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Grad Student Seminar

Graduate Student Seminar

The grad student seminar is a space for graduate students from the entire math department to gather every Friday afternoon and listen to one of our own give a 50 minute talk about the math they're doing or thinking about. It's an opportunity for the speakers to hone their presentation skills in front of a friendly, students-only audience, and aims to foster collaboration within the department through the sharing and discussing of mathematical ideas.

The talks will highlight some mathematical research: either your own original work, or an introduction to an area of research you are interested in (or a combination of the two!). For MSc students: if you're interested in giving a talk but you're not sure what you'd like to talk about, please reach out to any of the organizers. We'll pair you with a PhD student in your area of interest who can help you prepare a presentation. All talks will be 50 minutes in length with 10 minutes at the end for questions and discussion. We encourage everyone to attend and participate, so please consider giving a talk this term. During the seminar, we will serve pizza and refreshments. Afterwards, we invite you to join us for an informal social at the Grad Club. We hope to see you all there!

Practical information

Time: Fridays at 3:30 PM
Location: MC 108
Organizers: Aditya Rao, Harshith Alagandala and Thomas Thorbjørnsen.

Schedule

Winter 2025

March 21: Shubhankar

Title: Geometry of polar convex sets

Abstract: Polar convex sets can be seen as images of convex sets under some M\"obius transform. The notion of polar convex sets in the complex plane has been used to analyze the behavior of critical points of polynomials. In this talk, we go over the not too long history of polar convex sets in the plane. Following which, we extend the notion to finite dimensional Euclidean spaces. The goal of the talk is to build a theory of polar convexity and to show that the introduction of a pole creates a richer geometry compared to classical convex sets. Polar convexity enjoys a beautiful duality that does not exist in classical convexity. We will also formulate polar analogues of several classical results in convex optimization. Finally, we give a full description of the convex hull of finitely many points with respect to finitely many poles.

March 14: Mieke Fink

Title: Solving problems in matroid theory

Abstract: Many matroid invariants take values in polynomial rings. The study of the coefficients and roots of matroid invariants is an important topic in algebraic combinatorics. Examples are the characteristic polynomial, the Tutte polynomial or the Kazhdan-Lusztig polynomial of a matroid. Connections between matroid theory to the intersection theory of toric varieties has played an important role in the proof of several long-standing conjectures in recent years. In this talk I will explain techniques that have been successfully used to investigate properties of such polynomials.

March 7: Nathan Kershaw

Title: Efficient computations of discrete cubical homology

Abstract: We will present the fastest known algorithm for computing discrete cubical homology, a valuable graph invariant with a wide range of applications, including matroid theory, hyperplane arrangements, and topological data analysis. This invariant is capable of detecting certain types of "holes" within a graph, providing insight into its structure. We will begin by defining discrete cubical homology and outlining the standard approach to its computation. We will then present an algorithm designed to improve efficiency by using techniques such as faster generation of singular cubes, reducing chain complex dimensions through quotients over automorphisms, and preprocessing graphs using results from discrete homotopy theory. These advancements aim to make the invariant more accessible computationally for applications. We are now able to compute examples that were previously considered out of reach by experts. This talk is based on the paper: Kapulkin, Kershaw, Efficient computations of discrete cubical homology, arXiv:2410.09939.

February 28: Maximilian Petrowitsch

Title: Elementary Higher Toposes

Abstract: Elementary toposes are categories that share many properties of the category of sets. Every elementary topos has an internal language which is a version of typed intuitionistic higher-order logic obtained from the lattices of subobjects. It thus makes sense to speak of an elementary topos as a local set theory. The notion of elementary infinity-topos generalises this concept to infinity-categories. It is conjectured that the internal language of an elementary infinity-topos is a version of Homotopy Type Theory. Thus, it makes sense to speak of elementary infinity toposes as univalent dependent type theories, where instead of the lattice of subobjects we have a universal infinity-groupoid of all (small) objects. In the talk, I will introduce and motivate the notion of elementary infinity-topos, focussing on the concept of object classifiers and I will sketch the progress that has been made so far towards proving the conjecture. I will not assume any prior knowledge about toposes, logic or type theory.

February 7: Siyuan Yu

Title: The symplectic embeddings of closed balls into the complex projective plane $C P^{2}$

Abstract: In this presentation, we investigate the embeddings of five disjoint closed balls into the complex projective plane $C P^{2}$ . In 1985, Gromov found a new symplectic invariant called Gromov's invariant by studying the embedding of closed balls into a cylinder. This ground-breaking result opens a new era of the research of symplectic geometry. In this presentation, we explore the relationship between the homotopy type of the space of symplectic embeddings and the configuration space of $C P^{2}$ , focusing on the case of five balls.

January 31: Tao Gong

Title: Real toric surfaces modulo reflections

Abstract: A polygon is associated to a real toric surface. If the polygon admits some symmetry, then so does the variety. We will see that the quotient of the variety is homotopy equivalent to a wedge sum of circles.

