Upcoming Colloquia for Spring 2024

Science Hall 107

12:00PM refreshments | 12:30PM - 1:30PM

Zoom: Link

 

Spring 2024

 January 26

Speaker:  Sumeyra Sakalli, University of Arkansas

Title: 
Exotic 4-Manifolds and Complex Surfaces, Algebraic Fibrations, Singularities and Knots

Abstract: Exotic manifolds are smooth manifolds which are homeomorphic but not diffeomorphic to each other. After Donaldson showed that Dolgachev’s surface is an exotic complex projective plane blown-up at 9 points, constructing exotic 4-manifolds has been an active research area in symplectic and low dimensional topology over the last 35 years, with a lot of progress as well as many open problems.

In this talk, I will describe my research on exotic 4-manifolds in relation to complex surfaces, algebraic fibrations, singularities and also knots. In particular, I will present our constructions of the smallest exotic 4-manifolds with nonnegative signatures; deformation and splitting singularities in algebraic fibrations; and generalized chain surgeries, which are symplectic operations we defined, and are useful for building exotic 4-manifolds. I will also discuss slice knots in definite 4-manifolds. Some of these works are joint with Akhmedov, Karakurt, Kjuchukova-Miller-Ray, Park, Van Horn-Morris and Yeung. 

 January 22

Speaker:  Keegan Boyle, University of British Columbia

Title: 
Alexander Polynomials and Symmetric Knots

Abstract: 
The Alexander polynomial is an invariant of a knot in the 3-sphere, defined in terms of the homology of a particular covering space of the knot complement. It is easy to compute and contains significant geometric information about the knot. Consequently, it has received a lot of attention in the literature over the last century. In this talk, I will discuss several of my recent theorems about the Alexander polynomial. These theorems indicate how new geometric information about the knot can be extracted from the polynomial in the presence of a symmetry, and have applications to 4-manifold topology. The proofs of these theorems include a mix of classical tools and methods from algebraic and geometric topology, and modern inventions such as Heegaard Floer homology.

 January 19

Speaker:  William Balderrama, University of Virginia

Title:  Equivariant Maps Between Spheres

AbstractA fundamental problem in topology is to classify all continuous maps between two spheres up to continuous deformation, that is, up to homotopy. This problem becomes even richer when considering maps that are required to respect additional symmetries that may be present. Perhaps the first major result along these lines is the Borsuk--Ulam theorem, which asserts that there exists no continuous map from a sphere to a sphere of lower dimension that is moreover an odd function (in the usual calculus sense). In this talk, I will discuss how these ideas led to some of the early development of equivariant stable homotopy theory, and describe some recent work in this area.

Previous Colloquia

 

Fall 2023

 December 1

Speaker:  Søren Galatius, University of Copenhagen, Denmark

Title: 
Pontryagin Classes of Euclidean Bundles

Abstract:  
Pontryagin classes are invariants of a real vector bundle, usually defined as Chern classes of the complexified bundle, and appearing prominently in Hirzebruch's formula for the signature of a 4n-dimensional manifold. The Pontryagin classes can be defined more generally for euclidean bundles, that is, fiber bundles with fiber R^n and structure group the entire homeomorphism group of R^n. I will recall a bit of the classical theory and discuss some newer developments, including joint work with Randal-Williams (arXiv:2208.11507) on algebraic independence.

 December 1

Speaker:  Naoki Masuda, University at Buffalo (North Campus)

Title: 
Temporal Networks: State-Transition Dynamics, Embedding, and Switching Network Modeling

Abstract:  
Two salient features of empirical temporal (i.e., time-varying) network data are the time-varying nature of network structure itself and heavy-tailed distributions of inter-contact times. Both of them can strongly impact dynamical processes occurring on networks. In the first part of the talk, I introduce modeling of heavy-tailed distributions of inter-contact times by state-dynamics modeling approaches in which each node is assumed to switch among a small number of discrete states in a Markovian manner. This approach is interpretable, facilitates mathematical analyses, and seeds various related mathematical modeling, algorithms, and data analysis (e.g., embedding of temporal network data). The second part of the talk is on modeling of temporal networks by static networks that switch from one to another at regular time intervals. This approach facilitates analytical understanding of various dynamics (e.g., epidemic processes, evolutionary dynamics) on temporal networks as well as an efficient algorithm for containing epidemic spreading as convex optimization. Finally, I will touch upon my interdisciplinary collaborations on genomic and other data briefly.

