Claremont Center for the Mathematical Sciences

On Invariants for Surface-Links in Entropic Magmas via Marked Graph Diagrams (Seonmi Choi, Kyungpook Natl U, Korea)
2022-02-15 15:00 - 16:00 (America/Los_Angeles) | Zoom
M. Niebrzydowski and J. H. Przytycki defined a Kauffman bracket magma and constructed the invariant P of framed links in 3-space. The invariant is closely related to the Kauffman bracket polynomial. The normalized bracket polynomial is obtained from the Kauffman bracket polynomial by the multiplication of indeterminate and it is an ambient isotopy invariant for links. In this talk, we reformulate the multiplication by using a map from the set of framed links to a Kauffman bracket magma in order that P is invariant for links in 3-space. We define a generalization of a Kauffman bracket magma, which is called a marked Kauffman bracket magma. We find the conditions to be invariant under Yoshikawa moves except the first one and use a map from the set of admissible marked graph diagrams to a marked Kauffman bracket magma to obtain the invariant for surface-links in 4-space.

Solving the Race in Backgammon (Prof. Arthur Benjamin)
2022-02-16 16:15 - 17:30 | Zoom meeting, United States
Title: Solving the Race in Backgammon

Speaker: Prof. Arthur Benjamin
Smallwood Family Professor of Mathematics
Harvey Mudd College

Abstract: Backgammon is perhaps the oldest game that is still played today. It is a game that combines luck with skill, where two players take turns rolling dice and decide how to move their checkers in the best possible way. It is the ultimate math game, where players who possess a little bit of mathematical knowledge can have a big advantage over their opponents. Players also have the opportunity to double the stakes of a game using something called the doubling cube, which—when used optimally—leads to players winning more in the long run. Optimal use of the doubling cube relies on a player’s ability to estimate their winning chances at any stage of the game.

When played to completion, every game of backgammon eventually becomes a race, where each player attempts to remove all of their checkers before their opponent does. The goal of our research is to be able to determine the optimal doubling cube action for any racing position, and approximate the game winning chances for both sides. By calculating the Effective Pip Count for both players and identifying the positions’ Variance Types, we arrive at a reasonably simple method for achieving this which is demonstrably superior to other popular methods.

Arthur Benjamin, PhD, Smallwood Family Professor of Mathematics, is recognized nationally for his ability to perform rapid mental calculations. In 2020 he won the inaugural American Backgammon Tour Online (ABTO) with the best overall performance in a series of 17 national tournaments. He has published several books on how to make math both fun and easy. He is also a professional mathemagician and frequently performs at the Magic Castle in Hollywood and nationwide.

Modeling Zoonotic Infectious Diseases from Wildlife to Humans (Prof. Linda J. S. Allen)
2022-02-23 16:15 - 17:30 | Zoom meeting, United States
Title: Modeling Zoonotic Infectious Diseases from Wildlife to Humans
Speaker: Prof. Linda J. S. Allen, P. W. Horn Distinguished Professor Emeritus Texas Tech University
Abstract: Zoonotic infectious diseases are diseases transmitted from animals to humans. It is estimated that over 60% of human infectious diseases are zoonotic. The Centers for Disease Control and Prevention has identified eight priority zoonoses in the US. Three of the priority zoonoses are avian influenza, Lyme disease, and emerging coronaviruses. Spillover of infections from animals to humans depends on a complex pathway from the natural wildlife reservoir. The natural reservoir for avian influenza virus is wild birds but it is spread to humans from infected chickens. The natural reservoir for the bacterial pathogen causing Lyme disease is mice but it is transmitted to humans through the bite of an infected tick vector. In this presentation, we discuss a few of the modeling efforts to better understand the spread of infection in the natural reservoir and the spillover to humans as well as the impacts of demographic and environmental variability on timing of spillover.
___________________________________________________________________________________________________
Linda J. S. Allen received her PhD in Mathematics from University of Tennessee and was a Professor of Mathematics at Texas Tech University until 2019. She is currently an Adjunct Graduate Faculty at Texas Tech University. Her research interests are in mathematical ecology, epidemiology, and immunology.
https://www.math.ttu.edu/~lallen/
https://www.depts.ttu.edu/provost/scholars/lindaallen.php

