# Video Content by Date

**Jul, 26**: Condensation phenomena in random trees - Lecture 3

Speaker:

*Igor Kortchemski*

Consider a population that undergoes asexual and homogeneous reproduction over time, originating from a single individual and eventually ceasing to exist after producing a total of n individuals. What is the order of magnitude of the maximum number of children of an individual in this population when n tends to infinity? This question is equivalent to studying the largest degree of a large Bienaymé-Galton-Watson random tree. We identify a regime where a condensation phenomenon occurs, in which the second greatest degree is negligible compared to the greatest degree. The use of the "one-big jump principle" of certain random walks is a key tool for studying this phenomenon. Finally, we discuss applications of these results to other combinatorial models.

**Jul, 26**: Random walks and branching random walks: old and new perspectives - Lecture 16

Speaker:

*Perla Sousi*

This course will focus on two well-studied models of modern probability: simple symmetric and branching random walks in ℤd. The focus will be on the study of their trace in the regime that this is a small subset of the ambient space.

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

A branching random walk (or tree indexed random walk) in ℤd is a non-Markovian process whose time index is a random tree. The random tree is either a critical Galton Watson tree or a critical Galton Watson tree conditioned to survive. Each edge of the tree is assigned an independent simple random walk in ℤd increment and the location of every vertex is given by summing all the increments along the geodesic from the root to that vertex. When $d\geq 5$, the branching random walk is transient and we will mainly focus on this regime. We will introduce the notion of branching capacity and show how it appears naturally as a suitably rescaled limit of hitting probabilities of sets. We will then use it to study covering problems analogously to the random walk case.

**Jul, 26**: Subshifts with very low word complexity

Speaker:

*Ronnie Pavlov*

he word complexity function p(n) of a subshift X measures the number of n-letter words appearing in sequences in X, and X is said to have linear complexity if p(n)/n is bounded. It’s been known since work of Ferenczi that linear word complexity highly constrains the dynamical behaviour of a subshift.

**Jul, 26**: A computational phase transition for pressure approximation

Speaker:

*Raimundo Briceno*

In this talk, we will review some new techniques and limitations for achieving efficient approximation algorithms for entropy and pressure in the context of Gibbs measures defined over countable groups. Our starting point will be a deterministic formula for the Kolmogorov-Sinai entropy of measure-preserving actions of order-able amenable groups. Next, we will review techniques based on random orderings, mixing properties of Markov random fields, and percolation theory to generalize previous work. As a by-product of these results, we will obtain conditions for the uniqueness of the equilibrium state and the locality of pressure, among other implications that are not strictly algorithmic.

**Jul, 26**: Random matrix theory of high-dimensional optimization - Lecture 16

Speaker:

*Elliot Paquette*

Optimization theory seeks to show the performance of algorithms to find the (or a) minimizer x∈ℝd of an objective function. The dimension of the parameter space d has long been known to be a source of difficulty in designing good algorithms and in analyzing the objective function landscape. With the rise of machine learning in recent years, this has been proven that this is a manageable problem, but why? One explanation is that this high dimensionality is simultaneously mollified by three essential types of randomness: the data are random, the optimization algorithms are stochastic gradient methods, and the model parameters are randomly initialized (and much of this randomness remains). The resulting loss surfaces defy low-dimensional intuitions, especially in nonconvex settings.

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

how random matrices can be used to describe high-dimensional inference

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 25**: Condensation phenomena in random trees - Lecture 2

Speaker:

*Igor Kortchemski*

Consider a population that undergoes asexual and homogeneous reproduction over time, originating from a single individual and eventually ceasing to exist after producing a total of n individuals. What is the order of magnitude of the maximum number of children of an individual in this population when n tends to infinity? This question is equivalent to studying the largest degree of a large Bienaymé-Galton-Watson random tree. We identify a regime where a condensation phenomenon occurs, in which the second greatest degree is negligible compared to the greatest degree. The use of the "one-big jump principle" of certain random walks is a key tool for studying this phenomenon. Finally, we discuss applications of these results to other combinatorial models.

