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Eberly College of Science Mathematics Department

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September 1st, 2015 (11:15am - 12:05pm)
Seminar: Combinatorics/Partitions Seminar
Title: A Combinatorial Proof of a Relationship Between Maximal (2k-1,2k+1)-cores and (2k-1,2k,2k+1)-cores
Speaker: James Sellers, Penn State
Location: MB106

Integer partitions which are simultaneously t-cores for distinct values of t have attracted significant interest in recent years. When s and t are relatively prime, Olsson and Stanton have determined the size of the maximal (s,t)-core. When k > 1, a conjecture of Amdeberhan on the maximal (2k-1,2k,2k+1)-core has also recently been verified by numerous authors. In this work, we analyze the relationship between maximal (2k-1,2k+1)-cores and maximal (2k-1,2k,2k+1)-cores. In previous work, Nath noted that, for all k > 0, the size of the maximal (2k-1,2k+1)-core is exactly four times the size of the maximal (2k-1,2k,2k+1)-core and requested a combinatorial interpretation of this unexpected identity. Here, using the theory of abaci, partition dissection, and elementary results relating triangular numbers and squares, we provide such a combinatorial proof. This is joint work with Rishi Nath.

September 1st, 2015 (01:30pm - 02:20pm)
Seminar: Theoretical Biology Seminar
Title: Note special time: *Thursday* at 12:20pm
Speaker: Igor Aronson, Argonne National Laboratory
Location: MB106
September 1st, 2015 (02:30pm - 03:45pm)
Seminar: Logic Seminar
Title: Degrees of unsolvability. Part 1: introduction
Speaker: Stephen G. Simpson, Penn State
Location: MB315

This series of talks is based on my recent 3-hour tutorial at the Computability in Europe conference in Bucharest. An important 20th century discovery is that there are algorithmically unsolvable problems in virtually every branch of mathematics. Degrees of unsolvability are an attempt to measure the "amount" or "extent" of the unsolvability of such problems. They are based on the idea of "reducibility" of one problem to another. In this introductory talk we define and discuss two particular degree structures: the Turing degrees, and the Muchnik degrees. We point out that the Muchnik degrees are the completion of the Turing degrees, similarly to how the real numbers are the completion of the rational numbers. We discuss specific examples of Turing degrees, and we give at least one specific example of a Muchnik degree which is not a Turing degree. (There are many more such examples, and we shall discuss some of them in later talks in this series.) We point out how the Muchnik degrees provide a rigorous implementation of Kolmogorov's calculus of problems.

September 1st, 2015 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: Yang-Baxter and reflection equations: unifying structures behind quantum and classical integrable systems
Speaker: Vincent Caudrelier, City University of London
Location: MB106

The Yang-Baxter equation (YBE) is central in the theory of quantum integrable systems. For decades, together with its companion for problems with boundaries (the quantum reflection equation), it has been studied and used in the quantum realm, leading to the area of quantum groups. But it was suggested by Drinfeld in 1990 that the general study of the so-called «set-theoretical YBE » is also important. This can be understood as the problem of finding nonlinear representations of the braid group on arbitrary sets. It turns out that classical integrable PDEs provide a means to construct certain types of such representations, called Yang-Baxter maps, by looking at soliton collisions. I will use the vector nonlinear Schrödinger (NLS) equation as the main example to illustrate the idea. It has its origin in the physics of wave phenomena in fluid dynamics, nonlinear optics, plasma physics or quantum cold gases. After reviewing this, I will show how the new concept of set-theoretical reflection equation naturally emerges by studying solitons in integrable PDEs with a boundary. As before, the problem of finding solutions to this equation can be understood as the question of finding nonlinear representations of the finite Coxeter group of type BCn. I will show how to construct such representations using solitons.

September 3rd, 2015 (11:15am - 12:05pm)
Seminar: Algebra and Number Theory Seminar
Title: Some Open Problems Arising from my Recent Finite Field Research
Speaker: Gary Mullen, Penn State University
Location: MB106

We will discuss a number of my favorite open problems and conjectures which have arisen in my recent research related to finite fields. These discussions will focus on a variety of areas including some theoretical topics as well as some topics from combinatorics and coding theory.