January 24: Achraf Ben Said (Universidad Complutense de Madrid)

Title: The Norm of Hardy-type Oscillation Operators in the Continuous Settings

Abstract: In this presentation we intend to illustrate in a brief way the study of optimal constants in some inequalities in $L^{p}$ norm of operators involving the Hardy operator defined as $H f (x) = (1 / x) \int_{0}^{x} f (t) d t$ . In 1925, Hardy proves that $H$ is bounded from $L^{p}$ to $L^{p}$ , if 1 < p ≤ ∞, with norm exactly equal to p′. This result has given rise to a new field of study of great interest in Mathematical Analysis and Operator Theory. The aim of this talk is to go into this line of research by exposing the results that have been obtained recently, the open problems in this field, such as the problem formulated in 1996 by G. Bennett and, finally, to present some of our contributions giving solutions to several questions within this subject.

January 17: Meagan Brenda Marie James

Title: Mapping Class Groups, Homeomorphisms, Curve Graphs, and Ivanov’s Metaconjecture

Abstract: The curve graph of a surface encodes the intersection patterns of the isotopy classes of essential simple closed curves in the surface. In 1997, Ivanov showed that the curve graph can be used as a combinatorial model to study the extended mapping class group of a surface. Since then, with the objective of finding results analogous to that of Ivanov, several alternative versions of the curve graph have been defined and investigated in relation to certain algebraic invariants of a surface. For instance, Long–Margalit–Pham–Verberne–Yao successfully showed that the fine curve graph can be used as a combinatorial model for the group of homeomorphisms of a surface. In this expository talk, we will begin by introducing mapping class groups and several curve graphs associated to a surface. From here, we will investigate the relationship between curve graphs and homeomorphisms of a surface, stating several powerful theorems and even some preliminary results regarding separating curves. Finally, we will briefly discuss adjacent areas of research and the search for evidence supporting Ivanov’s infamous metaconjecture.

Fall 2024

December 6: Thomas Thorbjørnsen

Title: Constructive Homological Algebra

Abstract: Homological algebra is a powerful tool to differentiate between structures and study obstructions. For instance, homology of spaces is an invariant that is classically simple to compute. For these reasons, it is desirable to develop these tools constructively, that means without using the law of excluded middle and the axiom of choice. We will discuss the relevance of these axioms, what breaks, and different approaches to fix or side-step the problems all together.

November 29: Anif Shikder

Title: An Operator Approach to All-Pairs Shortest Path For Random Graphs

Abstract: We introduce an analytical model for Erdős–Rényi graphs called GRE and derive expressions approximating its adjacency matrix. We then find expressions approximating its l-path matrix. After formulating key graph measures such as the clustering coefficient, the distance matrix, and the average shortest distance in terms of the l-path matrix, we deduce analytical expressions approximating them. Our expressions do not contain any free parameters and show fast convergence valid non-asymptomatically over the entire probability space. The proposed approach is generalizable to other random graphs and can be used to approximate algebraically well-defined graph measures to investigate finite-sized real-world networks.

November 22: Harshith Alagandala

Title: Polynomial convexity and approximation by polynomials

Abstract: A compact subset K of C^n is said to be polynomially convex if every point in the complement of K can be separated from K by a polynomial. Oka-Weil theorem tells us that holomorphic functions can be approximated by polynomials on polynomially convex sets. This is a generalization of Runge approximation theorem to higher dimension. We will go through some approximation results in complex analysis, look at examples of polynomially convex sets, and analyze the local polynomial convexity of real n-dimensional manifolds embedded in C^n.

November 15: Theofanis Chatzidiamantis Christoforidis

Title: Formalising higher category theory

Abstract: Homotopy type theory (HoTT) is a formal system that allows us to do homotopy-invariant mathematics in a natural way, where types can be seen as representing topological spaces. In HoTT, every type has types of paths (identity types) between its terms, paths between paths, etc., giving types a higher groupoidal structure. Simplicial type theory is an extension of HoTT that adds extra categorical structure to types using constructions similar to those in simplicial sets, making it possible to state a definition of an infinity-category that does not depend on a particular set-theoretic model. Additionally, using type-theoretic foundations makes it easier to formalise parts of the theory of infinity-categories in proof assistants, and in particular the Rzk proof assistant developed specifically for this type theory. We present an introduction to HoTT and simplicial type theory, and discuss possible directions one can consider in synthetic higher category theory.

November 8: Nathan Pagliaroli

Title: The Free Central Limit Theorem

Abstract: Free Probability is used to study non-commutative random variables and has its roots in Voiculescu’s work on operator algebras in the 1980s. The notion of free independence serves as the non-commutative analogue of independence in classical probability theory. One of the most famous results in probability theory is the central limit theorem and in this talk we will prove the analogous result in free probability theory, known as the free central limit theorem. The proof is combinatorial in nature and all relevant notions of Free Probability will be introduced beforehand.

