DEPARTMENT OF MATHEMATICS

COLLOQUIUM


Unless stated otherwise, colloquia are scheduled for Tuesdays 3:00-3:50pm and Thursdays 3:00-3:50pm in Bachelor Hall Room 219 with refreshments served from 2:30-2:55 pm in Bachelor Hall Room 115A.


Spring 2017


  • Wednesday, April 5, 2017
    Analysis (Hosted by Ryan Causey)

    Speaker:Thomas Schlumprecht,
    Texas A&M University.
    Title:The algebra of bounded linear operators on \(\ell_p\oplus\ell_q\) and \(\ell_p\oplus c_0\), \(1\le p< q\le\infty\), has infinitely many closed subideals.
    Time: 4:00 - 4:50 pm
    Room: BAC 219.

    Abstract: For a Banach space \(X\) we consider \(\mathcal L(X)\), the algebra of linear bounded operators on \(X\). A closed subideal of \(\mathcal L(X)\), is a subideal which is closed in the operator norm. For very few Banach spaces \(X\) the structure of the closed subideals of \(\mathcal L(X)\) is well understood. For example it is known for a long time that the only non trivial closed subideals of \(\mathcal L(\ell_p)\) (other than the zero ideal and the entire algebra) is the ideal of compact operators. In his book ``Operator Ideals'' Albrecht Pietsch asked about thestructure of the closed subideals of \(\mathcal L(\ell_p\oplus\ell_q)\) and \(\mathcal L(\ell_p\oplus c_0)\), the space of operators on the complemented sum of \(\ell_p\) and \(\ell_q\) respectively \(c_0\), where\(1\le p< q\le \infty\). In particular he asked if there are infinitely many closed subideals. This question was recently solved affirmatively for the reflexive range \(1< p< q <\infty\), in a joint work by the author in collaboration with András Zsak, for \(p=1\) and \(1< q\le \infty\) and for \(1\le p<\infty \) and \(q=\infty\), by Sirotkin and Wallis, and for \(1< p<\infty\) and \(c_0\) by the author in collaboration with Dan Freeman and András Zsak. The solution of these three cases use quite different approaches, which we want to present in our talk.


  • Thursday, March 30, 2017
    Analysis (Hosted by Caleb Eckhardt)

    Speaker: Scott Atkinson,
    Vanderbilt University.
    Title:Minimal faces and Schur’s Lemma for embeddings into \(R^U\)
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: As shown by N. Brown in 2011, for a separable \(II_1\)-factor von Neumann algebra N, the invariant \(Hom(N,R^U)\) given by unitary equivalence classes of embeddings of \(N\) into \(R^U\)--an ultrapower of the separable hyperfinite \(II_1\)-factor--takes on a convex structure. This provides a link between convex geometric notions and operator algebraic concepts; e.g. extreme points are precisely the embeddings with factorial relative commutant (centralizer). The geometric nature of this invariant yields a familiar context in which natural curiosities become interesting new questions about the underlying operator algebras. For example, such a question is the following. "Can four extreme points have a planar convex hull?" The goal of this talk is to present a recent result generalizing the characterization of extreme points in this convex structure. After introducing this convex structure, we will see that the dimension of the minimal face containing an equivalence class \([\pi]\) is one less than the dimension of the center of the relative commutant of \(\pi\). This result also establishes the "convex independence" of extreme points, providing a negative answer to the above question. Along the way we make use of a version of Schur's Lemma for this context. A deep understanding of ultrapowers is not required, and no prior knowledge of this convex structure will be assumed.


  • Thursday, March 9, 2017
    Analysis (Hosted by Beata Randrianantoanina)

    Speaker: Florence Lancien,
    Laboratoire de Mathématiques de Besançon, France.
    Title: Analyticity of semigroups and geometry of Banach spaces
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: Semigroups of operators on Banach spaces have first been introduced for their connexions with solutions of PDE's, before their study became a subject in itself. The links between the properties of a semigroup, such as its analyticity, and the spectral properties of its generator have long been known. In 1986 G. Pisier used analytic semigroups of convolutions in order to prove famous results on the geometry of Banach spaces. In return it appeared that geometric properties of a Banach space \(X\) play an important role in the behavior of semigroups on \(X\) or their generators. I will present some of these "old" results about continous analytic semigroups as well as more recent questions about discrete analytic semigroups.