Research Experiences for Undergraduates at Texas Tech University “Mathematical, Statistical, and Computational Methods for Problems in the Life Sciences”
June 6-July 20, 2022

REU Applications Due: March 6, 2022:
https://www.math.ttu.edu/undergraduate/reu2022/

Applied Math Seminar -- Illia Karabash (IAMM of NAS of Ukraine and TU Dortmund)
2022-02-28 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: Pareto optimization of resonances and optimal control methods
Abstract:
First successes in fabrication of high-Q optical cavities two decades ago led to active applied physics and numerical studies of optimization problems involving resonances. The questions is how to design an open resonator that has an eigenvalue as close as possible to the real line under certain constraints. The analytic spectral optimization theory for such types of non-Hermitian eigenproblems is still in the stage of development. It is planned to explain briefly why the Pareto optimization settings are natural for non-Hermitian spectral problems, and how the associated nonlinear Euler-Lagrange eigenproblems can be rigorously derived for the case of resonances in 1-d photonic crystals. Then we concentrate on the recently developed optimal control approach of (Karabash, Koch, Verbytskyi `20) and show how it is related to Pareto frontiers and Hamilton-Jacobi-Bellman PDEs. An application of Pontryagin Maximum Principle and a special method of minimum-time shooting to a line of no-return will be also discussed.

Gap theorems for linear forms and for rotations on higher dimensional tori (Alan Haynes, University of Houston)
2022-03-01 12:30 - 13:20 | On Zoom
This talk is based on joint work with Jens Marklof, and with Roland Roeder. The three distance theorem states that, if x is any real number and N is any positive integer, the points x, 2x, … , Nx modulo 1 partition the unit interval into component intervals having at most 3 distinct lengths. We will present two higher dimensional analogues of this problem. In the first we consider points of the form mx+ny modulo 1, where x and y are real numbers and m and n are integers taken from an expanding set in the plane. This version of the problem was previously studied by Geelen and Simpson, Chevallier, Erdős, and many other people, and it is closely related to the Littlewood conjecture in Diophantine approximation. The second version of the problem is a straightforward generalization to rotations on higher dimensional tori which, surprisingly, has been mostly overlooked in the literature. For the two dimensional torus, we are able to prove a five distance theorem, which is best possible. In higher dimensions we also have bounds, but establishing optimal bounds is an open problem.

Two-Bridge Knots Admit no Purely Cosmetic Surgeries (Thomas Mattman, California State University, Chico)
2022-03-01 15:00 - 16:00 | Zoom
(Joint with Ichihara, Jong, and Saito). We show that two-bridge knots admit no purely cosmetic surgeries, ie no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument is based on a recent result by Hanselman and a study of signature and finite type invariants of knots as well as the SL(2,C) Casson invariant.

On sparse geometry of numbers (Prof. Lenny Fukshansky)
2022-03-02 16:15 - 17:30 | Shanahan B460 (HMC) and Zoom – Hybrid
Title: On sparse geometry of numbers

Speaker: Prof. Lenny Fukshansky, Department of Mathematics, Claremont McKenna College

Abstract: Geometry of Numbers is an area of mathematics pioneered by Hermann Minkowski at the end of the 19th century. He achieved stunning success introducing a novel geometric framework into the study of algebraic numbers, prompting mathematicians of later generations to compare his work to “the story of Saul, who set out to look for his father’s asses and discovered a Kingdom” (J. V. Armitage). In this talk, we will look at some contemporary variations of Minkowski’s classical results that will take us on a journey from linear algebra and convex analysis to algebraic number theory and arithmetic geometry. This is joint work with P. Guerzhoy and S. Kuehnlein.

Lenny Fukshansky is a Professor of Mathematics at Claremont McKenna College. His work is at the intersection of number theory, discrete geometry and geometric combinatorics. He is especially interested in lattices, quadratic forms, polynomials, height functions and Diophantine problems. When not doing math, Lenny loves biking in the mountains and drinking wine, although tries not to do it simultaneously.