 November 17

Speaker:  Chris Peterson, Colorado State University

Title: 
Stochastic Particle Systems with Redistribution and their Connection to Large Networks

Abstract:  
A general class of stochastic particle systems with a redistribution-triggering catalyst is considered.  In particular, we focus on their bulk behavior as the number of particles goes to infinity, as well as the asymptotic behavior (invariant measure) of the system.  At the end, we will reveal their connection to eigenvector centrality (measure of influence of a node in a network) on time-evolving graphs and the spectral clustering on large graphs.

 November 10

Speaker:  Chris Peterson, Colorado State University

Title: 
The Geometry of Digital Images

Abstract:  
There is a wide range of geometric tools that can be utilized and exploited in data analysis. The purpose of this talk is to provide specific examples concerning structure in collections of digital images.

 November 3

Speaker:  Boyu Li, NMSU

Title: 
Zappa-Szep Product, Self-Similar Action, and Imprimitivity Theorems

Abstract:  
Zappa-Szep product is a way to combine two groups together using a two-way interaction. It generalizes the semi-direct product construction, which encodes a one-way action. It is also connected to the notion of self-similar action arising from iterated function system and automata theory. Our goal is to extend the Zappa-Szep product construction to operator algebras and generalize results in crossed product of operator algebras to this new context. In this talk, I will present a generalized imprimitivity theorem for Zappa-Szep products that unified many similar theorems from group dynamics. This is a joint work with Anna Duwenig.

 October 27

 O'Donnell Hall 111

 1:00PM - 2:00PM

Speaker:  Agnès Beaudry, University of Colorado

Title: 
It's The Great Grading, Edgar Brown

Abstract: 
There aren't many theorems in topology as sincere as Thom's isomorphism for cohomology. Yet it's a whimsical result, requiring orientation or the choosing of sign-less coefficients. Even the smallest slip can cost you its visit. It takes the most serious of tumbles in the equivariant bubbles. But I will tell you a story of how Costenoble-Waner's theory fixes the thing with The Great Grading. 

The new tricks and treats presented today are joint with my fellow peanuts: Lewis, May (not Gaunce and Peter, but Chloe and Clover), Sabrina Pauli and Liz Tatum.

 October 27

 O'Donnell Hall 111

 2:30PM - 3:30PM

Speaker:  Katherine Poirier, New York City College of Technology

Title: 
Applying Chain-level Poincaré Duality to String Topology

Abstract:  
String topology studies algebraic structures that arise by intersecting loops, where a “loop” can mean something topological or algebraic. For example (on the topological side) on the homology of the free loop space of a closed, oriented manifold, there is a binary operation called the “loop product” and a unary operation called the “BV operator.” These two operations together give the homology of the free loop space the structure of a “BV algebra.” Separately (on the algebraic side) in the presence of an algebraic version of Poincaré duality, there is product and BV operator on the Hochschild cohomology of this algebra. These operations give the Hochschild cohomology of the algebra the structure of a BV algebra as well. Further, when the algebra is the cochain algebra of a closed, oriented, simply connected manifold there is an isomorphism between its Hochschild cohomology and the homology of the free loop space of the manifold. While Cohen and Jones showed that this isomorphism respects the product structure, subsequent work of Menichi suggested that, in the case of the 2-sphere with mod 2 coefficients, it did not respect the BV operator. In this talk, I will describe these operations and show that with an appropriate updated algebraic version of Poincaré duality for algebras—one involving higher homotopies—Hochschild cohomology can be given a BV operator that is, in fact, preserved by the isomorphism from the homology of the free loop space of the 2-sphere with mod 2 coefficients. This is joint work with Thomas Tradler.

 October 27

 O'Donnell Hall 111

 4:00PM - 5:00PM

Speaker:  Zhouli Xu, University of California at San Diego

Title: 
Stable Homotopy Groups of Spheres and Motivic Homotopy Theory

Abstract:  
The computation of stable homotopy groups of spheres is one the most fundamental problems in topology. It has connections to many topics in topology, such as cobordism theory and the classification of smooth structures on spheres.

In this talk, we will survey some classical methods, explain their difficulty via Mahowald’s Uncertainty Principles, and describe a new technique using motivic homotopy theory. This new technique yields streamlined computations through previously known range, and gives new computations through dimension 90.