November 1: Diego Tenoch Morales Lopez

Title: Mathematics applied to evolution: bridging the gaps between pure mathematics, biology and numerical analysis

Abstract: From the perspective of a pure mathematician, the application of mathematics to concrete problems (like evolution of the species) is not of primary importance. However, that does not imply that applied mathematics deals with problems that are trivial mathematically speaking, or that there is nothing to learn from applications. In this talk, I will showcase different ways in which mathematics are being used in the study of Evolution, ranging from systems of ODEs and PDEs to "simple" stochastic models, with some sprinkles of numerical analysis. My goal is to spark the interest of the pure math audience not only in applying math to biology, but also in further analyzing the mathematics that arise from biological problems for mathematics' sake .

October 25: Zack Dooley

Title: An Introduction to Proof Assistants

Abstract: Proof assistants are a variety of software tools which can be used to assist in proof writing and verifying mathematical statements. Recently, proof assistants have been gaining popularity not just for their ability to verify the correctness of complicated proofs, but also as a tool of collaboration for mathematicians. In this talk I will introduce the basics of what proof assistants are and how to use them, in particular, focusing on the proof assistant Coq. I will show how to write basic definitions and proofs in Coq and show how libraries can help us with collaboration and proof writing.

October 11: Alexander Zwart

Title: Rationality of Algebraic Tori

Abstract: Given a variety X a natural question to ask is whether it is birational to projective space. In general, this is quite a very hard question to answer. We restrict ourselves to a subclass of objects known as algebraic tori. It turns out that a slightly weaker notion related to rationality can be "cleanly" understood in terms of the character lattice for a given torus. I will give some background to state this result and then discuss the work that has been done on the rationality question for algebraic tori.

Winter 2024

April 5: Curtis Wilson

Title: The Golod-Shafarevich Theorem

Abstract: We introduce the class field tower infinite issue and state its solution, the Golod-Shafarevich theorem. We prove the theorem, and provide a refinement for the graded case. We discuss some examples and finish with an application involving finitely generated infinite torsion groups.

March 29: Yunhai Xiang

Title: An easy tour of Galois cohomology

Abstract: Galois cohomology is a topic that should interest a wide range of audiences: number theorists, algebraic geometers, homotopy theorists, etc. It is a wonderful example of an application of ideas from algebraic topology to study algebra and number theory. In this talk, we will discuss the basics of Galois cohomology, and we demonstrate its power by using it to prove the Mordell-Weil theorem for elliptic curves. If time permits, we might also discuss a little bit about its generalization: étale cohomology.

March 22: Esther Yartey

Title: Structural connectivity across datasets and species reveals community structure in cortex with specific connection features

Abstract: Advancements in neuroimaging technologies, particularly diffusion MRI, now allow reconstructing the long-range fiber connection patterns in the human brain. We study the network whose connection weights are determined by the number of fibers between individual brain regions. We study networks from the Human Connectome Project (HCP) and networks extracted from individual imaging subjects through a data processing pipeline developed in our group. By applying an algorithm to detect highly connected “communities” in these networks, we find a discrete set of communities appear robustly in the human brain. A specific community in the occipital lobe systematically displays high eigenvector centrality (EVC), a measure of the influence of nodes within a network. We explore the variations in these network structures among individuals and in retested subjects to isolate the sources of inter-individual variability. This result consistently appears across nearly all subjects and in a test-retest dataset. Similar community structure also appears in connectomes from macaque and marmoset brains, but the existence of an occipital lobe community with high EVC is specific to human connectomes. Taken together, these results reveal novel organization in the structural connectivity of the brain, derived from a fully data-driven approach, where clear community organization appears. This community organization relates to known functional divisions, such as visual and auditory sensory pathways, but also reveals community structure within higher-order areas, whose functional relevance can be studied in future work.

March 15: Thomas Thorbjornsen

Title: The Synthetic Fundamental Group of the Circle

Abstract: Homotopy type theory is a foundation for mathematics that is well-suited to do homotopy theory. Unlike ZFC, we no longer have access to the law of excluded middle or the axiom of choice. Many mathematicians find it daunting and off-putting to work without these axioms, but great opportunity is created amidst the challenge. In this talk we will investigate how we can do homotopy theory in homotopy type theory. We will construct the fundamental group of the circle by using the tools provided by Martin-Löf type theory, Voevodsky’s univalence axiom, and higher inductive types. In particular, we will look at the underlying language of type theory and its identity types, how fiber bundles are expressed as type families, and the essential role played by the univalence axiom.

March 8: Sayantan Roy Chowdhury

Title: Essential Dimension of Symplectic Sheaves over a curve

Abstract: Essential dimension of an algebraic object captures the minimum number of variables needed to parameterize it. Concrete as it sounds, this invariant is often hard to compute even in the most simplest of cases. In this talk, we will tackle this problem for the moduli space of symplectic sheaves on a curve. We will start out by defining (and drawing pictures) for these objects. Subsequently, we outline a roadmap for estimating the essential dimension of this moduli space enroute deformation theory.

