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Pure and Applied Mathematics Seminar

Spring 2024 (Thursdays from 12:00 p.m. to 12:50 p.m.)

i) January 25

Location: STV 136B

Speakers: Bob Skudnig (ISU), Christian Soltermann (ISU), Roberto Arturo Martinez Cerceno (ISU)

Title: An Application of Lie Algebra on Image Tracking

Abstract: Computer vision is a field of computer science concerned with extracting meaningful information from images and videos. An important subfield of computer vision is image tracking, which involves tracking the positions and orientations of objects over time for use in real-world applications. In this project, we apply concepts within the realm of Lie algebra to predict image transformations. Images were randomly transformed using affine matrices and then fed into various machine learning models with their respective transformation matrices. Then, the models predicted the affine matrix from the image with relatively high accuracy.

ii) February 1

Location: STV 136B

Speaker: Charlotte Ure (ISU)

Title: A common slot lemma over semiglobal fields

Abstract: Galois cohomology is an important invariant associated to any field. It captures many algebraic properties of the underlying field. For example, two torsion elements in the second cohomology group may be interpreted as equivalence classes of quaternion algebras. Tate’s common slot lemma states that, over a number field, any finite number of quaternion algebras may be split by a single quadratic extension of the base field. I will discuss a generalization of this result to semiglobal fields. This talk is based on joint work with Sarah Dijols, Raman Parimala, and Sujatha Ramdorai.

iii) February 8

Location:STV 136B

Speaker: Mehdi Karimi, Ph.D. (ISU)

Title: Efficient Optimization Methods for Quantum Relative Entropy

Abstract: In this talk, we start by defining the quantum relative entropy (QRE) cone and then discuss how we can optimize a convex function over this cone. Optimization over the QRE cone has many applications in quantum information processing, for example, calculating the key rates for quantum key distribution (QKD) protocols (we will see what it is). We have an optimization software package, Domain-Driven Solver (DDS), whose older versions could solve such optimization problems, with some limitations on the size. In this talk, we present new theoretical and computational results to improve our optimization methods’ performance for QRE significantly. These results were used to improve the new version of DDS (namely DDS 2.2). We present some numerical results using DDS, which lets us combine QRE constraints with many other function/set constraints. We finish with some open questions, such as how duality concepts like the dual cone and the Legendre-Fenchel conjugate can be used to improve performance. 

iv) February 22

Location: STV 136B

Speaker: Sunil Chebolu, Ph.D. (ISU)

Title: Additive Subgroups of Commutative Rings.

Abstract: In remembrance of the late Professor Wenhua Zhao, I will share a project that was an offshoot of some interesting discussions I had with him and Gail Yamskulna on Mathieu-Zhao subspaces. The investigation centers on a fundamental question: When does an additive subgroup of a commutative ring possess the property of being an ideal? We use elementary methods to address this problem, and it is tailored for students who completed a first course in abstract algebra.  This is joint work with my graduate student, Christina Negley.

v) March 7, 2024

Location: STV 136B

Speaker: Papa Sissokho, Ph.D. (ISU)

Title: Non-negative integer solutions of Linear Equations.

Abstract: Let A be a d x n-matrix of rank d with integer entries. Let S denote the set of all solutions 𝑥⃗ to the equation 𝐴𝑥⃗=0⃗⃗ such that the entries of 𝑥⃗ are nonnegative integers. The Hilbert basis of S is the minimal subset H of S with the property that any solution 𝑥⃗ in S can be written as a nonnegative integer combination of solutions in H. 

vi) March 21, 2024

Location: STV 136B

Speaker: Gaywalee Yamskulna, Ph.D. (ISU)

Title: Exploring the Frontier: Graded Traces in Vertex Operator Algebras Beyond the C_2 -Condition 

Abstract: Vertex operator algebras (VOAs) are the cornerstone of conformal field theory. Their profound ties to number theory and the representation theory of Lie algebras, finite simple groups, and quantum groups underscore their significance in contemporary mathematics. 

In Zhu’s landmark contribution, exploring graded traces within ℕ-gradable modules of a VOA V illuminated profound insights. Under the semisimplicity of module category and adherence to the C_2 condition, Zhu demonstrated the modular invariance of these graded traces. Further, Zhu unveiled a striking convergence: the graded dimensions of simple V-modules, as functions of 𝜏 converge to holomorphic functions on the complex upper half plane, and the linear space spanned by these holomorphic functions remains invariant under the action of 𝑆𝐿_2(ℤ). 

Building upon Zhu’s groundwork, Miyamoto expanded the scope of these findings. For VOAs whose module categories are no longer semisimple but still satisfy the C_2 condition, Miyamoto showcased the persistence of Zhu’s results by incorporating graded pseudo-traces into the analysis. 