September 3rd, 2015 (12:20pm - 01:10pm)
Seminar: Center for Interdisciplinary Mathematics Seminar
Title: Computational model of cell motility
Speaker: Igor Aronson, Argonne National Laboratory
Location: MB114
Abstract: http://

Cell motility and collective migration are among the most important themes in cell biology, mathematical biology, and bioengineering, and are crucial for morphogenesis, wound healing, and immune response in eukaryotic organisms. It is also relevant for the development of effective treatment strategies for diseases such as cancer, and for the design of bioactive surfaces for cell sorting and manipulation. Substrate-based cell motility is, however, a very complex process as both regulatory pathways and physical force generation mechanisms are intertwined. To understand the interplay between adhesion, force generation and motility, we develop a computational model based on the phase field method, which is especially suited to treat the moving and deformable boundaries involved in both individual and collective cell motility. The resulting system of coupled PDEs with the non-local volume-conservation constraint is solved by the quasi-spectral method in a periodic two-dimensional square domain. The model captures all essential phenomenology exhibited by moving cells, including the abrupt onset of motion and the response to external stimuli. We investigate by the means of large-scale GPU computations how cells navigate on substrates with patterned adhesion properties and modulated stiffness of substrate. Such substrates are currently under technological development to collect and sort cells. For multiple cells, the generalized multiphase-field model is able to predict that collective cell migration emerges spontaneously as a result of inelastic collision-type interactions of cells.

September 3rd, 2015 (01:25pm - 02:25pm)
Seminar: MASS Colloquium
Title: A tour of Pritchard Lab
Speaker: Diane Henderson, Penn State
Location: MB114

The MASS students will be introduced to the Pritchard Fluids Lab, a physics that is a part of the Mathematics Department.

September 3rd, 2015 (02:30pm - 03:30pm)
Seminar: Noncommutative Geometry Seminar
Title: Noncommutative Geometry 102: Asymptotics and spectral theory
Speaker: Nigel Higson, Penn State
Location: MB106

I shall give two lectures introducing some of the ideas that appear in the research of Penn State's noncommutative geometry group. In the first I shall discuss differential and Hilbert space operators, and various sorts of "symbols" that can be attached to them. In the second I shall examine how the general theory applies to Sturm-Liouville operators on a half-line, following some remarkable early work Hermann Weyl that has proved to be very influential in representation theory.

September 3rd, 2015 (03:30pm - 04:20pm)
Seminar: Department of Mathematics Colloquium
Title: Departmental Reception
Speaker: Departmental Reception
Location: MB114
September 4th, 2015 (03:00pm - 04:30pm)
Seminar: CCMA PDEs and Numerical Methods Seminar Series
Title: An introduction of the Allen-Cahn equation and the Cahn-Hilliard equation
Speaker: Yukun Li, Penn State University
Location: MB315

First we will briefly talk about the background in material science, then we will show the relation between these two equations and geometric problems. We will focus on the Cahn-Hilliard equation, and introduce some well-known facts, including existing PDE and numerical results. Besides, we will discuss the spectrum estimates, which play important roles in analyzing the asymptotic behavior of the solution of the Cahn-Hilliard equation. Finally, some numerical tests are given to verify the evolution of the solution, which is consistent with the corresponding geometric law.

September 4th, 2015 (03:35pm - 04:35pm)
Seminar: Probability and Financial Mathematics Seminar
Title: Algorithmic Stability in Adaptive Data Analysis
Speaker: Adam Smith, PSU: Computer Science and Engineering Department
Location: MB106

Adaptivity is an important feature of modern data analysis—often, the choice of questions asked about a dataset depends on previous interactions with the same dataset. Adaptivity can arise in a single study (say, when a researcher choses which model to fit based on some exploratory data analysis) or, more subtly, when data sets are shared and re-used across multiple studies. Unfortunately, most of the statistical inference theory used in empirical sciences to control false discovery rates, and in machine learning to avoid overfitting, assumes that the analyses to be performed are selected independently of the data. If the set of analyses run is itself a function of the data, much of this theory becomes invalid. Specifically, suppose there is an unknown distribution P and a set of n independent samples x is drawn from P. We seek an algorithm that, given x as input, “accurately” answers a sequence of adaptively chosen “queries” about the unknown distribution P. How many samples n must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions towards resolving this question: 1. We give upper bounds on the number of samples n that are needed to answer "statistical queries" that improve over the bounds in the recent work of Dwork et al. (2015). 2. We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and convex risk minimization queries. Our algorithms are based on a connection between generalization error and a distributional stability condition on inference algorithms, called "differential privacy". The talk will be self-contained. Based on joint work with Raef Bassily, Kobbi Nissim, Thomas Steinke, Uri Stemmer and Jon Ullman. http://arxiv.org/abs/1503.04843 For some nontechnical background reading, see Gelman and Lokem, "The Garden of Forking Paths". http://www.stat.columbia.edu/~gelman/research/unpublished/p_hacking.pdf