March 1: Marwa Tuffaha

Title: Mutator dynamics cannot be explained by mutation rates alone

Abstract: Mutators, cells with elevated mutation rates, are common in both natural microbial populations and in human cancers. Recent experiments have shown that mutators can invade a population, but the invasion dynamics and probability couldn’t be explained by mutation rates alone. Here we show, analytically and in simulation, that mutation bias (which types of mutations are likely to occur) can play an important role in the emergence of mutators. A mutator that reduces or reverses the historically prevailing mutational bias is shown to have an increased chance of invasion, while chances are reduced when the bias is reinforced. These findings are important when trying to understand natural populations or competition experiments with mutators.

February 16: Tao Gong

Title: What is the space of conjugacy classes?

Abstract: When G is a topological group, G acts on itself by conjugation. This leads to the natural question: what is the quotient space of conjugacy classes? In this talk, we will work out the answer to this question in the special case when G is a connected compact simple Lie group – the space is contractible!

February 9: Michelle Hatzel

Title: Continuation Methods for Numerical Problem Solving

Abstract: Informally, if two functions can be “continuously deformed” from one to the other, this is called a homotopy. Homotopy emerged from theory more than a century ago and was introduced as a numerical method for solving non-linear problems in the 1960s. The basic components of these early continuation algorithms built on earlier path-tracking methods, which exist in today’s “black box” solvers. We will look at the building blocks of continuation algorithms, how they work (or don’t), and how key insights from the 1970s and 1980s contributed to some powerful polynomial-solving software packages.

January 19: Blake Whiting

Title: Acyclic Models

Abstract: Acyclic models, as it's commonly seen today, is a proof technique used to show when two chain complexes are chain equivalent or have isomorphic homology. It originated as a theorem by Eilenberg and MacLane (1953), where it was immediately used to show the Eilenberg-Zilber theorem (1953). This theorem, proven directly via acyclic models, gives us a Künneth theorem and defines the cup product, which turns cohomology into a graded ring. This talk will be an exposition on (one version of) the acyclic models theorem, as given by Michael Barr in 2002. I will give the necessary definitions to understand Barr's modern formulation of acyclic models, and then prove it. Time permitting, I will also discuss how the Eilenberg-Zilber theorem follows directly from it and potential avenues to generalizing acyclic models.

Fall 2023

December 1: Prakash Singh

Title: The Hofer diameter problem for rational symplectic manifolds

Abstract: In general, Lie groups do not admit bi-invariant metrics, and infinite dimensional Lie groups should not admit such metrics either. But surprisingly, Ham admits one such metric (in fact, unique in a sense), called the 'Hofer metric', discovered by Hofer in the 90s. People have been studying the large-scale geometry properties of this metric for a long time, but such studies were restricted to either 2-dimensions, monotone symplectic manifolds, or to aspherical manifolds. In particular, it is widely conjectured that the hofer diameter is infinite for every closed symplectic manifold, and this conjecture has been settled for the above-mentioned manifolds. I will talk about the diameter problem associated with this metric for some rational ruled manifolds like CP2, S2 x S2, and their blow-ups, using methods from quantum homology and spectral invariants on them. I will prove the conjecture for CP2 and S2 x S2, and I will prove it under a mild assumption (but unproven) for S2 x S2 blown up once.

November 24: Elaine Murphy

Title: The Mathematical Structure of Point Mutations

Abstract: Mutation is the engine of evolution. By considering only single point mutations (SNPs) on DNA sequences, we see a natural group theoretic model of mutations acting on the set of nucleotides. In this talk, we will investigate the implications of this structure for synonymous mutations (mutations that do not change the encoded amino acids) and how this affects the notion of distance between two genetic sequences.

November 17: Manimugdha Saikia

Title: Analytic properties of quantum states associated with complex manifolds

Abstract: In Quantum Information Theory, there is a rich collection of analytic tools to study tensor product of Hilbert spaces. Geometric quantization attaches Hilbert spaces to symplectic manifolds. Study of these information theoretic measures on these specific Hilbert spaces leads to interesting insights. In our study, we look for invariants, or to what extent the geometry of the space influences the Quantum Information aspects of the Hilbert space and vice versa. In this talk, we shall present our asymptotic result for the average entropy over all the pure states on the (quantum) Hilbert space H_{1,N} \otimes H_{2,N} where H_{1,N}$ and H_{2,N} are the spaces of holomorphic sections of the N-th tensor powers of hermitian ample line bundles on compact complex manifolds. I shall also talk about how certain states associated with product submanifolds become separable.

November 10: Alejandro Santacruz Hidalgo

Title: Hardy's inequality: a brief review, some extensions and applications.

Abstract: In 1915 G.H Hardy needed an estimate for arithmetic means to find a proof of Hilbert's inequality for sequences, a continuous version of that inequality followed in 1925. Since then, extensions have been made in many directions; more general domains, weighted norm inequalities, general measures, among others. In this talk we will review the classic statement of Hardy's original inequality. We will explore some of the extensions of this important inequality and review some of its implications, such as, Sobolev inequalities and boundedness of the Fourier transform in weighted Lorentz spaces.