  • Tuesday, March 7, 2017
    Analysis (Hosted by Beata Randrianantoanina)

    Speaker: Gilles Lancien,
    Laboratoire de Mathématiques de Besançon, France.
    Title: A few questions in the metric analysis of Banach spaces
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: In the last fifteen years, the question of embedding metric spaces into ``nice'' Banach spaces has gathered researchers coming from very diverse origins, in particular from theoretical computer science, geometry of Banach spaces and geometric group theory. The starting point is to try to get a better understanding of complicated metric spaces, such as graphs with a high number of vertices and edges, by embedding them into well understood Banach spaces (like Hilbert spaces). In particular remarkable results of Yu and Kasparov assert that if the Cayley graph of a finitely generated group G coarsely embeds into a Hilbert space, or a super-reflexive Banach space, then some important conjectures on the homotopy invariance of the higher signatures for manifolds with fundamental group G are satisfied. Therefore a weak metric assumption on the group yields a very strong result in differential topology. In the so-called non linear geometry of Banach spaces, which is the subject of this talk, the approach is somewhat reversed, and people try to exhibit the linear properties of Banach spaces that are stable under some particular non linear maps. These non linear maps can be of very different nature: Lipschitz isomorphisms or embeddings, uniform homeomorphisms, uniform or coarse embeddings. Then, the next goal is to characterize these linear properties in purely metric terms. Usually, these characterizations are given by the (non)embeddability of special metric spaces, which are very often fundamental metric trees or graphs. Finally, this can motivate the study of these properties, whose linear character is now forgotten, in the larger setting of metric spaces. The study of metric embeddings into Banach spaces is thus a very rich meeting point attracting people and ideas from seemingly distant areas. We will try to illustrate these ideas with a few results (old or recent) and open questions.


  • Tuesday, February 28, 2017
    Analysis (Hosted by Beata Randrianantoanina)

    Speaker: Mikhail Ostrovskii,
    St. John's University.
    Title: Metric embeddings: embeddings of discrete metric spaces into Banach spaces
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: Embeddings of a discrete metric space into a Hilbert spaces or a “good'' Banach space have found many significant applications. At the beginning of the talk I plan to give a brief description of such applications. After that I plan to present three of my results: (1) On \(L_1\)-embeddability of graphs with large girth; (2) Embeddability of infinite locally finite metric spaces into Banach spaces is finitely determined; (3) A new metric characterization of superreflexivity (joint work with Beata Randrianantoanina).


  • Thursday, February 23, 2017
    Analysis (Hosted by Caleb Eckhardt)

    Speaker: Christopher Schafhauser,
    University of Waterloo, Canada.
    Title: Quasidiagonal \(C^*\)- algebras
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: A \(C^*\)- algebra is a sub algebra of the bounded linear operators on a Hilbert space which is stable under the adjoint operation and closed in the norm topology. Some of the fundamental examples of \(C^*\)- algebras arise from dynamical systems. In particular, given a group \(G\), there is a natural way to construct a \(C^*\)- algebra from the complex group ring of \(G\). More generally, given a group action on a topological space, a \(C^*\)- algebra can be obtained which encodes the given action. Quasidiagonality is an operator theoretic condition which roughly says an operator can be approximately decomposed as a direct sum of operators on finite dimensional spaces. Determining when a \(C^*\)- algebra is quasidiagonal has become an important, albeit difficult, problem in the structure and classification of \(C^*\)- algebras. I will discuss some of the recent progress on the quasidiagonality of these dynamically defined \(C^*\)- algebras.


  • Friday, February 17, 2017
    Differential Equations (Hosted by Anna R. Ghazaryan)

    Speaker: Ivan Sudakov,
    University of Dayton.
    Title:Arctic Tipping Elements and Bifurcations in the Climate System.
    Time: 4:00 - 4:50 pm
    Room: BAC-219

    Abstract: Different elements of the Arctic climate system show abrupt changes. Such tipping points are the visible "face of climate change". Once activated they are irreversible. I will give an overview of how the bifurcation theory are being used to study Arctic tipping elements and its role in the climate system.




