Fall 2024 (Thursdays from 12:00 p.m. to 12:50 p.m.)
i) September 5
Location: STV 314
Speaker: Sunil Chebolu (ISU)
Title: Fuchs Problem for Endomorphisms of Groups
Abstract:
In 1960, László Fuchs posed the problem of determining which groups G are realizable as the group of units in some ring R. In a series of paper, Keir Lockridge and I answered this question for a variety of groups. In our more recent work, we posed the following variant of this problem. Which groups G are realized by a ring R where every group endomorphism of G is induced by a ring endomorphism of R? Such groups are called fully realizable. I will discuss some results about this new problem that we published recently in the following two papers.
https://arxiv.org/abs/2308.11815
https://arxiv.org/abs/2408.08195
ii) September 12
Location: STV 314
Speaker: Dr. Perry Kleinhenz (ISU)
Title: The damped wave equation: overdamping and finite-time extinction
Abstract: The damped wave equation models a vibrating system in the presence of a friction force, which causes the energy of the system to tend towards 0. Counterintuitively, larger damping frequently produces slower energy decay, which is called overdamping. Because of this, the classical theory of bounded damping does not provide energy decay rates for unbounded damping. In this talk, I will present classical decay rates and explain how they change for unbounded damping. I will also discuss the surprising phenomenon of finite-time extinction for unbounded damping and a sharp condition on damping that rules it out. This is from joint work with Ruoyu P.T. Wang.
iii) September 19
Location: STV 314
Speaker: Bob Skudnig and Joseph Wittrock (ISU)
Title: Machine Learning in Building Sustainment: Data and Deterioration
Abstract:
Artificial intelligence (AI) has become a very powerful tool in recent years to assist with a wide range of tasks including data prediction, image classification, and language generation. Many organizations within the United States government, namely the Sustainability Management System Technical Center of Expertise (SMS-TCX), are currently looking into the potential applications and capabilities of AI models. Domain experts within SMS are researching applications of
• neural networks to predict the importance of military assets
• sequential decision optimization for component repair efficiency.
This seminar, given by two Student Research Mathematicians from SMS currently attending ISU, will discuss the mathematics under the hood of these robust models as well as their direct applications to Army sustainment.
This seminar, given by two Student Research Mathematicians from SMS currently attending ISU, will discuss the mathematics under the hood of these robust models as well as their direct applications to Army sustainment.
iv) October 3
Location: STV 314
Speaker: Charlotte Ure, Ph.D. (ISU)
Title: Quantum symmetries in algebras
Abstract: Classically, symmetries arise from group actions on polynomial rings. In contrast, quantum symmetries come from coactions of Hopf algebras on algebras. In this talk, I will discuss the notion of quantum-symmetric equivalence of algebras. I will investigate the equivalence classes of this relation. In particular, building on work of Raedschelders and Van den Bergh, I will discuss how Koszul Artin–Schelter regular algebras of a fixed global dimension form a single quantum-symmetric equivalence class. This is joint work with Hongdi Huang, Van C. Nguyen, Kent B. Vashaw, Padmini Veerapen, and Xingting Wang.
v) October 10
Location: STV 314
Speaker: Gaywalee Yamskulna, Ph.D. (ISU)
Title: Vertex Algebras associated with Gorenstein rings and Their Relation to the Free Boson Vertex Operator Algebra
Abstract: We consider vertex algebras, whose degree-zero component forms a Gorenstein ring, a structure arising naturally in commutative algebra with well-defined duality properties. The focus will be on understanding the implications of the Gorenstein condition on the algebraic structure of vertex algebras. Additionally, we will explore the connection to the free boson vertex operator algebra, a fundamental example in the theory of vertex algebras. This talk will demonstrate how the Gorenstein property imposes constraints and leads to new insights in the representation theory of vertex algebras.
This talk is based on joint work with Alex Keene and Christian Soltermann, both master’s degree students in pure and applied mathematics program at Illinois State University.
VI) October 24
Location: STV 314
Speaker: Lucian Ionesco, Ph. D. (ISU)
Title: On Some Programs in Mathematics and Physics – Part II: On Langlands programs and beyond
Abstract: We continue our historical Journey through Mathematics, continuing Pari I (if interested see WZ AGS, Fall 2024, 10/24 announcement; but not required as a background), with a far-reaching Program of the 60s aiming to relate Algebra and Geometry at even deeper levels: Langlands Program.