Equidistribution of norm 1 elements in cyclic number fields (Kate Petersen, University of Minnesota Duluth)
2022-03-08 12:30 - 13:20 | On Zoom
By Hilbert’s theorem 90, if K is a cyclic number field with Galois group generated by g, then any element of norm 1 can be written as a/g(a). This gives rise to a natural height function on elements of norm 1. I’ll discuss equidistribution problems and show that these norm 1 elements are equidistributed (in an appropriate quotient) with respect to this height.

Systematically Detecting Flypes and Hexagonal Mosaics (Hugh Howards, Wake Forest University)
2022-03-08 15:00 - 16:00 | Zoom
We talk about building knots using mosaics which were as introduced as a way of modeling quantum knots by Lomonaco and Kauffman and a newer variant, hexagonal mosaics, introduced by Jennifer McLoud-Mann. In the process we find a new bound on crossing numbers for hexagonal mosaics and find an infinite family of knots which do not achieve their hexagonal mosaic number while also in a projection which achieves their crossing number, extending a result of Lew Ludwig et al. In the process we introduce a new tool which makes it easier to systematically recognize when two knots differ by a sequence of Flypes (for example, giving a process to recognize that the Perko Pair were in fact the same knot). No background with mosaics or flypes is necessary. This is joint work with Jiong Li* and Xiotian Liu* (* indicates undergraduate students).

CCMS Field Committee Meeting
2022-03-09 16:00 - 17:45 | Shanahan B460, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
The Field Committee Meeting is our chance to socialize with our colleagues and coordinate our course offerings for the coming academic year (2022-2023). Please come to discuss course offerings and other synergistic items. Refreshments in the Shanahan sunken courtyard at HMC starting at 4:00, meeting in Shanahan B460 at 4:20.
We will be back in person for this meeting. A Zoom link will also be sent out, for those unable to attend physically.

Applied Math Seminar -- Jamie Haddock (HMC)
2022-03-21 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: Connections between Iterative Methods for Linear Systems and Consensus Dynamics on Networks
Abstract:
There is a well-established linear algebraic lens for studying consensus dynamics on networks, which has yielded significant theoretical results in areas like distributed computing, modeling of opinion dynamics, and ranking methods. Recently, strong connections have been made between problems of consensus dynamics on networks and classical iterative methods in numerical linear algebra. This talk will discuss an instance of these connections, in particular between the gossip methods in distributed computing and the Kaczmarz methods in numerical linear algebra. We will present theoretical convergence results, empirical and numerical simulation results, and discuss future work in applying these numerical linear algebraic techniques to broader and more complex consensus dynamics models, especially those coming from opinion dynamics and ranking.

Continuous extensions of Ramanujan-expandable arithmetic functions (Matthew Fox, Perimeter Institute for Theoretical Physics and Chai Karamchedu, Sandia National Labs)
2022-03-22 12:30 - 13:20 | On Zoom
We describe a natural way to continuously extend arithmetic functions that admit a Ramanujan expansion and derive the conditions under which such an extension exists. In particular, we show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization. We take the divisor function as a case study, and consider how to continuously extend it to the reals.

Towards Knot Homology for 3-Manifolds (Aaron Mazel-Gee, California Institute of Technology)
2022-03-22 15:00 - 16:00 | Zoom
The Jones polynomial is an invariant of knots in R^3. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin-Turaev using quantum groups. Khovanov homology is a categorification of the Jones polynomial of a knot in R^3, analogously to how ordinary homology is a categorification of the Euler characteristic of a space. It is a major open problem to extend Khovanov homology to knots in 3-manifolds. In this talk, I will explain forthcoming work towards solving this problem, joint with Leon Liu, David Reutter, Catharina Stroppel, and Paul Wedrich. Roughly speaking, our contribution amounts to the first instance of a braiding on 2-representations of a categorified quantum group. More precisely, we construct a braided (infinity,2)-category that simultaneously incorporates all of Rouquier’s braid group actions on Hecke categories in type A, articulating a novel compatibility among them.

