mosquito host. Because mosquitoes may be superinfected with parasites of different genotypes, this sexual stage of the parasite life-cycle presents the only opportunity in the full life cycle for genetic mixing of parasites. To investigate the role that mosquito biology plays for parasite diversity, we constructed a stochastic model of parasite development within the mosquito, generating a distribution of parasite densities at five parasite stages over the lifespan of a mosquito. We then coupled a model of sequence diversity generation via recombination and reassortment between genotypes to the population model. Our two-part model framework shows that early bottlenecks decrease diversity from the initial parasite population in a mosquito's blood meal, but diversity increases later in the parasite life cycle within the mosquito. Furthermore, beginning from only two distinct parasite genotypes, the probability of transmitting more than two unique genotypes from mosquito to human is high (> 65%) for many realistic infecting parasite densities.
• Maria-Veronica Ciocanel (Mathematical Biosciences Institute at The Ohio State University): Wave Propagation in Models of mRNA transport.
In many organisms, messenger RNA (mRNA) must accumulate at the egg cell periphery to ensure healthy development. The transport of these particles is not well understood, but is thought to depend on diffusion, bidirectional movement and anchoring mechanisms. We investigate these proposed mechanisms using linear and nonlinear PDE models and analysis, informed by numerical parameter estimation. Our results yield spreading Gaussian solutions for mRNA concentrations. We predict that accounting for the microtubule cytoskeleton in these transport models through spatially-dependent switching rates may be key in better understanding time and spatial scales of intracellular transport. I will also comment on our recent results using renewal rewards theory in a stochastic model formulation to understand effective transport properties of these models.
• Samantha Erwin (North Carolina State University): Understanding toxin production during Clostridioides difficile infection using high dimensional data.
The main virulence factors in C. difficile infection are toxins, yet little is known about the specific mechanism leading to it. In recent years, new technology has allowed us to achieve measurements of hundreds of metabolites and the expression of thousands of genes which maybe crucial to understand the driving factors in C. difficile colonization and toxin production. It is believed that toxin production is mediated by competition for nutrients in the gut metabolome which makes the large-scale metabolomics data useful. However, with this all-encompassing ‘omics’ data comes a critical need to reduce these datasets to the most functional elements so that we can discover key components driving toxin production. We use a recent animal model for C. difficile infection in which mice were antibiotic treated with cefoperazone and challenged with C. difficile 2 days following treatment. We develop sparse graphical networks to identify correlations between metabolites and toxins within high dimensional data sets and develop a mechanistic model of processes related to our network. We find the Stickland reaction to by critical in toxin production and suggest potential mitigation strategies for reducing toxin production.
• Katharine Gurski (Howard University): Sexually Transmitted Diseases: Lifetime Number of Partners and Longterm Partners.
Population models for sexually transmitted diseases are typically based on an infection transmission model that is better suited for measles or the flu by including only the risk that a susceptible partner can be infected as the probability per sexual act per partnership with an inherent partnership length of zero. We overcome this weakness by developing a population model that can account for the possibilities of an infection from either a casual sexual partner or a longtime partner who was uninfected at the start of the partnership. The model allows for multiple longterm partnerships, which adds the advantage that network models have, the means to include serially monogamous and concurrent relationships, within the traditional strengths of a population model for computational speed and understanding of how each parameter affects the disease spread in an analytic reproduction number. The model can be further diversified by including heterogeneous subpopulations, eg. populations divided by sexual behavior and other characteristics. We develop a model for sexually transmitted diseases with longterm partnerships using a SIR (Susceptible-Infected-Removed) population model with differential infectivity, i.e. infection stages with different infectiousness levels. We present a new treatment for contact numbers, the average number sexual partners per year with whom one has had a sexual encounter, in disease transmission rates which will result in a more realistic number of lifetime partners. Results include systems of equations for a homogeneous group (no sexual behavior differentiation) and a heterogeneous group (includes sexual behavior differentiation into two populations), along with the accompanying reproduction numbers, and numerical simulations using HIV and HSV-2 data.
• Leah Shaw (William & Mary): Stochastic model of social interactions in yeast biofilms
In a biofilm, microbes cooperate and produce beneficial chemicals (public goods) that are used by all members. They are therefore susceptible to cheaters who do not produce public goods yet benefit from them. However, some cooperators can exhibit kin recognition, in which they cooperate exclusively with themselves and not with cheater cell types. We develop a stochastic spatial simulation of a yeast biofilm and model social interactions between strains such as cooperation, competition, cheating, and kin recognition. We vary social interaction parameters and observe the effect on population sizes and on spatial distribution of the cell types.
Geometry and Low-Dimensional Topology Session:
• Danielle O’Donnol (Marymount University): Legendrian Realizations of Planar Graphs.
An embedded graph is Legendrian if it is everywhere tangent to the contact structure. We consider topologically planar Legendrian embeddings of planar graphs in the 3-sphere with the standard contact structure. I will talk about our recent work on classification problems for planar Legendrian graphs. We have results on Legendrian simplicity, the Legendrian mirror problem, and stabilization equivalence. This is joint with Peter Lambert-Cole (Georgia Tech).
