The math grad seminar serves as a space for graduate students to gather, give talks about math they’re doing and thinking about, and discuss ideas with one another. It’s meant to offer an opportunity for students to communicate their mathematical ideas and foster collaboration within the department.

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, we compiled a list of suggested topics for possible talks below. If you’re interested in one of the topics and need some guidance about resources or preparing the talk, please contact one of the organizers. You’re more than welcome to select a topic that is not on the list, please just reach out and let us know. 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!

# Schedule

## Winter 2023

### April 14: Reid Ciolfi

TitleWhy 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.

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.

# Suggested topics for MSc students

• Expander graphs and their applications
• Cayley graphs
• Graph invariants (e.g., Tutte polynomial, chromatic polynomial)
• Jacobian groups of graphs and the graph-theoretic Riemann-Roch theorem
• Caratheodory’s theorem in convex geometry
• The Birkhoff-von Neumann theorem in convex geometry
• Permutohedra and generalized permutohedra
• Galois groups of quartic polynomials
• Matroids and examples
• Lattice-based cryptography
• Game theory
• Dynamical systems
• Probability
• Models
• Math-bio (e.g., epidemiology, population dynamics...)
• Machine learning
• Neuroscience models
• Numerical simulations
• Free probability (central limit theorem)
• Weyl integration formula (in random matrix theory)
• Finite spectral triples (classification of)
• Fuzzy spectral triples (classification of)
• Fuzzy sphere
• Gelfand-Naimark theorem
• Spontaneous symmetry breaking (Higgs mechanism)
• Stone-Čech compactification theorem
• Urysohn’s lemma
• Classification of connected Riemann surfaces
• Frobenius theorem (differential topology)
• Flow box theorem
• Holomorphic differential equations and existence of their solutions
• Tychonoff's theorem