Fall 2016


  • Thursday, Novemebr 10, 2016
    Algebra (Hosted by Stephen Gagola)

    Speaker: L. C. Kappe,
    Binghamton University
    Title:TBA
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: TBA


  • Thursday, October 27, 2016
    Algebra (Hosted by Stephen Gagola)

    Speaker: Tony Evans, Write State University

    Title:TBA
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: TBA


  • Thursday, September 29, 2016
    Analysis
    (Hosted by Beata Randrianantoanina)

    Speaker:
    Michał Wojciechowski, Polish Academy of Sciences, Warsaw, Poland.
    Title:On the Pelczynski conjecture on Auerbach Bases
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract:We consider Auerbach bases in a Banach spaces of dimension \(n>2\). We show that there exists at least (\(n-1)n/2\)+1 such bases. This estimate follows from the calculation of the Lusternik-Schnirelmann category of the flag variety. A better estimate is obtained for generic Banach spaces by the Morse theory. We use tools from convex geometry, topology, and algebraic topology, but no advanced knowledge of these fields is required to understand the talk. The talk will be accessible to graduate students. (Joint work with A. Weber).


  • Tuesday, September 27, 2016
    Algebra
    (Hosted by Stephen Gagola)

    Speaker:
    Maria de Lourdes Merlini Giuliani, Universidade Fereral do ABC, Brazil.
    Title:General Linear Loop
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: The general linear loop, denoted by \(GLL(F)\), is the set of invertible elements of the split octonion algebra over the field \(F\). The elements of a split octonion algebra are known as Zorn vector matrices. Under multiplication, these matrices are nonassociative but do satisfy weaker laws of associativity first discovered by Ruth Moufang. Because of this, \(GLL(F)\) belongs to the class of Moufang loops. In this talk I will give an overview of definitions and results concerning the structure of \(GLL(F)\) given that the field \(F\) is finite, namely, \(F = F_q\).



Spring 2016


  • Tusday, April 26, 2016
    Logic and Set Theory


    Speaker:
    Shehzad Ahmed, Ohio University.
    Title:TBA
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: TBA


  • Thursday, April 21, 2016
    Algebra


    Speaker:
    Claudia Miller, Syracuse University.
    Title:TBA
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: TBA


  • Thursday, March 31, 2016
    Differential Equations


    Speaker:
    Stéphane Lafortune, College of Charlerston.
    Title:TBA
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: TBA


  • Thursday, March 10, 2016 Graph Theory and Combinatorics

    Speaker:
    Jozef Skokan, The London School of Economics and Political Science.
    Title:The Multicolor Ramsey Numbers of Cycles.
    Time: 4:00 - 4:50 pm (Note new time)
    Room: BAC-219

    Abstract: Graph Ramsey Theory, introduced by Erdős and his coauthors about 40 years ago, has quickly become one of the most active areas of Graph Theory. This area is not only a source of many fascinating problems but also serves as a testing ground for a variety of important combinatorial techniques. In this talk, we will illustrate this on the following multicolour Ramsey problem. For a graph \(G\), the \(k\)-color Ramsey number \(R_k(G)\) is the least integer \(N\) such that every \(k\)-colouring of the edges of the complete graph \(K_N\) contains a monochromatic copy of \(G\). Let \(C_n\) denote the cycle on \(n\) vertices. In 1973, Bondy and Erd\H{o}s conjectured that for fixed \(k\ge 3\) and \(n>3\) odd, \(R_k(C_n)=2^{k-1}(n-1)+1.\) We resolve this conjecture for every fixed \(k\) and large \(n\). We also establish a surprising correspondence between extremal \(k\)-colorings for this problem and perfect matchings in the hypercube \(Q_k\). This allows us to in fact prove a stability-type generalization of the above. The proof is analytic in nature and relates this problem in Ramsey theory to one in convex optimization.



Fall 2015


  • Thursday, September 17, 2015
    Analysis


    Speaker:
    Elizabeth Gillaspy, University of Colorado Boulder.
    Title: \(K\)-theory and twisted groupoid \(C^*\hspace{-.1cm}\)- algebras
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: In the first part of the talk, I will define the objects in the title and explain how the \(K\)-theory of groupoid \(C^*\hspace{-.1cm}\)- algebras can tell us about other mathematical objects, from dynamical systems to string theory. My hope is that this will be a gentle introduction to the topic(s) at hand; no prior familiarity with groupoids or with \(C^*\hspace{-.1cm}\)- algebras will be assumed. In the second part of the talk, I'll discuss the particular question I've investigated about the \(K\)-theory of twisted groupoid \(C^*\hspace{-.1cm}\)- algebras, why I chose it, and the progress that I've made so far. Time permitting, I will sketch some of the proofs, so this part of the talk will be more technical than the first part.