October 27: Nathan Pagliaroli

Title: The Gaussian Unitary Ensemble and the Enumeration of Maps

Abstract: In this talk I will introduce the notation of a matrix ensemble with a focus on the Gaussian Unitary Ensemble (GUE) as an example. I will introduce its basic properties in connection with map enumeration. In particular, I will outline a proof of the Genus Expansion Formula for moments of the GUE. Time permitting we will discuss the famous Harer-Zagier formula .

October 20: Alan Flatres

Title: Hamilton's rule: basic concept and extension

Abstract: Altruism can seem at first glance to be counterintuitive: why would I spend my energy and resources for someone else's profit? To elucidate this mystery, in 1964, Hamilton wrote a simple rule that describes the evolutionary trajectory of a trait that is costly for the individual having it but that brings benefits to another individual. This formula sums the cost of bearing this trait and the benefits received by the other individual, weighted by the genetic linkage between the two individuals. Thanks to Hamilton's work, we can better understand the evolution of costly behavior such as cooperative breeding. However, the simplicity of Hamilton's rule makes it hard to apply in nature. Indeed, the diversity of species and their life histories requires extensions of Hamilton's rule to study the evolution of altruism in different contexts. In this talk, I will present Hamilton's rule from its basic form to more complex extensions that help us to understand the evolution of costly behavior in different species.

October 13: Nathan Kershaw

Title: Closed symmetric monoidal structures on the category of graphs

Abstract: Discrete homotopy theory is a relatively new area of mathematics, concerned with applying methods from homotopy theory in topology to the category of graphs. In order to do this, a notion of a product between graphs is required. Classically two products have been considered, the box product and the categorical product. These products lead to two different homotopy theories, namely A-theory and X-theory, respectively. This leads us to the question of why these two products are considered, and if one can define other products to study discrete homotopy theory with instead. In this talk, we will answer this question by fully characterizing all closed symmetric monoidal products on the category of graphs. This talk will be based on joint work with C. Kapulkin (arxiv:2310.00493).

October 6: Oussama Hamza

Title: On extensions of number fields with given quadratic algebras and cohomology

Abstract: At the beginning of the century, Labute and Minac introduced a criterion, on presentations of pro-p groups, ensuring that the cohomological dimension is two. Groups with presentations satisfying this condition are called mild. For that purpose, they mixed gradations and filtrations techniques, originally introduced by Golod and Shafarevich to construct infinite pro-p groups, with Anick's results on graded algebras.

Recently, Hamza introduced a new criterion on the presentation of finitely presented pro-p groups which allows us to compute their cohomology groups and infer quotients of mild groups of cohomological dimension strictly larger than two. Hamza still used previous techniques and enrich them using Graph and Right Angled Artin Groups/Algebras Theory, Groebner basis, etc.

Hamza applied previous criterion to obtain new Galois groups over p-rational fields with prescribed ramification and splitting and cohomological dimension larger than two.

In this talk, we discuss previous results with their motivations and techniques if time permits.

September 29: Tenoch Morales

Title: Adaptation reshapes the distribution of fitness effects

Abstract: Evolution is driven by mutations that change the reproductive success of mutant individuals with respect to their unmutated counterparts. These effects on the "fitness" of the organisms can be measured empirically and using mathematical models, which has been increasingly successful in the past decade. In these studies, the effects of the novo mutations in the fitness is measured by a probability distribution, known as the Distribution of Fitness Effects (the DFE). This distribution is predicted to differ for better- or worse-adapted organisms, thus the DFE must change dynamically during the process of adaptation, a fact highlighted in recent studies.

In this work, we analyze the change in the DFE during an adaptive process across a fitness landscape. First, we derive analytical approximations for the DFE and the underlying distributions for the allele's fitness contributions. Then, we compare these results with independent simulations that relax several simplifying assumptions made in the analysis. This computational work confirms that our analytical expressions provide a good approximation to the dynamically changing DFE during adaption.

We observe that as de novo mutations accumulate, the DFE is shaped in two meaningful ways: by increasing the fraction of deleterious mutations and by decreasing the variance of the distribution.

Winter 2023

April 14: Reid Ciolfi

Title: Why is the derivative a non-injective operator?

Abstract: The simplest answer to this question is very well known: it is because the derivative maps all constants to zero. In this talk, we shall not be satisfied with the simplest answer. We will ask a series of additional questions about why and how the derivative maps constants to zero. In order to delve deeper, we shall introduce and briefly motivate the concept of fractional calculus. Armed with this additional structure, and with some of the rich properties of the Gamma function, we will answer our questions to the fullest possible extent. However, we will have to take the Gamma function apart in the process; an integral representation, a product representation, and an exponential representation will each be used in turn. Along the way, we will also discover an eloquent proof of why the exponential function is a fixed point of the derivative, and why Stirling's approximation for the Gamma function is effective.

March 31: Mojgan Ezadian

Title: Quantitative estimates of selection bias in bacterial mutation accumulation experiments

Abstract: Mutation accumulation (MA) experiments play a crucial role in understanding the processes underlying evolution. In microbial populations, MA experiments typically involve a period of population growth between severe bottlenecks, such that a single individual can form a visible colony. In this study, we quantify the impact of positive and negative selection on MA experiments. Our results demonstrate that selective effects can significantly bias the distribution of fitness effects (DFE) and mutation rates estimated from MA experiments in microbes. Furthermore, we propose a straightforward correction for this bias that can be applied to both beneficial and deleterious mutations. The outcomes of this research emphasize the importance of positive selection in microbial MA experiments to obtain a more accurate understanding of fundamental evolutionary parameters.