Developed further as a team effort by the Math community, recently reached a high point to be presented.
While quite deep and technical, the ideas can be understood by examples and with pictures, like a commentator of an exciting mountain climb, say of Everest.
Less known, newer developments in Math-Physics led to an even more ambitious Program: a correspondence between Natural Physics Laws and Algebraic-Geometric Periods, with a predecessor Noether Theorems as a historical breakthrough (Part I). It is of course related to Riemann Hypothesis and the duality between Riemann zeros and prime numbers.
The emerging conclusion: Is “Reality” Math or Physics? The answer is … Come to the talk to find out.
VII) October 31, 2024
Location: https://illinoisstate.zoom.us/j/89922146872
Speaker: Christy Bui, Ph.D. (Machine Learning Software Engineer, Google)
Title: A Game Theoretic Approach To Search For A Moving Target
Abstract: We present a zero-sum game between a Hider and a Searcher. The Hider places a target in one of n boxes. The target then moves among boxes according to a time-homogeneous Markov chain. Each box i has a detection probability qi. That is, given the target is in box i, a search of that box will find the target independently with probability qi. The Searcher picks a search sequence of length m corresponding to the order in which the boxes are searched. The payoff is the probability of finding the target in m searches, which the Searcher wishes to maximize, and the Hider wishes to minimize. We prove the game has a value and each Player has an optimal strategy. We also present the solution of the game for some classes of Markov chains.
VIII) November 7, 2024
Location: https://illinoisstate.zoom.us/j/89922146872
Speaker: Tung Nguyen, Ph.D. (Lake Forest College)
Title: Algebraic graph theory, computational number theory, and non-linear dynamics
Abstract: Network theory plays a fundamental role in the study and understanding of collective behavior in biological and physical systems. Typically, we consider these systems to be a single network. However, many real-world systems are multi-leveled; namely, the system can have two levels of connections. On the first level, there are connections within each sub-network, and on the higher level, there are connections between the sub-networks. While having multi-level networks provides a more realistic and rich modeling option, it can lead to intrinsic difficulty in analyzing their dynamics. This observation leads to the following natural question: given a network, can we develop an algorithm to detect whether it has a multi-layered structure?
In this talk, using tools from algebraic graph theory, we will provide an answer to this question when the underlying graph is a Cayley graph. In particular, we provide a complete classification of unitary Cayley graphs which are not multi-layered. Time permitting, we will discuss some related problems in computational number theory.
IX) November 14, 2024
Location: https://illinoisstate.zoom.us/j/89922146872
Speaker: Devin Akman (Washington University in St. Louis )
Title: A-Polynomials and Zero Loci of Regulator Maps
Abstract: The A-polynomial is an invariant of 3-manifolds with toric boundary, such as knot exteriors. We view it as an algebraic curve and place it in a family by varying coefficients. I will discuss the criterion in algebraic K-theory that characterizes A-polynomial curves. New finiteness results about the R-split locus of an admissible normal function (equivalently, an extension of admissible variations of mixed Hodge structures) will then have implications for the finiteness of the “A-polynomial locus.”
X) November 21, 2024
Location: STV 314
Speaker: Papa Sissokho, Ph.D. (ISU)
Title: THE MINIMUM SIZE OF A FINITE VECTOR SPACE PARTITION
Abstract: Let V = V (d, q) denote the vector space of dimension d over the finite field with q elements. A vector space partition (VSP) of V, is a collection of nontrivial subspaces of V such that each nonzero vector is in exactly one subspace in the collection. In the first part of the talk, we will survey the main research problem in the area of VSP and discuss its applications to coding theory. In the second part, we will discuss the minimum size of a VSP of V (d, q) whose largest subspace has a fixed dimension t < d. We also explore variants of this concept and its relevance to some classical combinatorial notions (e.g., blocking sets).
The second part is based on a joint work with Esmeralda Nastase (Xavier University).
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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: https://illinoisstate.zoom.us/j/94859720062 (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
Abstract:
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 networks. This 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.