The 6 Cs - Covid and the 5 Claremont Colleges (Prof. Maryann E. Hohn)
2022-03-23 16:15 - 17:30 | Shanahan B460 (HMC) and Zoom – Hybrid
Title: The 6 Cs – Covid and the 5 Claremont Colleges
Speaker: Maryann E. Hohn, Department of Mathematics and Statistics, Pomona College
Abstract: The Claremont Colleges’ (5Cs) environment consists of students, faculty, and staff that congregate together in indoor spaces, creating a higher risk for possible COVID-19 infection. Additionally, a majority of the students live on campus, presenting a relatively closed campus environment that limits students’ interactions with their greater community. However, the close knit quarters in which students live may contribute to a rise in infections that may ultimately reach other more vulnerable populations on the campuses such as faculty and staff.

In this talk, we present several models of COVID-19 spread at the 5Cs. We start with an early model consisting of several interconnected modified SEIR differential equations to investigate the dynamics between different populations at the 5Cs and the influence of mitigation techniques such as students adhering to health protocols and contact tracing. With the addition of vaccines, we show how the model changed, how student researchers are contributing to our models, and how a few students created their own.

Dr. Maryann Hohn is a Visiting Assistant Professor of Mathematics and Statistics at Pomona College. Her research interests lie in mathematical modeling and data analysis to solve societal problems. She utilizes a variety of mathematical tools such as stochastic processes, PDEs, numerical analysis, and graph theory. She also actively supports groups like AWM that support students in underrepresented groups, mentors both undergraduate and graduate students, and advises undergraduate researchers.

Peg solitaire in multiple colors on graphs (Tara Davis, Hawaii Pacific University and Roberto Soto, Cal State Fullerton)
2022-03-29 12:30 - 13:20 | On Zoom
Peg solitaire is a popular one person board game that has been played in many countries on various board shapes. Recently, peg solitaire has been studied extensively in two colors on mathematical graphs. We will present our rules for multiple color peg solitaire on graphs. We will present some student and faculty results classifying the solvability of the game on several graceful graphs, as well as discuss open questions.

Kauffman Bracket Skein Modules and their Structure (Rhea Palak Bakshi, ETH Zurich)
2022-03-29 15:00 - 16:00 | Zoom
Skein modules were introduced by Jozef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is notoriously hard, especially over the ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years for KBSMs. We show that some of these conjectures, and even theorems, are not true. In this talk I will briefly discuss a counterexample to Marche’s generalisation of Witten’s conjecture. I will show that a theorem stated by Przytycki in 1999 about the KBSM of the connected sum of two handlebodies does not hold. I will also give the exact structure of the KBSM of the connected sum of two solid tori.

Voronoi Tessellations: Optimal Quantization and Modeling Collective Behavior (Prof. Rustum Choksi)
2022-03-30 16:15 - 17:30 | Zoom
Title: Voronoi Tessellations: Optimal Quantization and Modeling Collective Behavior
Speaker: Prof. Rustum Choksi, Department of Mathematics and Statistics, McGill University
Abstract: Given a set of N distinct points (generators) in a domain (a bounded subset of Euclidean space or a compact Riemannian manifold), a Voronoi tessellation is a partition of the domain into N regions (Voronoi cells) with the following property: all points in the interior of the ith Voronoi cell are closer to the ith generating point than to any other generator. Voronoi tessellations give rise to a wealth of analytic, geometric, and computational questions. They are also very useful in mathematical and computational modeling.
This talk will consist of three parts:

We begin by introducing the basic definitions and geometry of Voronoi tessellations, centroidal Voronoi tessellations (CVTs), and the notion of optimal quantization.
We will then address simple, yet rich, questions on optimal quantization on the 2D and 3D torus, and on the 2-sphere. We will address the geometric nature of the global minimizer (the optimal CVT), presenting a few conjectures and a short discussion on rigorous asymptotic results and their proofs.
We will then shift gears to address the use Voronoi tessellations in modeling collective behaviors.