September 7th, 2015 (03:35pm - 04:35pm)
Seminar: Dynamical systems seminar
Title: Labor Day
Speaker: NO SEMINAR
Location: MB114
September 8th, 2015 (11:15am - 12:05pm)
Seminar: Combinatorics/Partitions Seminar
Title: A Combinatorial Proof of a Relationship Between Maximal (2k-1,2k+1)-cores and (2k-1,2k,2k+1)-cores, II
Speaker: James Sellers, Penn State
Location: MB106

This talk (the second of this two-part series) will be a continuation of last week's Combinatorics/Partitions Seminar talk.

September 8th, 2015 (01:30pm - 02:20pm)
Seminar: Theoretical Biology Seminar
Title: TBA
Speaker: Casian Pantea, West Virginia University
(Host: Carina Curto)
Location: MB106
September 8th, 2015 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: Geometry of broken Lefschetz fibrations
Speaker: Ramon Vera, Penn State
Location: MB106

In this talk, we will discuss two geometric structures that are naturally associated to broken Lefschetz fibrations: near-symplectic forms and singular Poisson structures. Broken Lefschetz fibrations, or bLfs, were introduced by Auroux, Donaldson, and Katzarkov as mappings from a 4-manifold to the 2-sphere with two types of singularities. The natural geometric structure of these mappings is a near-symplectic form. This connects to an idea of Taubes of studying near-symplectic structures on 4-manifolds as a generalization of symplectic topology. Since then, these fibrations have found applications in low-dimensional topology and symplectic geometry. One reason is that every 4-manifold admits a bLf. After introducing bLfs, we will describe near-symplectic forms and present the generalization in higher dimensions. Using bLfs, we will also see a link to Poisson geometry. Any homotopy class of maps from a 4-manifold to the 2-sphere admits a singular Poisson structure of rank 2, whose foliation is compatible with the fibres of a bLf.

September 8th, 2015 (02:30pm - 03:29pm)
Seminar: Center for Dynamics and Geometry Colloquium
Title: Flexibility of entropies and Lyapunov exponents for smooth dynamical systems. An overview of a new program and first results.
Speaker: Anatole Katok, The Pennsylvania State University
Location: MB114
September 8th, 2015 (03:00pm - 05:00pm)
Seminar: Complex Fluids Seminar
Title: Mini-course on general diffusion and applications in molecular biology (I)
Speaker: Chun Liu, Penn State University
Location: MB216
September 8th, 2015 (03:30pm - 06:00pm)
Seminar: Working Seminar: Dynamics and its Working Tools
Title: Slow entropy for smooth flows on surfaces. Proofs. ATTENTION: THIS TALK WILL START AT 3;45pm
Speaker: Adam Kanigowski, Penn State
Location: MB114
September 8th, 2015 (04:00pm - 05:00pm)
Seminar: Applied Analysis Seminar
Title: A Priori and A Posteriori Estimates for Consistent Atomistic/Continuum Coupling Methods
Speaker: Lei Zhang, Shanghai Jiao Tong University
Location: MB106

Atomistic/Continuum(AtC) coupling methods combine the accuracy of potential based atomistic models with the efficiency of coarse grained continuum elasticity models by applying the former only in a small neighborhood of defects such as vacancies, dislocations and crack tips. We develop a framework for the a priori and a posteriori error estimates of AtC methods. This framework can be applied to various AtC methods in 2D, in particular, consistent methods which has no artificial 'ghost force', and corresponding error estimates can be established. Our numerical experiments indicate optimal convergence rates of the resulting algorithms. Adaptive mesh refinement techniques, which can be applied to general AtC mesh generation and refinement will also be discussed.

September 10th, 2015 (11:15am - 12:05pm)
Seminar: Algebra and Number Theory Seminar
Title: On sums of three cubes
Speaker: Robert C. Vaughan, Penn State University
Location: MB106

Estimates for the size of the set of positive integers which are the sum of three non-negative cubes has s long history, going back to Dvanport and even Hardy and Littlewood. Advances in our knowledge have and the underlying questions have often stimulated fundamental advances in our knowledge of additive number theory, especially the theory of Waring's problem and diagonal forms. We will give an overview of the substantial history of the subject and a brief description of the recent work of Trevor Wooley.