Spring 2015


  • Thursday, April 16, 2015
    Graph Theory and Combinatorics


    Speaker:
    Gyula Katona, Alfréd Rényi Institute of Mathematics
    Title: TBA
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: TBA


  • Thursday, April 9, 2015
    Logic/Set Theory


    Speaker:
    Will Boney, University of Illinois at Chicago
    Title: Ultraproducts in Nonelementary Classes
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: Ultraproducts are a standard and powerful tool in the study of elementary classes, those that are models of a first order theory. However, many natural mathematical classes (complete metric spaces, locally finite groups, etc.) are not elementary. In this talk, we give an introduction to ultraproducts and explore variations that can be used to get powerful results in some nonelementary classes.


  • Thursday, April 2, 2015
    Logic/Set Theory


    Speaker:
    Liuzhen Wu, Institute of Mathematics, Chinese Academy of Sciences
    Title: A Lindelof group with non-Lindelof square
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: In this talk, I will present a \(ZFC\) construction of a Lindelof group with non-Lindelof square, answering a question of Arhangelskii. This is a joint work with Yinhe Peng.


  • Thursday, March 19, 2015
    Graph Theory and Combinatorics


    Speaker:
    Zoltan Furedi, University of Illinois Urbana-Champaign
    Title: Superimpose Codes
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: There are many instances in Coding Theory when codewords must be restored from partial information, like defected data (error correcting codes), or some superposition of the strings. These lead to superimposed codes, a close relative of group testing problems. There are lots of versions and related problems, like Sidon sets, sum-free sets, union-free families, locally thin families, cover-free codes and families, etc. We discuss two cases cancellative and union-free codes. A family of sets \(\mathcal F\) (and the corresponding code of 0-1 vectors) is called union-free if \(A\cup B\neq C\cup D\) and \(A,B,C,D\in \mathcal F\) imply \(\{ A,B\}=\{ C, D \}\). \(\mathcal F\) is called \(t\)- cancellative if for all distinct \(t+2\) members \(A_1, \dots, A_t\) and \(B,C\in \mathcal F\), \(A_1\cup\dots \cup A_t\cup B \neq A_1\cup \dots A_t \cup C.\) Let \(c_t(n)\) be the size of the largest \(t\)-cancellative code on \(n\) elements. We significantly improve the previous upper bounds of Körner and Sinaimeri, e.g., we show \(c_2(n)\leq 2^{0.322n}\) (for \(n> n_0\)). We introduce a method to deal with such problems, namely we first investigate the constant weight case (i.e., uniform hypergraphs).


  • Friday, March 13, 2015 - New date and location
    The Differential Equations Group

    Speaker:
    Hung Tran, University of Chicago
    Title: Selection problems for a discounted degenerate viscous Hamilton-Jacobi equation
    Time: 3:00 - 3:50 pm
    Room:BAC 245

    Abstract: I will give first a brief overview on the selection problem for solutions of Hamilton—Jacobi equations, which leads to the theory of viscosity solutions. Then I will describe the cell/ergodic problem of interest and its interesting phenomena. Finally, I will state the corresponding selection problem, the main result, and explain some key ideas. This is a joint work with Hiroyoshi Mitake.


  • Friday, March 6, 2015 - New date and location
    The Differential Equations Group
    Miami University Undergraduate Research Award recipients: Eli Thompson and Robert Doughty


    Speaker:
    Miranda Teboh-Ewungkem, Lehigh University
    Title: The dynamics of malaria transmitting mosquitoes and their impact on the disease malaria-A mathematical study.
    Time: 3:00 - 3:50 pm
    Room: BAC 245.