March 24: Marios Velivasakis

Title: Representations of the symmetric group and Z-forms

Abstract: In representation theory, we try to connect groups with invertible matrices (usually over the complex numbers). Our most powerful tool is character theory, i.e., looking at the trace of the corresponding matrices, because it is a full invariant (two representations are equivalent if and only if they have the same character). An interesting question is: “What happens if we consider matrices over smaller fields or even the integers? Do we still have the same invariants?”. In this talk, we will discuss representation theory in general, and how we can produce all representations for the symmetric group combinatorially. In addition, we will talk about what happens if we consider our matrices over the integers and how the standard tools fail to describe these representations.

March 17: Jacqueline Doan

Title: Neural Network Powered Recommender System: Restricted Boltzmann Machine

Abstract: How did Spotify know our music taste so well? Recommendation algorithms are widely implemented on entertainment platforms like Netflix and Spotify to provide users with a more personalized experience online. Collaborative filtering is the idea that a target user is more likely to like an product if others with the same interests highly rated that product. In order to analyze and process large and sparse data sets often associated with users’ data, Restricted Boltzmann Machine (RBM), a stochastic neural network model, was implemented as a model for users’ ratings of products by Salakhutdinov et al. in 2007. We will discuss the implementation of RBM and the ethics of recommender systems in this talk.

March 10: Kumar Shukla

Title: Poincaré duality and enumerative geometry

Abstract: How many lines intersect 4 given lines in 3-dimensional space? Poincaré duality gives us a geometric interpretation of the cup product. We can use this to compute the cohomology ring of the Grassmanians. This interpretation is also central to the subject of enumerative geometry. Using Poincaré duality as the starting point, we will give a brief introduction to enumerative geometry and answer some of the classical problems in this subject like the one posed above or the problem of counting the number of lines in a cubic surface.

March 3: Chirantan Mukherjee

Title: Model structure on simplicial categories

Abstract: In the first part of the talk, we review the axioms of model categories and define the Kan-Quillen model structure on simplicial sets. We then move on to define a model structure on the category enriched over simplicial sets. This, further forms a model of (∞, 1)-categories.

February 17: Tao Gong

Title: Cohomology rings of classifying spaces

Abstract: The classifying space of a Lie group is used to classify the principle bundle, hence computation on cohomology of classifying spaces is quite important. In this lecture, we will focus on cohomology over integers. For a Kac-Moody Lie group of finite type, the homotopy colimit of classifying spaces of parabolic groups is the base space of a sphere bundle, where the total space is exactly the classifying space of the original group. This provides a systematic way of computation. We will see examples of the exceptional Lie group and the projective unitary group.

February 10: Alan Flatres

Title: Evolution of cooperative breeding with group augmentation effects and ecological feedbacks

Abstract: Cooperative breeding occurs when an individual helps to raise the offspring of others. It is typically considered to be costly for helpers who lose or postpone the opportunity of personal fitness gains. This behaviour is widespread, occurring in a variety of different taxa, and ecological settings. Moreover, phylogenetic data suggest that environmental conditions play a role in promoting and hindering cooperative breeding. The complex interplay between environmental constraints and population interaction makes it challenging to model cooperative breeding in a satisfying way.

In order to better understand the influence of the environment on cooperative breeding while having reasonable computations, we built a coarse-grained model to study the group augmentation effect. That way, this population model allows us to have more complex relations between the environment and the population and thus to understand their role better.

Specifically, by computing the inclusive fitness of this kin selection model, we were able to show that environment-individuals relations, for instance, the probability of establishment, have an influence over the emergence and development of altruistic behaviour in the population.

February 3: Curtis Wilson

Title: Classifying diagram algebras

Abstract: We introduce the representation theory of quivers with a focus on their indecomposable representations. We provide a nice criterion for an algebra to be indecomposable, and finish by proving the remarkable fact that quiver representations of finite type are exactly those with underlying Dynkin type A, D, and E.

January 27: Shubhankar

Title: Analytic Theory of Polynomials and Polar Convexity

Abstract: Traditionally, polynomials have been treated as objects of algebra. However, over the years people realized their excellent analytic properties and big names like Chebyshev, Weierstrass, Fourier spent a chunk of their careers studying them in this context. Indeed, the study of their extremal properties and critical points is of interest in more than one way. The Gauss-Lucas theorem is one such celebrated result. Polar convexity is a relatively new notion that exploits properties of Möbius transforms and convex analysis to give a new outlook on such analytic problems. The tools, even in their infancy, seem powerful and give promising results. The goal of this talk is to introduce the notion of polar convexity and time-permitting, prove a few of these results.