 October 20

Speaker:  Andrew Comech, Texas A&M

Title: 
Stability Solitary Waves in the Nonlinear Dirac Equation

Abstract:  
The electron is known to physicists to be stable, and Wikipedia gives a one-line proof: the electron is the least massive particle with non-zero electric charge, so its decay would violate charge conservation. On the mathematical side, the (classical) Dirac--Maxwell system is known to have localized solitary waves; their stability is still not known. Mathematicians can not claim whether these solutions have any relation to electrons or not, but at least we hope to eventually understand their stability; no stability -- no relation to electrons.

The main result of the talk is the spectral stability (absence of linear instability) of solitary waves in the nonlinear Dirac equation with the scalar self-interaction, known as the Soler model. The stability takes place when certain relation between powers of nonlinearity and the spatial dimension is satisfied. We will illustrate some ideas of the proof on simpler models: nonlinear wave equation and nonlinear Schrodinger equation.

This is a joint research with Nabile Boussaid, University Franche—Comte (Besancon).

The results are available in:

Nabile Boussaid, Andrew Comech, Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity, Journal of Functional Analysis 277 (2019), 108289.

http://arxiv.org/abs/1705.05481

 October 6

Speaker:  Arvind Kumar, NMSU

Title: 
Regularity Bound of Generalized Binomial Edge Ideal of Graphs

Abstract:  
I will discuss the Castelnuovo-Mumford regularity of generalized binomial edge ideals. This class of ideals arises in the study of conditional independence ideals and was introduced by Johannes Rauh in 2011. I will cover Saeedi Madani and Kiani's conjecture about the regularity of this class of ideals.

 September 22

Speaker:  Frank Sottile, Texas A&M University

Title: 
Galois Groups in Enumerative Geometry

Abstract: 
In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that  a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.

I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.

 September 1

Speaker:  Yang Hu, NMSU

Title: 
A Calculus Approach to Vector Bundle Enumerations

Abstract:  
In the unstable range, topological vector bundles over finite CW complexes are difficult to classify in general. Over complex projective spaces, such bundles are far from being fully classified, or even enumerated, except for a few small dimensional cases studied in the 1970’s using classical tools from homotopy theory, and more recently using the modern tool of chromatic homotopy theory. We apply another modern tool, Weiss calculus, to enumerate topological complex vector bundles over complex projective spaces with trivial Chern class data, in the first two cases of the metastable range.

 August 25

Speaker:  Michael DiPasquale, NMSU

Title: 
Two Perspectives on Affine Semigroup

Abstract: 
A numerical semigroup is the set of all integers which can be written as a non-negative integer combination of a fixed set of positive integers which are relatively prime.  The non-negative integers which are not in the numerical semigroup is called the set of holes of the semigroup.  In 1978, Herbert Wilf posed a question about the density of this set of holes which is still widely open and has become known as Wilf's conjecture.  In the first portion of this talk we will discuss Wilf's conjecture and an extension of it to higher dimensions, where a numerical semigroup is replaced by a certain type of affine semigroup.  An affine semigroup is the set of non-negative integer combinations of a fixed set of vectors with integer entries.

In the remainder of the talk we will focus on the representation of an affine semigroup as a quotient of a polynomial ring by a so-called toric ideal. A Gr\"obner basis for the toric ideal possesses a wealth of information about the semigroup, and it is of particular interest both in pure and applied algebraic geometry to find a Gr\"obner basis of low degree. Time permitting we will touch on two projects where we find a Gr\"obner basis of quadrics for combinatorially meaningful classes of toric ideals.

 September 22

Speaker:  Frank Sottile, Texas A&M University

Title: 
Galois Groups in Enumerative Geometry

Abstract: 
In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that  a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.

I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.

 September 1

Speaker:  Yang Hu, NMSU

Title: 
A Calculus Approach to Vector Bundle Enumerations

Abstract:  
In the unstable range, topological vector bundles over finite CW complexes are difficult to classify in general. Over complex projective spaces, such bundles are far from being fully classified, or even enumerated, except for a few small dimensional cases studied in the 1970’s using classical tools from homotopy theory, and more recently using the modern tool of chromatic homotopy theory. We apply another modern tool, Weiss calculus, to enumerate topological complex vector bundles over complex projective spaces with trivial Chern class data, in the first two cases of the metastable range.