    Abstract: Mathematical models for malaria often treat the population of the mosquito that transmits the disease as a constant. However, there is a reproductive gain that accrue to the mosquito's population as a result of its lifestyle, feeding and reproductive habits, and its interaction with the human population. In particular, the interaction between mosquitoes and humans introduce variability in the density of the mosquito's population and this impacts the dynamics of the malaria disease. This variability is affected by the mosquito's ability to interact with humans and animals in order to draw blood needed for the maturation of the mosquito's eggs, a process essential for reproduction. Using a continuous time mathematical model, I will highlight how the reproductive lifestyle of the Anopheles mosquito and the interaction between the mosquito and humans affect malaria transmission dynamics, and introduce complex phenomena such as backward bifurcation, as well as oscillations rarely observed in unforced continuous time models. Implications for disease control will also be discussed.


  • Thursday, February 26, 2015
    Analysis


    Speaker:
    Jonathan Brown, University of Dayton
    Title: Injective homomorphisms from a graph \(C^*\)-algebra
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: A \(C^*\)-algebra is an infinite dimensional analogue of the \(n\) by \(n\) matrices with complex entries. \(C^*\)-algebras have an extremely rich structure and have been used to describe quantum processes and unitary group representations. In 1997 Kumjian, Pask, Raeburn and Renault show how to construct \(C^*\)-algebras from directed graphs and it has since been shown that much of the \(C^*\)-algebras structure can be read from the combinatorial properties of the graph.Thus, in some cases, the existence of \(C^*\)-algebras satisfying certain properties amounts to constructing special directed graphs.The driving force behind the theory of graph \(C^*\)-algebras is that surjective homomorphisms from a graph \(C^*\)-algebra are particularly easy to construct. In this talk, I will present a result which provides checkable conditions on a general homomorphism from a graph \(C^*\)-algebra that guarantee the homomorphism is injective; thus helping deciding if a constructed surjective homomorphism is an isomorphism. This result is due to Nagy and Reznikoff 2012 (also Syzmanski 2002) but has been recently generalized to groupoid \(C^*\)- algebras by me and my collaborators Nagy, Reznikoff, Sims and Williams.




Fall 2014


  • Thursday, September 25, 2014
    Graph Theory and Combinatorics


    Speaker:
    Allan Lo, University of Birmingham (UK)
    Title: Edge-decompositions of graphs with high minimum degree
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: A fundamental theorem of Wilson states that, for every graph \(F\), every sufficiently large \(F\)-divisible clique has an \(F\)-decomposition. Here a graph \(G\) is \(F\)-divisible if \(e(F)\) divides \(e(G)\) and the greatest common divisor of the degrees of \(F\) divides the greatest common divisor of the degrees of \(G\), and \(G\) has an \(F\)-decomposition if the edges of \(G\) can be covered by edge-disjoint copies of \(F\). We extend this result to graphs which are allowed to be far from complete: we show that every sufficiently large \(F\)-divisible graph \(G\) on \(n\) vertices with minimum degree at least \((1- |F|^{-4})n\) has an \(F\)-decomposition. Our main contribution is a general method which turns an approximate decomposition into an exact one. This is joint work with Ben Barber, Daniela Kühn and Deryk Osthus.


  • Thursday, October 23, 2014
    Graph Theory and Combinatorics


    Speaker:
    Theo Molla, University of Illinois at Urbana-Champaign
    Title: TBA
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: When \(G\) and \(H\) are graphs, a collection of copies of \(H\) in \(G\) is an \(H\)-factor if every vertex of \(G\) is contained in exactly one copy of \(H\). So, when \(H\) is the complete graph on \(2\) vertices, a perfect matching is equivalent to an \(H\)-factor. When \(H\) is any complete graph, \(H\)-factors are closely related to equitable \(k\)-colorings; that is, vertex colorings such that no two adjacent vertices are given the same color and the number of vertices given a particular color is either the ceiling or the floor of the number of vertices divided by \(k\). In 1970, Hajnal and Szemerédi proved that every graph with maximum degree less than \(k\) has an equitable \(k\)-coloring. This theorem is easily seen to be best possible: complete graphs, odd cycles and balanced complete bipartite graphs with odd sized parts are all not equitably \(k\)-colorable when \(k\) is the maximum degree. In 1994, Chen, Lih and Wu conjectured that these are the only connected graphs that are not equitably \(k\)-colorable and have maximum degree at most \(k\). This conjecture is still open, but a few special cases have been proved. In this talk, we will discuss a result which implies the Chen--Lih--Wu conjecture for \(k\)-equitable colorings of graphs with \(3k\) vertices as well as other results related to the Hajnal--Szemerédi Theorem and the \(H\)-factor problem.