**Jul, 25**: Subset Problems in Combinatorial Number Theory Coding Problems in Symbolic Dynamics

Speaker:

*Sophie Morin*

TBA

**Jul, 25**: Random matrix theory of high-dimensional optimization - Lecture 15

Speaker:

*Elliot Paquette*

Optimization theory seeks to show the performance of algorithms to find the (or a) minimizer x∈ℝd of an objective function. The dimension of the parameter space d has long been known to be a source of difficulty in designing good algorithms and in analyzing the objective function landscape. With the rise of machine learning in recent years, this has been proven that this is a manageable problem, but why? One explanation is that this high dimensionality is simultaneously mollified by three essential types of randomness: the data are random, the optimization algorithms are stochastic gradient methods, and the model parameters are randomly initialized (and much of this randomness remains). The resulting loss surfaces defy low-dimensional intuitions, especially in nonconvex settings.

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

how random matrices can be used to describe high-dimensional inference

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 25**: Divisibility of integer Laurent polynomials and dynamical systems

Speaker:

*Douglas Lind*

Let f, p, and q be Laurent polynomials in one or several variables with integer coefficients, and suppose that f divides p + q. In joint work with Klaus Schmidt, we establish sufficient conditions to guarantee that f individually divides p and q. These conditions involve a bound on coefficients, a separation between the supports of p and q, and, surprisingly, a requirement on f called atorality about how the complex variety of f intersects the multiplicative unit torus. The proof uses an algebraic dynamical system related to f and the fundamental dynamical notion of homoclinic point. Without the atorality assumption this method fails, the validity of our result in this case remains an open problem. We have recently learned that if the general case could be proved (even a very special version of it), there would be important consequences in determining whether certain upper triangular groups have trivial Poisson boundary, but already the proven case does have implications for this.

**Jul, 25**: Shifts of finite type defined by one forbidden block and a Theorem of Lind

Speaker:

*Chengyu Wu*

We focus on shift of finite type (SFTs) obtained by forbidding one word from an ambient SFT. Given a word in a one-dimensional full shift, Guibas and Odlyzko, and Lind showed that its auto-correlation, zeta function and entropy (of the corresponding SFT) all determine the same information. We first prove that when two words both have trivial auto-correlation, the corresponding SFTs not only have the same zeta function, but indeed are conjugate to each other, and this conjugacy is given by a chain of swap conjugacies. We extend this result to the case when the ambient SFT is the golden mean shift, with an additional assumption on the extender set of the words. Then, we introduce SFTs obtained by forbidding one finite pattern from a higher dimensional ambient SFT. We prove that, when two patterns have the same auto-correlation, then there is a bijection between the language of their corresponding SFTs. The proof is a simple combination of a replacement idea and the inclusion-exclusion principle. This is joint work with Nishant Chandgotia, Brian Marcus and Jacob Richey.

**Jul, 25**: Random walks and branching random walks: old and new perspectives - Lecture 15

Speaker:

*Perla Sousi*

This course will focus on two well-studied models of modern probability: simple symmetric and branching random walks in ℤd. The focus will be on the study of their trace in the regime that this is a small subset of the ambient space.

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

A branching random walk (or tree indexed random walk) in ℤd is a non-Markovian process whose time index is a random tree. The random tree is either a critical Galton Watson tree or a critical Galton Watson tree conditioned to survive. Each edge of the tree is assigned an independent simple random walk in ℤd increment and the location of every vertex is given by summing all the increments along the geodesic from the root to that vertex. When $d\geq 5$, the branching random walk is transient and we will mainly focus on this regime. We will introduce the notion of branching capacity and show how it appears naturally as a suitably rescaled limit of hitting probabilities of sets. We will then use it to study covering problems analogously to the random walk case.