A natural inquiry arises: what unfolds when V no longer abides by the C_2 condition? This question, ripe with intrigue, is a focal point for our talk. 

In this talk, we will first lay the groundwork with a thorough exposition of vertex operator algebras and their modules. Along the way, we elucidate key concepts through illustrative examples. Our narrative traverses the seminal contributions of Zhu and Miyamoto, weaving together their profound insights. 

Finally, we pivot to discuss my collaborative work with Katrina Barron, Karina Batistelli, and Florencia Orosz Hunziker. Together, we delve into graded pseudo-traces for vertex operator algebras that defy the confines of the C_2 condition and elude classification within a semisimple module category. Our joint efforts illuminate new pathways, advancing our understanding of these enigmatic structures. For instance, we introduced the notion of strongly interlocked generalized modules for vertex operator algebra and showed that the notion of graded pseudo-trace is well-defined. In addition, we proved that graded pseudo-trace is a symmetric linear operator that satisfies the logarithmic derivative property. 

vii) March 27, 2024

Location: STV 136B

Speaker: Dr. Ünal Ufuktepe (University of Missouri-Columbia)

Title: Discrete Wolbachia Diffusion in Mosquito Populations with Allee Effects

Abstract: We explore the stability analysis of a discrete-time dynamical system involving the diffusion of Wolbachia in mosquito populations, incorporating Allee effects on the native mosquito population. Our investigation delves into the competition dynamics between released and wild mosquitoes. The study encompasses an examination of the local and global stabilities of fixed points, as well as an exploration of bifurcation types based on varying parameters.

viii) April 4, 2024

Location STV 136B

Speaker: Dr. Nick Rekuski (Wayne State University)

Title: Stability of Syzygy Bundles

Abstract: Vector bundles are an algebraic tool to study geometric spaces. In other words, algebraic properties of a vector bundle detect subtle geometric properties of a space. For example, the hairy ball theorem says the tangent bundle on a sphere is indecomposable. That is to say, the tangent bundle cannot be written as the sum of two vector bundles. In contrast, the tangent bundle on the 3-sphere can be written as the sum of two vector bundles. In general, on a fixed space, it is a difficult problem to construct indecomposable vector bundles with given invariants. In this talk, we show a certain class of vector bundles, called syzygy bundles, are stable—a stronger property than indecomposable.

ix) April 11, 2024

Location  (Meeting ID: 948 5972 0062)

Speaker: Dr. Jianqi Liu from the University of Pennsylvania.

Title: Twisted conformal blocks of vertex operator algebras.

Abstract: Twisted representations arose naturally in the theory of vertex operator algebras. They give rise to twisted conformal blocks on orbifold curves.

In this talk, after introducing vertex operator algebras and related constructions, I will present some recent progress in finding the relations between twisted conformal blocks, correlation functions, and fusion rules among twisted modules. This talk is based on joint work with Xu Gao and Yiyi Zhu.

x) April 11, 2024

Speaker: Dr. Lucian Ionescu (ISU)

Title: Rethinking “Real Numbers”

Abstract: Graded rings and filtrations are structures for defining differential equations frameworks. We look for additional structures (grading?) for Real Numbers and the continued fractions representation is the key.

An introduction to Real Numbers for GS (abstract, existence and uniqueness, constructions) will precede our joint work with Anurag Kurumbail, with its main goal “to structure the continuum”, and of new insights into the theory of algebraic continued fractions, from Galois Theory viewpoint.

The last part of the presentation will survey the Math structures “behind” numbers, especially the class of algebraic periods, including what “pi is”, and what “e is”.

The talk aims to provide some new ideas, some old important math concepts every mathematician should know (eventually!) and emphasize the Math Design aspects of the profession, with historical “feedback” and modern “updates” as part of the mathematician’s job description.

xi) Date: April 25, 2024

Location: STV 136B

Speaker: Dr. Fusun Akman (ISU)

Title: Partitions of a Group into Cosets of Many Subgroups

Abstract: This is joint work with Papa Sissokho (2024), and part of our research program on subspace partitions of finite dimensional vector spaces. The Herzog-Schönheim Conjecture states that there is no partition of any group into finitely many left cosets of a list of subgroups where the subgroups have different indices (repeated indices may come from the same subgroup). We proved the conjecture for up to 7 distinct subgroups and proposed a stronger one for a certain class of groups/subgroups: if the subgroups of a group whose cosets contribute to a partition of the group mutually commute (i.e., HK=KH), then at least one subgroup must have two or more cosets in the partition. This conjecture holds for the cases of two and three subgroups, and with a certain restriction, for four subgroups; it is also true when a certain non-inclusion rule holds for all subgroups, and when the subgroups form a chain. We furthermore studied the inner structure of coset partitions. The products of mutually commuting subgroups are also subgroups, so an obvious/standard way to cover a group G with cosets (say, for three subgroups H, K, L) is to partition it into HKL-cosets, then partition each HKL-coset into HK-, HL-, or KL-cosets, and finally, each double coset into H-, K-, or L-cosets. We showed that for up to three contributing subgroups this standard construction is the ONLY way to achieve a coset partition of a group, but starting with four commuting subgroups, non-standard constructions are possible.