  • November 6, 2014
    Graph Theory and Combinatorics


    Speaker:
    Landon Rabern, LBD Data
    Title: A common generalization of Hall's theorem and Vizing's edge-coloring theorem
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: We introduce a two-player card game and demonstrate a simple winning strategy. As corollaries, we get Vizing's edge-coloring theorem as well as other classical edge-coloring results. A generalized version of this game gives a unifying framework for various adjacency lemmas.


  • November 13, 2014
    Graph Theory and Combinatorics


    Speaker:
    Csaba Biro, University of Louisville
    Title:Stability questions on the dimension of posets TBA
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: For a finite poset \(P\), the least positive integer \(n\) such that \(P\) can be embedded into \(\mathbb{R}^n\) is called the dimension of \(P\). A classical theorem states that the dimension never exceeds half of the number of elements. If the dimension of \(P\) is exactly half of the number of elements, the structure of \(P\) can be described exactly. The goal of this talk is to introduce recent results that describe the structure of the poset if the dimension is slightly less than the possible maximum. This is joint work with Peter Hamburger, Attila Por, and William T. Trotter


  • November 20, 2014
    Analysis


    Speaker:
    Bhishan Jacelon, Purdue University
    Title:Self adjoint unitary orbits in pure \(C^*\)-algebras
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: A classical result of Hermann Weyl says that the distance between the unitary orbits of self adjoint matrices is exactly the distance between their eigenvalues. Suitably reinterpreted, this was extended in the 1980s to self adjoint operators on infinite dimensional Hilbert space, or more generally in a semifinite von Neumann factor. We exploit recent developments in the Elliott classification program to show that an analogous result holds in certain well behaved \(C^*\)-algebras. This is joint work with Karen Strung and Andrew Toms.


  • November 25, 2014
    Applied Mathematics


    Speaker:
    Stephane Lafortune, College of Charleston
    Title:Vortex Filaments and torus knots
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: By the term vortex filament, we mean a mass of whirling fluid or air (e.g. a whirlpool or whirlwind) concentrated along a slender tube. The most spectacular and well-known example of a vortex filament is a tornado. A waterspout and dust devil are other examples. In more technical applications, vortex filaments are seen and used in contexts such as superfluids and superconductivity. One system of equations used to describe the dynamics of vortex filaments is the Vortex Filament Equation (VFE). The VFE is a system giving the time evolution of the curve around which the vorticity is concentrated. In this talk, we develop a framework for studying the stability solutions of the VFE, based on the correspondence between the VFE and the NLS provided by the Hasimoto map. This framework is applied to VFE solutions that take the form of torus knots. We show that these torus knots they are stable only in the unknotted case. We also establish the spectral stability of soliton solutions.


  • Thursday, December 4, 2014
    Analysis


    Speaker:
    Michael Brannan, University of Illinois at Urbana-Champaign
    Title: Connes' Embedding Property For Quantum Groups
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: In this talk, I will begin with a light introduction to the concept of a quantum group. From a functional analysis perspective, quantum groups are interesting because they yield important examples of \(C^*\)- algebras and von Neumann algebras. After discussing some basic examples, our main focus will be on the question of whether a given quantum group von Neumann algebra satisfies Connes' Embedding Property (an important finite dimensional approximation property in operator algebra theory). Using some techniques developed in collaboration with Benoit Collins and Roland Vergnioux, we show that the von Neumann algebras associated to Wang's free orthogonal quantum groups have Connes' Embedding Property. As an application, we compute Voiculescu's free entropy dimension of the generators of these algebras.





Spring 2014


  • Tuesday, February 18, 2014
    Algebra


    Speaker:
    Janet Striuli, Fairfield University
    Title: Finite invariants of infinite free resolutions
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: Free resolutions linearize properties of finitely generated modules so that techniques from linear algebra find applications in studying the category of modules. On the other hand, over a singular ring, most free resolutions are infinite in length making harder the task to decode the properties of the modules. In this talk we will present finite invariants related to infinite resolutions with particular attention to finite bounds of Artin-Rees type.


  • Thursday, May 1, 2014
    Dynamical Systems


    Speaker:
    Steve Schecter, North Carolina State University
    Title: TBA
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: TBA.