 August 25

Speaker:  Michael DiPasquale, NMSU

Title: 
Two Perspectives on Affine Semigroup

Abstract: 
A numerical semigroup is the set of all integers which can be written as a non-negative integer combination of a fixed set of positive integers which are relatively prime.  The non-negative integers which are not in the numerical semigroup is called the set of holes of the semigroup.  In 1978, Herbert Wilf posed a question about the density of this set of holes which is still widely open and has become known as Wilf's conjecture.  In the first portion of this talk we will discuss Wilf's conjecture and an extension of it to higher dimensions, where a numerical semigroup is replaced by a certain type of affine semigroup.  An affine semigroup is the set of non-negative integer combinations of a fixed set of vectors with integer entries.

In the remainder of the talk we will focus on the representation of an affine semigroup as a quotient of a polynomial ring by a so-called toric ideal. A Gr\"obner basis for the toric ideal possesses a wealth of information about the semigroup, and it is of particular interest both in pure and applied algebraic geometry to find a Gr\"obner basis of low degree. Time permitting we will touch on two projects where we find a Gr\"obner basis of quadrics for combinatorially meaningful classes of toric ideals.

 

 

 

Spring 2023

 April 28

Speaker:  Samuel Scarpino, Northeastern University

Title: 
Behavior-induced Phase Transitions in Contagion Models on Networks

Abstract:  
Seemingly trivial modifications to the classical model of contagion spreading can dramatically alter its phenomenology.  For example, discontinuous phase transitions can occur due to complex or interacting contagions, accelerating transmission and hysteresis loops can occur when individuals modify their behavior after becoming infectious, and double phase transitions can emerge in the presence of asymmetric percolation. In this talk, I will present theoretical work on the effects of behavior on contagion spreading and discuss empirical support for these new models. Our findings demonstrate the inherent complexity of biological contagion, and we anticipate that our methods will advance the emerging field of disease forecasting.

 April 21

Speaker:  Boris Choy, University of Sydney Business School

Title: 
Non-elliptical Student-t Distributions with Computations and Applications

Abstract: 
In this talk, an extension from the elliptical to non-elliptical Student-t distribution is proposed to improve modelling capability. This extension is based on the mean-variance structure of the t and skew-t distributions. Although the probability density distributions of these non-elliptical distributions do not have a close form, the MCMC and EM algorithms have proven to be very efficient to handle these distributions. Financial data and insurance loss data are used for illustration purposes.

 April 14

Speaker:  Stephen Brazil, Actuary, USA

Title: 
What's an Actuary?

Joint with the Actuarial Seminar Business College, New Mexico State University

 March 24

Speaker:  Marco Aiello, University of Stuttgart, Germany

Title: 
Data is Nothing Without Control: On Building IoT Architectures for a Sustainable Society

Abstract:  
The availability of data in digital form is at unprecedented levels and is enabling a new way of building systems that are transforming our society. But data by itself is not enough to provide useful systems. In this talk, I will overview recent and current research efforts at the Service Computing Department of the Institute of Architecture of Application Systems of the University of Stuttgart. The research efforts revolve around the design of architectures that go beyond the mere collection of data and can support humans in smart ways. The applications of these systems are in the field of energy and sustainability, in particular: Smart Buildings, Smart Grids, and Smart Data Centers.

 March 3

Speaker:  Eugene Bilokopytov from University of Alberta, Canada

Title: 
Locally Solid Convergences on a Vector Lattice

Abstract: 
In this talk we will discuss convergences on a vector lattice which are not necessarily induced by a topology, but are compatible with the order and algebraic structure of the vector lattice. An example of such convergence is the convergence almost everywhere on the spaces of measurable functions. The word "convergence" will be given a precise meaning. We will also look at some canonical convergences associated with a vector lattice and characterize certain completeness-like properties of vector lattices in terms of the completeness of the corresponding convergence.

 February 24

Speaker:  Alexander Shen, University of Montpellier 2, France

Title:  Information Inequalities

Abstract:  The intuitive notion of "amount of information" can be formalized in may ways. The algorithmic information (Kolmogorov complexity) theory says that the amount of information in a finite object x is the minimal length of a program that produces x. Some intuitive properties of information (say, the amount of information in pair (x,y) is bounded by the sum of the amounts of information in x and y) can be proven. Another way to measure information is suggested by Shannon information theory. It turns out that the "laws of information theory" (understood as linear inequalities) are the same for both cases (Romashchenko), and the same class of inequalities appears in group theory, database theory, geometric dimension theory. A survey of these results will be given.