• Svetlana Katok (Penn State University): Coding of geodesics via continued fractions and their generalizations.
I will discuss a method of coding of geodesics on quotients of the hyperbolic plane by Fuchsian groups using boundary maps and “reduction theory”. For the modular surface these maps are related to a family of (a,b)-continued fractions, and for compact surfaces they are generalizations of the Bowen-Series maps, also studies by Adler and Flatto. The boundary maps are given by the generators of the group and have a finite set of discontinuities. We study the two forward orbits of each discontinuity point and show that for a family of such maps the cycle property holds: the orbits coincide after finitely many steps. We also show that for an open set of discontinuities the associated two-dimensional natural extension maps possess global attractors with finite rectangular structure to which (almost) every point is mapped after finitely many iterations. These two properties belong to the list of “good” reduction algorithms, equivalence or implications between which were suggested by Don Zagier. I will also explain how the geodesic flow can be represented symbolically as a special flow over a cross-section of “reduced” geodesics parametrized by the corresponding attractor. The talk is based on joint works with Ilie Ugarcovici and Adam Zydney.
• Heather Russel (University of Richmond): Ineffective Sets and Region Crossing Change.
Region crossing change (RCC) is an operation performed on link diagrams in which one reverses all crossings incident to a specific region of the diagram. Such operations are interesting to knot theorists because they can aid in the study of knot invariants. We call a collection of regions such that performing RCCs on all of them changes no crossings of the diagram an ineffective set for that diagram. Ineffective sets are a key tool in the literature exploring RCC. In this talk, we explain why ineffective sets are useful and give an easy method for finding all ineffective sets for any diagram.
• Radmila Sazdanovic (North Carolina State University): Torsion in Khovanov homology.
In this talk we give a brief overview of Khovanov homology, the developments in low dimensional topology it has inspired, and relations with other link homology theories. Special emphasis is placed on the role of torsion in Khovanov homology.
• Lisa Traynor (Bryn Mawr): Legendrian Torus Links.
Legendrian knots and links are smooth knots and links that satisfy a geometrical condition imposed by a contact structure. Due to this extra structure, there are multiple (in fact, infinitely many) Legendrian representatives of any knot or link. In this talk, we focus on Legendrian torus knots and links. Etnyre and Honda gave a classification of Legendrian torus knots. I will discuss the classification of unordered and ordered Legendrian torus links. This is joint work with Jennifer Dalton and John Etnyre.
History of Mathematics Session:
• Amy Ackerburg-Hastings (Smithsonian, National Museum of American History): Charles Davies as a Philosopher of Mathematics Education.
Charles Davies (1798–1876), who taught at West Point, Hartford's Trinity College, New York University, and Columbia, was one of the most prolific and popular compilers of mathematics textbooks in the United States in the 19th century. This talk explores his 1850 The Logic and Utility of Mathematics, With the Best Methods of Instruction Explained and Illustrated, which James K. Bidwell and Robert G. Clason (1970) and Phillip S. Jones and Arthur F. Coxford, Jr., (1970) called the "first American book on mathematics teaching methods." I will provide an overview of the contents of this volume. I will also consider how Davies and this book contributed to the gradual professionalization (and feminization) of American schoolteaching.
• Nancy Hall (University of Delaware): Gertrude Cox - A Modern Statistical Pioneer.
Certrude Cox (1900-1972) was an Iowa farm girl. Working her way through college at Iowa State so she could qualify to run an orphanage, she assisted in the Statistics Department, and eventually got a Masters Degree in Statistics. In 1940 she went to Raleigh, N.C. where she singlehandedly founded the Statistics Department at North Carolina State, training and mentoring many, a Statistics Department at the the University of North Carolina at Chapel Hill, a Department of Biostatistics, and an Institute of Statistics. She was a founding participant in RTI (Research Triangle Institute). After five years there she started working abroad under the auspices of the U.S. State Department and the Rockefeller Foundation. She worked extensively in Egypt and in Thailand, teaching courses in experimental design and advising on experiments. She was the author or co-author of forty-one journal articles and books. Her 1950 book Experimental Designs, with William Cochran, is still on many shelves today.
• Karen Parshall (University of Virginia): Americans Take the International Stage?: The Aborted 1940 International Congress of Mathematicians.
In the 1930s, some within the American mathematical community had begun to think seriously about hosting an International Congress of Mathematicians (ICM) in the United States, and it seemed like 1940 would be their year. The American mathematical community had something to prove. They aimed both to showcase American mathematical accomplishments and to play a more important and visible international leadership role. This talk will examine the planning for what was ultimately—owing to the outbreak of World War II in Europe—the aborted 1940 ICM to have been held in Cambridge, Massachusetts.
• Cathy Shi (University of Richmond): Solomon Lefschetz: The right person at the right place at the right time.