**Jul, 24**: Shift equivalence implies flow equivalence for reducible shifts of finite type

Speaker:

*Mike Boyle*

Explain the implication stated in the title.

**Jul, 24**: Variations on Krieger's Embedding Theorem and a Theorem of Boyle

Speaker:

*Brian Marcus*

Krieger's celebrated embedding theorem gives necessary and sufficient conditions, in terms of periodic points, for proper embedding of a subshift into a mixing shift of finite type (SFT). The result does not generalize to mixing sofic shifts. Boyle introduced the notion of a receptive periodic point, which is the main obstruction. He also proved that a condition involving receptive periodic points is sufficient for proper embedding of a subshift into a mixing sofic shift. We introduce a stronger notion of embedding, called factorizable embedding, which requires the embedding to factor through a mixing SFT. We show that Boyle's sufficient condition for proper embedding into a mixing sofic shift is necessary and sufficient for a factorizable embedding into a mixing sofic shift. We also give a new characterization of receptive periodic points, and we give a characterization of mixing for irreducible sofic shifts in terms of receptive periodic points. This is joint work with Tom Meyerovitch, Klaus Thomsen and Chengyu Wu.

**Jul, 23**: Condensation phenomena in random trees - Lecture 1

Speaker:

*Igor Kortchemski*

Consider a population that undergoes asexual and homogeneous reproduction over time, originating from a single individual and eventually ceasing to exist after producing a total of n individuals. What is the order of magnitude of the maximum number of children of an individual in this population when n tends to infinity? This question is equivalent to studying the largest degree of a large Bienaymé-Galton-Watson random tree. We identify a regime where a condensation phenomenon occurs, in which the second greatest degree is negligible compared to the greatest degree. The use of the "one-big jump principle" of certain random walks is a key tool for studying this phenomenon. Finally, we discuss applications of these results to other combinatorial models.

**Jul, 23**: Trading linearity for ellipticity: a nonsmooth approach to Einstein's theory of gravity and Lorentzian splitting theorems

Speaker:

*Rober McCann*

While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while and Ricci curvature is reformulated using entropic convexity along geodesics of probability measures.

This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. In work in progress, we aim to establish a low regularity splitting theorem by sacrificing linearity of the d’Alembertian to recover ellipticity. We exploit a negative homogeneity $p-$ d’Alembert operator for this purpose. The same technique yields a simplified proof of Eschenberg (1988) Galloway (1989) and Newman’s (1990) confirmation of Yau’s (1982) conjecture, bringing all three Lorentzian splitting results into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry.

**Jul, 23**: Random walks and branching random walks: old and new perspectives - Lecture 14

Speaker:

*Perla Sousi*

This course will focus on two well-studied models of modern probability: simple symmetric and branching random walks in ℤd. The focus will be on the study of their trace in the regime that this is a small subset of the ambient space.

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

A branching random walk (or tree indexed random walk) in ℤd is a non-Markovian process whose time index is a random tree. The random tree is either a critical Galton Watson tree or a critical Galton Watson tree conditioned to survive. Each edge of the tree is assigned an independent simple random walk in ℤd increment and the location of every vertex is given by summing all the increments along the geodesic from the root to that vertex. When $d\geq 5$, the branching random walk is transient and we will mainly focus on this regime. We will introduce the notion of branching capacity and show how it appears naturally as a suitably rescaled limit of hitting probabilities of sets. We will then use it to study covering problems analogously to the random walk case.

**Jul, 23**: Random matrix theory of high-dimensional optimization - Lecture 14

Speaker:

*Elliot Paquette*

Optimization theory seeks to show the performance of algorithms to find the (or a) minimizer x∈ℝd of an objective function. The dimension of the parameter space d has long been known to be a source of difficulty in designing good algorithms and in analyzing the objective function landscape. With the rise of machine learning in recent years, this has been proven that this is a manageable problem, but why? One explanation is that this high dimensionality is simultaneously mollified by three essential types of randomness: the data are random, the optimization algorithms are stochastic gradient methods, and the model parameters are randomly initialized (and much of this randomness remains). The resulting loss surfaces defy low-dimensional intuitions, especially in nonconvex settings.