For our students: This talk should be very accessible if you are taking or have taken a course on basic group theory, and you can look up quotient groups, products of groups, cosets, and group actions beforehand if you like (I will define most terms; if not, ask!).

Fall 2023 (Thursdays from 12:00 p.m. to 12:50 p.m.)

i) September 21

Speaker: Charlotte Ure (ISU)

Talk Title: Twisting of Algebras and Comodules

Abstract: Deformations of algebra structures arise naturally in the study of quantum symmetries and have many applications in representation theory and noncommutative algebra. For example, one may twist the algebra structure by a graded isomorphism. In this talk, I will introduce and compare different methods of deformation of algebras such as twisting by an automorphism or by a 2-coycle. The talk is based on joint work with Hongdi Huang, Van C. Nguyen, Kent B. Vashaw, Padmini Veerapen, and Xingting Wang.

ii) October 12 (10:00am-10:50am)

Speaker: Samantha Kirk (Bradley University)

Talk Title: A Vertex Algebra Construction of Representations of Twisted Toroidal Lie Algebras 


 Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one can construct a twisted toroidal Lie algebra. Similar to twisted affine Lie algebras, which are well-studied in the literature, we can construct representations of twisted toroidal Lie algebras with the help of vertex algebras. In this talk, I will discuss twisted modules of vertex algebras, and I will show how representations of twisted toroidal Lie algebras can be constructed from such twisted modules. 

Joint work with Bojko Bakalov 

iii) October 19

Speaker: Lucian Ionescu (ISU)

Talk Title: Real Numbers: A New (Quantum) Look

Abstract: The continuum of Real Numbers is central in Math, but in conflict with modern Physics, Sciences, Alg. Geom., Number Theory etc. I will present a Number Theory oriented completion of the rational numbers, using sequential topology (no metric/distance used) based on continued fractions and modular group PSL(2;Z) (Mobius transformations of the rational circle).

The background material presented is interesting in itself: Continued Fractions and Modular Group (2D-congruence arithmetic), fractions and Farey sequence, providing a deeper understanding of Pythagorean triples, algebraic numbers etc. 

Applications and implications to Mathematics in general (and Physics) will be briefly mentioned at the end; for example, Platonic solids are just “dessins d’enfant” on modular curves with Belyi maps, as previously presented in the seminar.

This is work in progress together with Anurag Kurumbail, just opening a new direction of research .

iv) November 16

Speaker: Benton Duncan (ISU)

Title: What is an operator algebra?

Abstract: I will discuss the definition of operator algebras as well as their abstract characterizations. Teh abstract characterization for C*-algebras is much cleaner, and I will survey work on connecting arbitrary operator algebras with associated C*-algebras.

v) November 30

Speaker: Thuy (Christy) Bui (Ph.D. Candidate in Operations Research at Rutgers Business School)

Title: A game-theoretic approach to patrolling problems

Abstract: This research studies the problems of finding optimal patrols to prevent adversarial attacks on a network, for example, addressing how to patrol a border to guard against infiltration or how to optimally patrol an airport or shopping mall to minimize the risk of terrorist attacks. We consider the problems (games) in continuous time and space where the attack could take place anywhere on the network at any time. The game is modeled as a zero-sum game between an Attacker and a Patroller. The Attacker will choose a time and a point to attack the network for a fixed amount of time. The Patroller picks a patrol of the network. The Patroller wins if he visits the attacked point while the attack is occurring; otherwise, the Attacker wins. We present (i) a solution for any arbitrary networks as long as the attack time is sufficiently short, (ii) a solution for all tree networks with any attack time, and (iii) a solution in some cases for complete networksThis is joint work with Steve Alpern, Thomas Lidbetter, and Katerina Papadaki.

vi) December 7

Speaker: Matt Speck (Ph.D. Candidate at Auburn University)

Title: Progress on the Marcus-de Oliveira Conjecture

Abstract: Marcus (1972) and de Oliveira (1982) independently conjectured bounds on the determinantal range of a sum of normal matrices given their respective eigenvalues. We will propose a proof method rooted in Lie theory and representation theory and discover some interesting combinatorics along the way. This talk will be accessible to anyone with mathematical curiosity, regardless of background. This is joint work with Luke Oeding.

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