Fall 2022

December 2: Yanni Zeng

Title: Bifurcation analysis on a predator-prey model with Allee Effect

Abstract: The dynamics of a population is greatly affected by its interaction with other populations. There exist many kinds of interaction among populations, such as competition, predation, parasitism and mutualism. The predator-prey interaction is one of the most fundamental interactions and one of the most fascinating interactions to investigate. In 1931, the concept 'Allee effect' was put forward referring to a decrease in population growth rate at low population density since the growth of the species will also be affected by factors: difficulties in mating, unable to defence as a group, social felicitation of reproduction, etc. We apply bifurcation theory to consider a predator-prey model including the Allee effect and show that the species having a strong Allee effect may affect their predation and hence extinction risk. In this talk, I will introduce the related model and present methods analyzing the complex dynamical behaviors of the models with the Allee effect.

November 25: Gunjeet Singh

Title: Classification of compact, connected topological surfaces

Abstract: Topology as an independent subject in mathematics was started by Poincaré at the end of nineteenth century but the notion of surfaces is quite ancient than topology itself. Surfaces were studied extensively by many mathematicians such as Gauss, Riemann, Mobius, Jordan, etc in various contexts like in analysis, differential geometry, etc. Naturally enough, people wanted to classify surfaces. One of the earliest attempts were by Mobius and Jordan in 1860s even after being devoid of the definition of a 'topological surface'. It was only in 1907 when Dehn and Heegaard gave a rigorous enough proof of the statement using 'polygonal presentations' of the surfaces. In this talk, I will present the main ideas of the proof and some interesting and important examples of it.

The classification theorem says that every compact, connected 2-manifold is homeomorphic either to a sphere, or a connected sum of one or more toriz or a connected sum of projective planes. The proof uses 'polygonal presentations' which are a special class of cell complexes, in which spaces are represented as quotients of polygons (with even number of sides) with their edges identified.

November 18: Mahan Moazzeni

Title: Introduction to Khovanov homology

Abstract: A knot is a smooth embedding of circle in R3. We are essentially interested in looking at knots, up to an ambient isotopy and see whether knot K1 can be ”distorted” into the other knot, K2. One of the most important problems in knot theory, is the classification problem, which roughly is providing a list of all of the existing knots, up to ambient isotopy. In order to classify them, we need a collection of powerful invariants that recognise each knot from the others. Khovanov Homology (KH) is one of the few topological invariants which at least detects a collection of knots from the other ones. KH is combinatorial in nature and it uses (1 + 1)-topological quantum field theory (TQFT) to move from the category of 1-manifolds to the category of vector spaces in its construction. The construction of KH requires lots of works and kind of ”boring” computations but the result, is one of the most interesting and powerful tools in knot theory that we have, as an instance, KH can detect unknot from any other knots, using KH we can find a combinatorial proof for the celebrated Milnor’s Conjecture without using Seiberg-Witten theory for torus knots. Our main goal for this talk is to introduce the KH rigorously and prove some of its basic properties. If time allows, we will proceed and show some of the interesting results about KH for alternating knots. Our ultimate goal would be to go through the J. Rasmussen’s proof of Milnor’s Conjecture on torus knots using KH, but it requires a lot works. We will most certainly cover the whole idea of his proof.

October 28: Tedi Ramaj

Title: Investigating the Spread of an Invasive Weed, Tradescantia fluminensis, via Partial Differential Equation Modelling and Dynamical Systems Techniques

Abstract: A species is typically defined to be invasive to an ecosystem if it is a non-native species which threatens the ecosystem and its native species. Invasive species may include animals, plants, fungi, and other living organisms. Invasive species have historically been implicated as the one of the greatest drivers of biodiversity loss. We consider the invasion of an ecosystem by invasive plant species, Tradescantia fluminensis (T. fluminensis), an invasive weed which has been implicated in native forest depletion in countries such as Australia, New Zealand, and parts of the United States. We explore the dynamics of T. fluminensis spreading via partial differential equation (PDE) modelling and the application of nonlinear dynamical systems and phase portrait techniques. We propose a competition model, modelling the impact of competition between the invasive weed and a pre-existing native plant species, based on previous models. We are able to use some results from basic existing PDE theory in order to obtain some insights on the biological system. We also explore the existence of travelling wave solutions (TWS) of the PDE systems which represent transitions of the state of the ecosystem. In this talk, we explore both the mathematical theory necessary to obtain the results and the policy decisions which the results may help guide.

These results have been published in the Bulletin of Mathematical Biology and may be found here in greater detail:
Ramaj, T. On the Mathematical Modelling of Competitive Invasive Weed Dynamics. Bull Math Biol 83, 13 (2021).

October 14: Alejandro Santacruz Hidalgo

Title: Monotonicity in ordered measure spaces

Abstract: Monotone functions defined on the real numbers are very simple and straightforward objects to understand, yet a rich theory of monotone (or decreasing) functions has been developed and has proven to provide new insight on seemingly unrelated problems like characterization of weighted Hardy's inequalities or boundedness for the Fourier transform between Lorentz spaces.