 January 25

Speaker:  Yairon Cid-Ruiz, KU Leuven, Belgium

Title:  Rees algebras of determinantal ideals

Abstract:  We will give an introduction to SAGBI basis theory and its application to the Rees algebra. We then present a couple of recent results using SAGBI basis theory to find the presentation equations of Rees algebras of determinantal ideals and their properties.

 January 23

Speaker:  Boyu Li, University of Waterloo

Title:  Analysis on Semigroups Through the Lens of Dilation Theory

Abstract:  The analysis of algebraic structures using bounded linear operators on Hilbert spaces traces back to the pioneering work of Murray and von Neumann. For example, the abstract harmonic analysis of groups relates various properties of a group to its unitary representations. For semigroups, however, the lack of inverse leads to much richer classes of representations. These representations are often interrelated via dilation theorems. In this talk, I will briefly go over the history of the analysis of semigroups, with a focus on the role of dilation theory. I will then explain how our recent dilation theorem on semigroup dynamical systems unifies many earlier results in dilation theory.

This is partly based on joint work with Marcelo Laca.

 January 20

Speaker:  Michael DiPasquale, University of South Alabama

Title:  A Bridge Between the Algebra and Geometry of Hyperplane Arrangements

Abstract:  A hyperplane arrangement is a union of codimension one linear spaces. These simple objects provide fertile ground for interactions between combinatorics, algebra, algebraic geometry, topology, and group actions. The combinatorics of an arrangement is encoded by the pattern of intersections among the hyperplanes, called its intersection lattice. On the other hand, a key algebraic object is the module of vector fields tangent to the arrangement, called the module of logarithmic derivations, which was introduced by Saito in 1980 to study the singularities of hypersurfaces. An enduring mystery in the theory of hyperplane arrangements is which algebraic properties of the module of logarithmic derivations can be determined from the intersection lattice, and which properties depend fundamentally on the geometry (i.e. the exact hyperplanes). At the center of this mystery is Terao's conjecture, which proposes that the algebraic property of freeness can be determined only from the intersection lattice. In this talk we explain how rigidity of planar frameworks (dating back to Maxwell) plays a key role in connecting the geometry and algebra of line arrangements in the projective plane.

This is joint work with Jessica Sidman and Will Traves.

 January 18

Speaker:  Timothy Rainone, Occidental College

Title:  Paradoxical decompositions, dichotomy, and Finite-Approximability in Operator Algebras

Abstract:  Notions of finiteness are prevalent in mathematics, not least in operator algebras. For instance, with matrix algebras we observe Dedekind-finiteness; that is every isometry is unitary. This also holds for large classes of C*-algebras as well as their stabilization. Such algebras are called stably finite. At the opposite end of the spectrum are the purely infinite C*-algebras whose internal structure admit paradoxical decompositions; especially when the algebras in question are crossed-product constructions. In this talk we will study the stably finite/purely infinite dichotomy for certain C*-algebras that arise from dynamical systems. We will then move our discussion to matricial field (MF) C*-algebras; a subclass of stably finite algebras that admit external matrix approximations. The C*-analogue of the Connes Embedding Conjecture proposes that every stably finite C*-algebra should admit such an approximation. We show that is is indeed the case for large classes of crossed products of nuclear algebras by free groups. This talk is aimed at a general math audience; we will carefully introduce the theory of C*-algebras as well as all the notions alluded to above.

January 16

Speaker:  Alessandra Costantini, Oklahoma State University

Title:  Rees Algebras: An Algebraic Tool to Study Singularities

Abstract:  How many tangent lines does a plane curve have at a given point? And how can we effectively solve polynomial equations with integral coefficients? These seemingly unrelated problems from algebraic geometry and number theory can be both solved using tools from abstract algebra, through the notion of Rees algebras.

In this talk, I will first discuss the fundamental role of Rees algebras in the study of commutative rings and singularities of algebraic varieties. I will then focus on the problem of describing Rees algebras in terms of generators and relations, presenting some of my recent results on this topic, which were obtained using techniques from algebraic combinatorics.

January 13

Speaker:  Sudan Xing, University of Alberta, Canada

Title:  Dual Orlicz-Minkowski Problem

Abstract:  The classical Minkowski problem is a central problem in convex geometry which asks that given a nonzero finite Borel measure, what are the necessary and sufficient conditions on this measure such that it equals to the surface area measure of a convex body. In this talk, I will present the general dual extension of the classical Minkowski problem—the generalized dual Orlicz-Minkowski problem.