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

how random matrices can be used to describe high-dimensional inference

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 22**: Random matrix theory of high-dimensional optimization - Lecture 13

Speaker:

*Elliot Paquette*

ptimization theory seeks to show the performance of algorithms to find the (or a) minimizer x∈ℝd of an objective function. The dimension of the parameter space d has long been known to be a source of difficulty in designing good algorithms and in analyzing the objective function landscape. With the rise of machine learning in recent years, this has been proven that this is a manageable problem, but why? One explanation is that this high dimensionality is simultaneously mollified by three essential types of randomness: the data are random, the optimization algorithms are stochastic gradient methods, and the model parameters are randomly initialized (and much of this randomness remains). The resulting loss surfaces defy low-dimensional intuitions, especially in nonconvex settings.

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 22**: Random walks and branching random walks: old and new perspectives - Lecture 13

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 19**: Random walks and branching random walks: old and new perspectives - Lecture 12

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 19**: Random matrix theory of high-dimensional optimization - Lecture 12

Speaker:

*Elliot Paquette*

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 18**: Permutations in random geometry - Lecture 3

Speaker:

*Jacopo Borga*

I will introduce a new universal family of random permutons, called the skew Brownian permutons, describing the scaling limit of various natural models of random constrained permutations. After that, the main goal will be to discuss some connections between random permutations and random geometry. In particular, we will focus on the problem of the longest increasing subsequence in permutations sampled from the skew Brownian permuton and its connection with the study of certain directed metrics on planar maps, which conjecturally should converge in the limit to a notion of "directed Liouville quantum gravity metric.

**Jul, 18**: Random matrix theory of high-dimensional optimization - Lecture 11

Speaker:

*Elliot Paquette*

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 18**: Random walks and branching random walks: old and new perspectives - Lecture 11

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 16**: Permutations in random geometry - Lecture 2

Speaker:

*Jacopo Borga*

andom permutons, called the skew Brownian permutons, describing the scaling limit of various natural models of random constrained permutations. After that, the main goal will be to discuss some connections between random permutations and random geometry. In particular, we will focus on the problem of the longest increasing subsequence in permutations sampled from the skew Brownian permuton and its connection with the study of certain directed metrics on planar maps, which conjecturally should converge in the limit to a notion of "directed Liouville quantum gravity metric.

**Jul, 16**: Random walks and branching random walks: old and new perspective - Lecture 10

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 16**: Random matrix theory of high-dimensional optimization - Lecture 10

Speaker:

*Elliot Paquette*

Optimization theory seeks to show the performance of algorithms to find the (or a) minimizer x∈ℝd of an objective function. The dimension of the parameter space d has long been known to be a source of difficulty in designing good algorithms and in analyzing the objective function landscape. With the rise of machine learning in recent years, this has been proven that this is a manageable problem, but why? One explanation is that this high dimensionality is simultaneously mollified by three essential types of randomness: the data are random, the optimization algorithms are stochastic gradient methods, and the model parameters are randomly initialized (and much of this randomness remains). The resulting loss surfaces defy low-dimensional intuitions, especially in nonconvex settings.

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

* how random matrices can be used to describe high-dimensional inference

* nonconvex landscape properties

* high-dimensional limits of stochastic gradient methods.

**Jul, 15**: Permutations in random geometry - Lecture 1

Speaker:

*Jacopo Borga*

I will introduce a new universal family of random permutons, called the skew Brownian permutons, describing the scaling limit of various natural models of random constrained permutations. After that, the main goal will be to discuss some connections between random permutations and random geometry. In particular, we will focus on the problem of the longest increasing subsequence in permutations sampled from the skew Brownian permuton and its connection with the study of certain directed metrics on planar maps, which conjecturally should converge in the limit to a notion of "directed Liouville quantum gravity metric.