Collective behavior in biological systems, in particular the contrast and connection between individual and collective behavior, has fascinated researchers for decades. A well-studied paradigm entails the tendency of groups of individual agents to form flocks, swarms, herds, schools, etc. We will first review some well-known and widely used models for collective behavior. We will then present a new dynamical model for generic crowds in which individual agents are aware of their local Voronoi environment — i.e., neighboring agents and domain boundary features –and may seek static target locations. Our model incorporates features common to many other active matter models like collision avoidance, alignment among agents, and homing toward targets. However, it is novel in key respects: the model combines topological and metrical features in a natural manner based upon the local environment of the agent’s Voronoi diagram. With only two parameters, it captures a wide range of collective behaviors. The results of many simulations will be shown.

Rustum Choksi received the PhD degree in mathematics from Brown University, in 1994. He held post-doctoral positions with the Center for Nonlinear Analysis, Carnegie Mellon University and the Courant Institute, New York University. From 1997 to 2010, he was a faculty member with the Department of Mathematics, Simon Fraser University. In 2010, he joined McGill University where he is currently a full professor with the Department of Mathematics and Statistics. His main research interests include interaction of the calculus of variations and partial differential equations with pattern formation.

Applied Math Seminar -- Kathryn G. Link (UC Davis)
2022-04-04 16:15 - 17:30 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: Viscoelastic Effects of Spontaneous Oscillations of Elastic Filaments in the Follower-Force Problem.
Abstract: It is well know that microorganisms, such as bacteria and eukaryotes, often move in intricate environments experiencing mechano-chemical dynamics. These environments consist of rheologically complex substances such as mucus and other biofilms that are more complicated than water. Spermatozoa (sperm), for example, swim in viscoelastic mucus via deformations of their flagella, which are slender threadlike structures that are powered by internal molecular motors. The motor activity generates flagellar bending, resulting in an undulatory beat. The effects of a fading-memory fluid on emergent properties of these spontaneous oscillations are not entirely known. Here we combine analysis with numerical simulations of finite-length, small-amplitude pinned filaments subject to a compressive follower force to elucidate the Hopf bifurcation that occurs with increasing forcing on the filament. Additionally, we determine characteristics of the flapping motion, specifically frequency and amplitude changes and how those changes depend on follower force strength as well as fluid elasticity.

Covering by polynomial planks (Alexey Glazyrin, University of Texas Rio Grande Valley)
2022-04-05 12:30 - 13:20 | On Zoom
In 1932, Tarski conjectured that a convex body of width 1 can be covered by planks, regions between two parallel hyperplanes, only if the total width of planks is at least 1. In 1951, Bang proved the conjecture of Tarski. In this work we study the polynomial version of Tarski’s plank problem.
We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the Bang plank covering theorem.
Using the polynomial approach, we also prove the strengthening of the Fejes Tóth zone conjecture on covering a sphere by spherical segments, closed parts of the sphere between two parallel hyperplanes. In particular, we show that the sum of angular widths of spherical segments covering the whole sphere is at least π.
This is a joint work with Roman Karasev and Alexandr Polyanskii.

Applied Math Seminar -- Applied Attractions at Claremont Colleges
2022-04-11 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
During this student-centered Applied Math Seminar, there will be discussion and presentation about upcoming courses in applied mathematics to help students make their enrollment choices for Fall 2022 and beyond.

Geometrization of Markov numbers (Oleg Karpenkov, University of Liverpool)
2022-04-12 12:30 - 13:20 | TBA
In this talk we link discrete Markov spectrum to geometry of continued fractions. As a result of that we get a natural generalization of classical Markov tree which leads to an efficient computation of Markov minima for all elements in generalized Markov trees.

Cusps in Convex Projective Geometry (Martin Bobb, IHES)
2022-04-12 15:00 - 16:00 | Zoom
Convex real projective structures generalize hyperbolic structures in a rich way. We will discuss a class of manifolds introduced by Cooper Long and Tillmann, which include finite-volume cusped hyperbolic manifolds and other manifolds with well-controlled ends. These manifolds have nice deformation theoretic properties, and we will conclude with an existence theorem for novel structures on some hyperbolic manifolds.