September 10th, 2015 (01:25pm - 02:25pm)
Seminar: MASS Colloquium
Title: Congruences for Fishburn Numbers
Speaker: James Sellers, Penn State
Location: MB114

The Fishburn numbers, originally considered by Peter C. Fishburn, have been shown to enumerate a variety of combinatorial objects. These include unlabelled interval orders on n elements, (2+2)--avoiding posets with n elements, upper triangular matrices with nonnegative integer entries and without zero rows or columns such that the sum of all entries equals n, non--neighbor--nesting matches on [2n], a certain set of permutations of [n] which serves as a natural superset of the set of 231--avoiding permutations of [n], and ascent sequences of length n. In December 2013, Rob Rhoades (Stanford) gave a talk in the Penn State Algebra and Number Theory Seminar in which he described, among other things, the relationship between Fishburn numbers, quantum modular forms, and Ramanujan's mock theta functions. Motivated by Rhoades' talk, George Andrews and I were led to study the Fishburn numbers from an arithmetic point of view - something which had not been done prior. In the process, we proved that the Fishburn numbers satisfy infinitely many Ramanujan--like congruences modulo certain primes p (the set of which we will easily describe in the talk). In this talk, we will describe this result in more detail as well as discuss how our work has served as the motivation for a great deal of related work in the last year by Garvan, Straub, and many others.

September 10th, 2015 (02:30pm - 03:30pm)
Seminar: Noncommutative Geometry Seminar
Title: There and back again: from the Borsuk-Ulam Theorem to quantum spaces
Speaker: Piotr Hajac, IMPA
Location: MB106

Assuming that both temperature and pressure are continuous functions, we can conclude that there are always two antipodal points on Earth with exactly the same pressure and temperature. This is the two-dimensional version of the celebrated Borsuk-Ulam Theorem which states that for any continuous map from the n-dimensional sphere to n-dimensional real Euclidean space there is always a pair of antipodal points on the sphere that are identified by the map. Our quest to unravel topological mysteries in the Middle Earth of quantum spaces will begin with gentle preparations in the Shire of elementary topology. Then, after riding swiftly through the Rohan of C*-algebras and Gelfand-Naimark Theorems and carefully avoiding the Mordor of incomprehensible technicalities, we shall arrive in the Gondor of compact quantum groups acting on unital C*-algebras. It is therein that the generalized Borsuk-Ulam-type statements dwell waiting to be proven or disproven. Time permitting, we shall pay tribute to the ancient quantum group SUq(2), and show how the proven nontrivializability of the SUq(2)-principal instanton bundle is a special case of two different noncommutative Borsuk-Ulam-type conjectures. (Based on joint work with Paul F. Baum and Ludwik Dabrowski.)

September 10th, 2015 (03:30pm - 04:20pm)
Seminar: Department of Mathematics Colloquium
Title: Faculty Meeting
Speaker: Faculty Meeting
Location: MB114
September 14th, 2015 (03:35pm - 04:35pm)
Seminar: Dynamical systems seminar
Title: Coarse hyperbolicity and closed orbits for quasigeodesic flows
Speaker: Steven Frankel, Yale University
Location: MB114

A flow on a 3-manifold is called quasigeodesic when each flowline is coarsely comparable to a geodesic. In this talk we will show that every quasigeodesic flow on a closed hyperbolic 3-manifold has closed orbits, resolving a question of Calegari's. A quasigeodesic flow is defined by a tangent condition. In contrast, a pseudo-Anosov flow is defined by a transverse condition; the transverse structure is hyperbolic in the sense that nearby flowlines diverge in some directions and converge in others. A priori, these transverse and tangent conditions are unrelated. However, when the ambient manifold is hyperbolic, a quasigeodesic flow admits a transverse structure that is remarkably similar to that of a pseudo-Anosov flow, except that it is only *coarsely* hyperbolic. Nevertheless, we can prove a sort of closing lemma for coarsely hyperbolic flows, allowing us to approximate recurrent orbits by closed orbits.