In this talk, we will give an introduction to the development of a theory of ordered measures spaces and generalize the theory of monotone functions to this setting. In a general measure space, we assume no order among its elements, instead we rely on a totally ordered collection of measurable sets to carry all the monotonicity properties, with this collection we define what a monotone function is. Next, we explore two different partial orders on the set of decreasing functions and show that there is an optimal upper bound in these partial orders. A collection of function spaces called 'Down spaces' defined by decreasing functions will be introduced and their relationship with the partial orders explained.

October 21: Nathan Pagliaroli

Title: Random matrices and Tutte’s recursion

Abstract: In the 1950’s, W.T. Tutte found a recursive formula for counting a combinatorial object known as a planar map: a 2-cell embedding of a connected planar graph into the oriented sphere, considered up to orientation preserving homeomorphisms of the sphere. In the 1970’s, maps and Tutte’s Recursion were first used as powerful tools in the context of random matrix theory. Both the theory of maps and random matrix theory have benefited from this connection, with methods of proof lending themselves between these areas.

In this talk I will introduce the concept of maps, their generating functions, and their connection to random matrices, with the goal of deriving Tutte’s recursive formula.

October 7: Alexandra Busch

Title: Neural sequences in primate prefrontal cortex encode working memory in naturalistic environments

Abstract: Working memory is the ability to briefly remember and manipulate information after it becomes unavailable to the senses. A specific region of the brain - the lateral prefrontal cortex (LPFC) - has been widely implicated in working memory performance in primates. Despite decades of study, how neurons in LPFC coordinate their activity to hold sensory information in working memory remains controversial. In this talk, I will give a brief overview of the traditional model for working memory, and discuss how it is impacted by recent advances in neural recording techniques and more complex experimental paradigms. I will then focus on results from a recent project in which we analyzed the activity of hundreds of neurons recorded from LPFC of non-human primates during a naturalistic working memory task involving navigation in virtual reality. We found that selective sequential activation across neurons encoded specific items held in working memory. Administration of ketamine distorted neural sequences, selectively decreasing working memory performance. Our results indicate that neurons in the lateral prefrontal cortex causally encode working memory in naturalistic conditions via complex and temporally precise activation patterns.

September 30: Tenoch Morales

Title: Using Fitness Landscapes to understand the shifts in mutation biases

Abstract: Mutations are the engine that drives evolution and adaptation forward in that it generates the variation on which natural selection acts. Although mutations are considered to occur randomly in the genome, we see that in many organisms some types of mutations occur more often than expected under uniformity; these deviations are called mutation biases.

Even though there is no clear description of the biological mechanisms governing the formation of mutation biases, theoretical and experimental work has shown that a shift in mutation biases during the evolutionary process could grant an adaptive advantage to an organism by increasing the sampling of previously poorly explored types of mutations.

In this talk, we will explore the most popular Fitness Landscape models, which map the genotypic space of an organism to its adaptive fitness. With these models, we can simulate the evolutionary process of a population as a walk through the genotypic space towards genotypes with higher fitness, which will help us understand the adaptive effect of shifts in mutation biases at different points on the evolutionary path.

September 23: Jarl Taxerås Flaten

Title: The moduli space of multiplications on a space

Abstract: Since the mid-50s, various topologists have been interested in counting homotopy classes of multiplications (i.e. H-space structures) on certain spaces. For example, there's a unique multiplication on the circle (complex multiplication), and James showed that there are 12 multiplications on the 3-sphere and 120 on the 7-sphere. No other spheres admit a multiplication, barring the 0-sphere.

We present a formula for the moduli space of multiplications on a pointed object of an ∞-topos. By specializing to the ∞-topos of spaces and counting the path components of these moduli spaces, we recover the numbers just mentioned. These results have been shown in Homotopy Type Theory, which I will give a brief introduction to, and have been formalized in the Coq proof assistant, which I will demonstrate with some live-coding.

September 16: Oussama Hamza

Title: Filtrations, arithmetic, and explicit examples in an isotypical context

Abstract: Pro-p groups arise naturally in number theory as quotients of absolute Galois groups over number fields. These groups are quite mysterious. During the 60's, Koch gave a presentation of some of these quotients. Furthermore, around the same period, Jennings, Golod, Shafarevich and Lazard introduced two integer sequences (a_n) and (c_n), closely related to a special filtration of a finitely generated pro-p group G, called the Zassenhaus filtration. These sequences give the cardinality of G, and characterize its topology. For instance, we have the well-known Gocha's alternative (Golod and Shafarevich): There exists an integer n such that a_n=0 (or c_n has a polynomial growth) if and only if G is a Lie group over p-adic fields. In 2016, Minac, Rogelstad and Tan inferred an explicit relation between a_n and c_n. Recently (2022), considering geometrical ideas of Filip and Stix, Hamza got more precise relations in an isotypical context: when the automorphism group of G admits a subgroup of order a prime q dividing p-1. In this talk, we will mostly review some results of Golod, Shafarevic, Koch, Lazard, Minac, Tan and Hamza. We also give several explicit examples in an arithmetical context.

Graduate Program