Geometry of continued fractions (Prof. Oleg Karpenkov)
2022-04-13 16:15 - 17:30 | Shanahan B460 (HMC) and Zoom – Hybrid
Title: Geometry of continued fractions

Speaker: Oleg Karpenkov, Department of Mathematical Sciences, University of Liverpool

Abstract: In this talk we introduce a geometrical model of continued fractions and discuss its appearance in rather different research areas:
— values of quadratic forms (Perron Identity for Markov spectrum)
— the 2nd Kepler law on planetary motion
— Global relation on singularities of toric varieties

Oleg Karpenkov is a mathematician at the University of Liverpool (UK), working in the general area of discrete geometry. Specifically, his interests include geometry of numbers, discrete and semi-discrete differential geometry and self-stressed configurations of graphs. He completed his Ph.D. at Moscow State University under the supervision of Vladimir Arnold in 2005. He held several postdoctoral positions in Paris (Fellowship of the Mairie de Paris), Leiden, and Graz (Lise Meitner Fellowship) before arriving in Liverpool in 2012. In 2013 he published a book “Geometry of Continued Fractions” (its extended second edition will be available soon). His Erdos number is 3.

A conjugacy criterion for two pairs of 2 x 2 matrices over a commutative ring (Bogdan Petrenko, Eastern Illinois University)
2022-04-19 12:30 - 13:20 | On Zoom
I will explain how to apply presentations of algebras (together with some classical results from non-commutative algebra) to obtain some 5 polynomial invariants telling us when two pairs of 2×2 matrices over a commutative ring are conjugate, assuming that each of these pairs generate the matrix algebra. This talk is based on my joint paper with Marcin Mazur (Binghamton University): Separable algebras over infinite fields are 2-generated and finitely presented, Arch. Math. 93 (2009), 521-529.

Linear independence, counting, and Hilbert's syzygy theorem (Prof. Youngsu Kim)
2022-04-20 16:15 - 17:30 | Shanahan B460 (HMC) and Zoom – Hybrid
Title: Linear independence, counting, and Hilbert’s syzygy theorem
Speaker: Youngsu Kim, Department of Mathematics, Cal State San Bernardino
Abstract: Linear independence is an essential concept in mathematics and one of the most fundamental notions in linear algebra.

Linear algebra studies the solutions of linear equations. Algebraic geometry studies the solutions of polynomial equations (of arbitrary degree). In this talk, we explore how linear independence can help study algebraic geometry and Hilbert’s syzygy theorem.

Youngsu Kim earned his Ph.D. from Purdue University. He had visiting positions at UC Riverside and the University of Arkansas. Currently, he works at Cal State San Bernardino, and his primary research interest is in commutative algebra.

Applied Math Seminar -- Alona Kryshchenko (CSUCI)
2022-04-25 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: Data science and applications in dynamic topic modeling
Abstract:
The shockwaves of the big data boom have thrown into sharp relief the critical need for domain-driven, large-scale data analytic techniques across the fields of, among others, finance, political science, economics, psychology, and medicine. It is not simply the size of data sets that contributes to the extreme challenges of data analysis in these fields, but the inherent complexity of this data. Often this data is multi-modal, with modes representing measurements along different dimensions (e.g., spatial, and temporal dimensions of video data, or word and document dimensions of text corpora data). This data is often naturally formatted as a tensor, a higher-order generalization of a matrix. In this talk, we will explore nonnegative tensor decompositions and their applications in dynamic topic modeling.