Fall 2013


  • Thursday, August 29, 2013
    Logic/Set Theory


    Speaker:
    Nam Trang, Carnegie Mellon University
    Title: Generalized Solovay Measures
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: It's well-known that \(AD\) (Axiom of Determinacy) and its strengthening \(AD_R\) contradict \(AC\) (Axiom of Choice). Nevertheless, one can construct natural models of \(AD\) and \(AD_R\) from \( ZFC\) \(+\) large cardinals. Solovay shows, as part of his axiomatic study of \(AD_R\), that there is a supercompact measure on \(P_{\omega_1}(R)\), the set of countable subsets of \(R\). We discuss constructions from large cardinals of minimal models containing such measures and show that the existence of such measures is much weaker than that of \(AD_R\). We also discuss generalizations of the Solovay measure and some applications. Some background on basic set theory and large cardinals needed for the talk will be briefly reviewed.


  • Thursday, September 5, 2013
    Geometry/Topology


    Speaker:
    Frank Connolly, University of Notre Dame
    Title: On the Classification of Pseudo-free Cocompact Group Actions on Contractible Manifolds.
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: An action of a group \(\Gamma\) on a manifold \(M\) is pseudo-free if the set of singular orbits forms a discrete subset of \(M\). In the case when \(M\) is contractible, and \(M/\Gamma\) is compact, we show how to classify these actions provide that there is at least one such action by isometries on a CAT(0) manifold. The classification is quite explicit. This is joint work with Jim Davis and Qayum Khan


  • Thursday, September 19, 2013
    Logic/Set Theory


    Speaker:
    Julia F. Knight, University of Notre Dame
    Title: How hard is it to compute the floor function?
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: For the field of real numbers, we have the usual floor function, with range equal to the set of integers. If we expand the reals, adding the function \(2^x\), then for a positive integer \(x\), \(2^x\) is also a positive integer. Mourgues and Ressayre showed that every real closed field has an ``integer part.'' The construction is complicated. Ressayre showed that every real closed exponential field has an ``exponential integer part''. This construction is even more complicated. The talk will describe results on the complexity of these constructions, from the point of view of computable structure theory.


  • * Note the two colloquiums on Thursday, September 26.
  • Thursday, September 26, 2013
    Analysis


    Speaker:
    David Sherman, University of Virginia
    Title: Model theory for functional analysis
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: Model theory studies the interplay between mathematical structures and their logical properties. Some of its most beautiful theorems involve a construction called an ultrapower. Many standard objects in functional analysis, such as Banach spaces and operator algebras, carry useful notions of ultrapower, but this does not interact well with classical model theory. An elegant recent solution, very natural for analysts, is to switch to a logic in which truth values are drawn from the interval \([0,1]\). I will give a "big picture" survey of this approach and its prehistory, not assuming that the audience has any familiarity with logic or ultrapowers.


  • Thursday, September 26, 2013
    Buckingham Scholar Speaker


    Speaker:
    Joseph Gallian, University of Minnesota Duluth
    Title: Using groups and graphs to create symmetry patterns
    Time: 5:30 - 6:30 pm
    Room: BAC-219

    Abstract: We use video animations to explain how Hamiltonian paths, spanning trees, cosets in groups, and factor groups can be used to create computer generated symmetry patterns in hyperbolic and Euclidean planes. These methods were used to create the image for the 2003 Mathematics Awareness Month poster.


  • Thursday, October 10, 2013
    Logic/Set Theory


    Speaker:
    Justin Moore, Cornell University
    Title: A geometric solution to the von Neumann-Day problem for finitely presented groups
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: Shortly after isolating the notion of an amenable group, von Neumann asked whether containing the free group on two generators is the only obstruction to the amenability of a discrete group. Olshanskii eventually constructed a counterexample in 1980 and Olshanskii and Sapir constructed a finitely presented counterexample in 2003. It was only recently, however, that Monod produced an example which was easy to describe and of a more gemetric nature. We will construct a finitely presented counterexample to the von Neumann-Day problem inside Monod's group. This is joint work with Yash Lodha.