Solomon Lefschetz played a critical role in the American mathematical community in the early twentieth century. He contributed significantly to algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. He not only exhibited academic excellence in mathematics, but he also demonstrated leadership as a faculty member at the University of Princeton and as President of the American Mathematical Society. He edited the Annals of Mathematics and revised American engineering education. Even with all of his contributions to mathematics and the broader community, Lefschetz is often described as a man with rather unpleasant characteristics. Yet, Lefschetz alone wrote the letter that would bring Emil Artin and his young family to America to escape the situation in Nazi Germany. This talk offers a richer understanding of Lefchetz the man and his work in mathematics and shows the powerful combination of the right person at the right place at the right time.
• Brit Shields (University of Pennsylvania): Solving the “Shortage of Trained Brains”: The Engineering, Science and Management War Training Program During the Second World War.
With the outbreak of the Second World War, many US mathematicians, scientists and engineers began research work for the war effort. Those at universities also participated in one of the largest government-subsidized educational programs in US history. The Engineering, Science and Management War Training Program, funded by the US Office of Education, operated on over 200 campuses nationwide. The program included opportunities for students to train in engineering or for current engineers to gain new skills. This talk will discuss the role of the mathematical sciences within this program, focusing on how the courses developed at New York University.
• Ying Wu (University of Richmond): Aiming for high standards: Solomon Lefschetz as editor of the Annals of Mathematics.
Lefschetz served as the main editor for the Annals of Mathematics from 1928 to 1958, an important period for the journal. During this time, it became an increasingly well-known and respected journal. Its rise, in turn, stimulated American mathematics. This work specifically looks at Lefschetz’a role as editor of the Annals, the papers that were published in the journal, the papers that were not published in the journal, the editorial boards, the presence---or not---of international authors, etc. In particular, we hope to address rumors related to him “favoring” contributions from his students and contributions in topology.
Topology Session:
• Rebecca Fields (James Madison University): Finite generation of group cohomology
The fact that the derived functor of invariants of a finite group G acting on a finitely generated module over a Noetherian ring is finitely generated is an important classic result. We will discuss two of the standard proofs of this result, in the language of algebraic topology, and how they relate to contemporary research.
• Kelli Karcher (Virginia Tech): The Space of Biorders on Solvable Groups.
A group is said to be biorderable if it has a total order invariant under left and right multiplication. These orders can be given a topology that is called the space of biorders on this group. I will focus on solvable groups to show that under certain conditions the space of biorders is either finite or homeomorphic to a Cantor set.
• Martina Rovelli (Johns Hopkins University): What features are detected by characteristic classes?
We propose a uniform interpretation of characteristic classes as complete obstructions to the reduction of the structure group of a principal bundle, and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber. By plugging in the correct parameters, we recover several classical theorems. This leads to the definition of a family of invariants for principal bundles that detect the number of group reductions associated to characteristic classes that a principal bundle admits.
• Sarah Yaekel (University of Maryland): Isovariant cell complexes.
An isovariant map between G-spaces is an equivariant map which preserves isotropy groups. Isovariant maps appears in situations where homotopy theory is applied to geometric problems, for example, in surgery theory. We will describe some new results in isovariant homotopy theory with plenty of examples. This is work in progress, based on conversations with Cary Malkiewich, Mona Merling, and Kate Ponto.
• Alexandra Yarosh (Penn State): Topological detection of monotone rank
Monotone rank of a matrix is the minimal possible rank of a matrix obtained by applying arbitrary monotone functions to each row of M. The problem of finding the non-linear rank often arises in neuroscience context, in particular in estimating the dimension of the space of stimuli, sampled by neurons. While an exact algorithm for computing the nonlinear rank is unknown, it can be efficiently estimated using topological methods, as it is closely related to the geometry and topology of hyperplane arrangements.
A natural tool that captures the essence of the problem is directed complexes -- a generalization of simplicial complexes that allows us to keep track of the order of vertices. Directed complexes capture much of the relevant geometric information. For example, the nonlinear rank, as well as other geometric properties of data can be estimated from the homology of an associated directed complex. I will present results and conjectures about the directed complexes associated to some problems of estimating the dimension of the space of stimuli.
Emily Riehl (Johns Hopkins University): Categorifying cardinal arithmetic.
In this interactive talk we’ll prove, with help from the audience, the distributivity of multiplication over addition — a x (b + c) = a x b + a x c — not via the usual methods but by diving deeper into the question of what cardinal numbers really mean. The first deep idea is categorification, where we understand cardinal numbers as describing sizes of sets. The second step involves the Yoneda lemma, which tells us that any set can be characterized by the collection of functions for which it serves as the domain. The third deep idea describes operations + and x on sets via their universal properties, that is, by characterizing the functions whose inputs are drawn from the sets so-construted. The final step involves the notion of adjunction, in this case an operation known as “currying” in computer science. In an epilogue, we will discover that the proof just described applies in vastly more general contexts and try to understand why one would want to bother describing mathematical objects in this abstract fashion.
Kim Sellers (Georgetown University): Making Statistics that Count!
Discrete data (particularly counts) introduce an added layer of complexity to any sort of statistical analysis. While their is historically a classical distribution (namely the Poisson distribution) used to describe and/or model such data, constraining qualities bring its use into question. This talk will introduce alternative models, particularly one with which I have worked heavily to advance the field of discrete or count data modeling.