September 15th, 2015 (02:30pm - 03:45pm)
Seminar: Logic Seminar
Title: TBA
Speaker: Jan Reimann, Penn State
Location: MB315
September 15th, 2015 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: TBA
Speaker: Aissa Wade, Penn State
Location: MB106
September 15th, 2015 (03:30pm - 06:00pm)
Seminar: Working Seminar: Dynamics and its Working Tools
Title: Kakutani Equivalence theory I. An Overview
Speaker: Adam Kanigowski, Penn State
Location: MB114
September 16th, 2015 (12:00pm - 01:30pm)
Seminar: Geometry Luncheon Seminar
Title: Products of two symmetries of order two
Speaker: Oleg Viro, Stony Brook
Location: MB114

Many groups (e.g., the isometry groups of all classical space forms, i.e. Euclidean, hyperbolic and elliptic spaces) are generated by involutions. Moreover, each element can be presented as a product of two involutions. In fact, any group admits an embedding into a group in which any element is a product of two involutions. The presentations of isometries in this form gives rise to a fast pictorial calculation of their compositions. It brings in a geometry into the set of involutions. One can extract a complete set of relations among the symmetries.

September 17th, 2015 (01:25pm - 02:25pm)
Seminar: MASS Colloquium
Title: Real enumeration problems
Speaker: Oleg Viro, Stony Brook University
Location: MB114

We will consider problems of mixed setup, in which the initial object belongs to the elementary differential geometry (say, a smoothly immersed generic planar or spherical curve) and we are counting with certain weights the simplest algebraic curves in special position to the original curve. Say, bitangent lines or tritangent circles. The resulting quantity happens to be a topological invariant of the curve, which can be calculated combinatorially.

September 17th, 2015 (02:30pm - 03:30pm)
Seminar: Noncommutative Geometry Seminar
Title: Symbols and K-theory
Speaker: John Roe, Penn State
Location: MB106

I will give two talks explaining how the calculus of symbols (as developed in Nigel's two talks) can be used as a guiding example in the development of K-theory and K-homology, two standard topological tools in Noncommutative Geometry.

September 17th, 2015 (03:30pm - 04:20pm)
Seminar: Department of Mathematics Colloquium
Title: Hyperfields
Speaker: Oleg Viro - Joint MASS Speaker (Host: Sergei Tabachnikov), Stony Brook University
Location: MB114

In order to define hyperfields, one should allow in the definition of field the addition to be multivalued and adjust the axioms minimally. Hyperfields are not just exotic creatures produced by this abstract play with axioms. They appear naturally and pervade secretly the whole classical mathematics. For example, triangle inequalities describe a multivalued addition of non-negative real numbers. Equipped with this addition and the usual multiplication, this set is a hyperfield. Multiplicative norms in a ring are nothing but homomorphisms to this hyperfield. Hyperfields appear as natural quotient sets of fields and as dequantizations of fields (degenerate members in continuous families of fields). Mysterious fields of characteristic one may be hyperfields. Hyperfields provide an algebraic framework for the tropical geometry, a dequantized algebraic geometry.

September 18th, 2015 (03:35pm - 04:35pm)
Seminar: Probability and Financial Mathematics Seminar
Title: TBA
Speaker: Guodong (Gordon) Pang, Penn State University
Location: MB106
September 22nd, 2015 (02:30pm - 03:45pm)
Seminar: Logic Seminar
Title: TBA
Speaker: Jan Reimann, Penn State
Location: MB315
September 22nd, 2015 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: TBA
Speaker: John Roe, Penn State
Location: MB106
September 22nd, 2015 (03:30pm - 06:00pm)
Seminar: Working Seminar: Dynamics and its Working Tools
Title: Kakutani Equivalence theory II. Some new developments
Speaker: Adam Kanigowski, Penn State
Location: MB114
September 24th, 2015 (02:30pm - 03:30pm)
Seminar: Noncommutative Geometry Seminar
Title: Symbols and K-homology
Speaker: John Roe, Penn State
Location: MB106

I will give two talks explaining how the calculus of symbols (as developed in Nigel's two talks) can be used as a guiding example in the development of K-theory and K-homology, two standard topological tools in Noncommutative Geometry.

September 29th, 2015 (02:30pm - 03:45pm)
Seminar: Logic Seminar
Title: TBA
Speaker: Jake Pardo, Penn State
Location: MB315
September 29th, 2015 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: TBA
Speaker: Paul Baum, Penn State
Location: MB106