Bounds for nonzero Littlewood-Richardson coefficients (Müge Taskin, Boğaziçi University, Turkey)
2022-04-26 12:30 - 13:20 | On Zoom
As $lambda$ runs through all integer partitions, the set of Schur functions ${s_{lambda}}_lambda$ forms a basis in the ring of symmetric functions. Hence the rule $$s_{lambda}s_{mu}=sum c_{lambda,mu}^{gamma} s_{gamma}$$ makes sense and the coefficients $c_{lambda,mu}^{gamma}$ are called textit{Littlewood-Richardson (LR) coefficients}. The calculations of Littlewood-Richardson coefficients has been an important problem from the first time they were introduced, due to their important role in representation theory of symmetric groups and enumerative geometry.
In this talk we will explain some of the main features of these coefficients and provide a summary of the characterizations given by Littlewood and Richardson (1934), Berenstein- Zelevinsky ()1988) and Knutson-Tao (1999). Then we will explain our approach to a seemingly easier problem, that is, the determination of triples $(lambda,mu,gamma)$ of partitions for which $c_{lambda,mu}^{gamma}$ is non zero. Our method describes some upper and lower bounds for triples $(lambda,mu,gamma)$ with nonzero $c_{lambda,mu}^{gamma}$, by using Young diagram combinatorics and especially, the indispensable Dominance order. This is joint work with R. Bedii Gümüş and supported by Tübitak/1001/115F156.

Contact topology and geometry in high dimensions (Prof. Bahar Acu)
2022-04-27 16:15 - 17:30 | Shanahan B460 (HMC) and Zoom – Hybrid
Title: Contact topology and geometry in high dimensions
Speaker: Bahar Acu, Department of Mathematics, Pitzer College
Abstract: A very useful strategy in studying topological manifolds is to factor them into “smaller” pieces. An open book decomposition of an n-manifold (the open book) is a special map (fibration) that helps us study our manifold in terms of its (n-1)-dimensional submanifolds (i.e. fibers=the pages) and (n-2)-dimensional boundary of these submanifolds (the binding). Open books provide a natural framework for studying topological properties of certain geometric structures on smooth manifolds such as “contact structures”. Thanks to open books, contact manifolds, odd dimensional manifolds carrying these geometric structures, can be studied from an entirely topological viewpoint. For example, every contact 3-manifold can be presented as an open book whose pages are surfaces and binding is a knot/link. In this talk, we will talk about higher-dimensional contact manifolds and provide a setting where we study these manifolds in terms of 3D open books. We present various results along with examples concerning geometric and topological aspects of these manifolds.

Dr. Bahar Acu (pronounced: Ah-Joo) is an Assistant Professor of Mathematics at Pitzer College since Spring 2022. Prior to joining Claremont Colleges, Dr. Acu held positions at UCLA, Northwestern, ETH Zürich, and IAS Princeton following a Ph.D. degree from the University of Southern California in 2017. Dr. Acu’s primary research interests are in the field of geometric topology, more precisely contact and symplectic topology in high dimensions and their relations with low-dimensional topology. While doing so, Dr. Acu actively thinks about ways in which the math community at large can improve and promote the presence and visibility of more first-gen, womxn, queer, and many other historically underrepresented individuals in math in various mathematical events and projects. Dr. Acu continues to hope that more of the math colleagues join these efforts in their day-to-day navigation in math in any beneficial way they can.

Applied Math Seminar -- Almut Burchard (U. Toronto)
2022-05-02 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: What is the best shape? Geometric
problems arising in aggregation models
Abstract: How do pair interactions shape the large-scale
behaviour of a cloud of particles (animals,
social agents …) ? In the most basic
models, the shape of the cloud is determined
by minimizing an attractive-repulsive interaction
energy under suitable geometric constraints.
When can we expect aggregation to occur? what
is the shape of the resulting flock? I will
describe recent work on optimal shapes in capacitor
problems that occur as limiting cases.

Beran’s tests of uniformity for discrete data (Michael Orrison, HMC)
2022-05-03 12:30 - 13:20 | On Zoom
Suppose you are given a data set that can be viewed as a nonnegative integer-valued function defined on a finite set. A natural question to ask is whether the data can be viewed as a sample from the uniform distribution on the set, in which case you might want to apply some sort of test of uniformity to the data. In this talk, I will share some work Anna Bargagliotti (Loyola Marymount University) and I have been doing to better understand a particular class of tests of uniformity first described in a 1968 paper written by R.J. Beran. Our approach uses tools from harmonic analysis on finite groups, and in this talk I will introduce those tools and then show how they can easily be used when working with discrete circular data.