  • Friday, November 1, 2013 - New date and location
    Analysis


    Speaker:
    Dan Freeman, St. Louis University
    Title: Moving redundant frames on manifolds
    Time: 3:00 - 3:50 pm
    Room: BAC-245

    Abstract: A frame for \(\mathbb R^n\) is essentially a coordinate system that can represent any element of \(\mathbb R^n\) as a linear combination of the frame vectors. The main difference between a frame and a basis is that a frame can be redundant in that the particular linear combination used to reconstruct a vector may not be unique. This redundancy can be useful in application as it makes the coordinate system more resilient to error. This is partly because losing a basis coefficient would result in the loss of an entire dimension but the loss of a frame coefficient can be made up for by using the other vectors. The word frame has a different meaning in differential geometry. In this context, a moving frame for a smooth manifold is a basis for the tangent space at each point on the manifold which moves continuously over the manifold. In this talk we will combine the two notions of frames to obtain a moving redundant frame on a manifold. We will show that some of the fundamental theorems about frames for \(\mathbb R^n\) can be extended to move continuously over a manifold. We will also consider cases when certain manifolds do not have a particular moving basis for their tangent space, but do have a nice moving redundant frame. This talk will cover one paper with Ryan Hotovy and Eileen Martin, and a different paper with Daniel Poore, Rebecca Wei, and Madelyn Wyse. I will be speaking on work with undergraduates that I did through an REU program, so the talk will be accessible to undergraduates.


  • Tuesday, November 5, 2013
    Geometry/Topology


    Speaker:
    Kun Wang, The Ohio State University
    Title: On passage to over-groups of finite index of the Farrell-Jones isomorphism conjecture.
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: The Farrell-Jones isomorphism conjecture (FJIC) plays an important role in manifold topology as well as computations in algebraic \(K\)- and \(L\)-theory. It implies, for example, the Borel conjecture on topological rigidity of closed aspherical manifolds and the Novikov conjecture on homotopy invariance of higher signatures. It is a conjecture about groups and has some nice inheritance properties. For example, if FJIC holds for a group, then it holds for any of its subgroup. In this talk, I will present some results obtained concerning the converse question: if a group has a subgroup of finite index satisfies FJIC, whether the group itself satisfies FJIC. The first half of the talk will be an introduction to the conjecture and I’ll try to make it accessible.


  • Thursday, November 21, 2013
    Geometry/Topology


    Speaker:
    Patrick Reynolds, University of Utah
    Title: Recent progress on outer automorphism groups of free groups
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: Use \(Out(F)\) to denote the group of outer automorphisms of a finite rank free group \(F\). In the past 25 years, substantial progress has been made in understanding the geometric and dynamical structure of elements, and certain subgroups, of \(Out(F)\), as well as the large scale structure of \(Out(F)\) as a whole. This progress has come primarily from adapting, to the context of free groups, the Nielsen-Thurston approach for studying surface diffeomorphisms, as well as a resultant program for studying groups of mapping classes of a surface developed by many researchers, with pioneering work due to Masur and Minsky. The goal of this talk is to survey some main ideas of this approach to \(Out(F)\) and to mention some recent theorems.


  • Thursday, December 5, 2013
    Combinatorics/Graph Theory


    Speaker:
    Bernard Lidický, University of Illinois
    Title: Flag algebras and its application
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: Flag algebras is a method, recently developed by Razborov, designed for attacking problems in extremal graph theory. There are recent applications of the method also in discrete geometry or permutation patterns. The aim of talk is to give a gentle introduction to the method and show some of its applications. The talk is based on joint works with J. Balogh, P. Hu, H. Liu, O. Pikhurko, B. Udvari, and J. Volec.


  • Thursday, December 12, 2013
    Geometry/Topology


    Speaker:
    Stratos Prassidis, University of the Aegean/SUNY Binghamton
    Title: The Isomorphism Conjecture for some subgroups of the automorphism group of the free group.
    Time: 3:00 - 3:50 pm
    Room: BAC-219

    Abstract: Starting from the fact that the automorphism group of the free group of two generators satisfies the Isomorphism Conjecture, we derive that the same is true for the holomorph (the universal split extension) of this group. There is a natural embedding of the holomorph into the automorphism group of the free group on three generators. We can repeat the process and get a sequence of groups into the automorphism group of the free group of higher rank. We will show that the Isomorphism Conjecture holds for all these groups and we will use it to compute its lower \(K\)-theory. This is joint work with Vassilis Metaftsis.


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