**Jul, 15**: Random matrix theory of high-dimensional optimization - Lecture 9

Speaker:

*Elliot Paquette*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 15**: Random walks and branching random walks: old and new perspectives - Lecture 9

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 12**: Random walks and branching random walks: old and new perspectives - Lecture 8

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 12**: Random matrix theory of high-dimensional optimization - Lecture 8

Speaker:

*Elliot Paquette*

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 11**: Probabilistic techniques in number theory - Lecture 3

Speaker:

*Emma Bailey*

In this mini-course I will cover some classical theorems from probabilistic number theory, and then discuss some recent developments in the distribution of values of L-functions (focussing on the simplest L-function: the Riemann zeta function). I will emphasise surprising connections to random matrix theory. Many of these ideas can be visualised numerically, giving the course a computational flavour as well.

**Jul, 11**: Random walks and branching random walks: old and new perspectives - Lecture 7

Speaker:

*Elliot Paquette*

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 11**: Random walks and branching random walks: old and new perspectives - Lecture 7

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 9**: Probabilistic techniques in number theory - Lecture 2

Speaker:

*Emma Bailey*

In this mini-course I will cover some classical theorems from probabilistic number theory, and then discuss some recent developments in the distribution of values of L-functions (focussing on the simplest L-function: the Riemann zeta function). I will emphasise surprising connections to random matrix theory. Many of these ideas can be visualised numerically, giving the course a computational flavour as well.

**Jul, 9**: Random walks and branching random walks: old and new perspectives - Lecture 6

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 9**: Random matrix theory of high-dimensional optimization - Lecture 6

Speaker:

*Elliot Paquette*

*Please note: Due to a problem with the zoom configuration, there is no video associated with this lecture, only the audio was recoreded.*

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 8**: Probabilistic techniques in number theory - Lecture 1

Speaker:

*Emma Bailey*

In this mini-course I will cover some classical theorems from probabilistic number theory, and then discuss some recent developments in the distribution of values of L-functions (focussing on the simplest L-function: the Riemann zeta function). I will emphasise surprising connections to random matrix theory. Many of these ideas can be visualised numerically, giving the course a computational flavour as well.

**Jul, 8**: Random matrix theory of high-dimensional optimization - Lecture 5

Speaker:

*Elliot Paquette*

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 8**: Random walks and branching random walks: old and new perspectives - Lecture 5

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 5**: Random walks and branching random walks: old and new perspectives - Lecture 4

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 5**: Random matrix theory of high-dimensional optimization - Lecture 4

Speaker:

*Elliot Paquette*

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 4**: Random matrix theory of high-dimensional optimization - Lecture 3

Speaker:

*Elliot Paquette*

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 4**: Random walks and branching random walks: old and new perspectives - Lecture 3

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 3**: Random walks and branching random walks: old and new perspectives - Lecture 2

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jul, 3**: Random matrix theory of high-dimensional optimization - Lecture 2

Speaker:

*Elliot Paquette*

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 2**: Random matrix theory of high-dimensional optimization - Lecture 1

Speaker:

*Elliot Paquette*

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

**Jul, 2**: Random walks and branching random walks: old and new perspectives - Lecture 1

Speaker:

*Perla Sousi*

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

**Jun, 21**: Almost periodicity and large oscillations of prime counting functions

Speaker:

*Jan-Christoph Schlage-Puchta*

If we assume the relevant Riemann hypotheses, after a suitable rescaling many functions counting certain primes become almost periodic. There are different notion of almost periodicity in use; here we consider the notion induced by the norm $||f|| = \sup_{x∈\mathbb{R}} \int_x^{x+1} |f(t)|^2\,dt$. We show that if a function $f$ can be approximated by linear combinations of periodic functions with respect to this norm, then the level sets $\left\{x: f(x) \geq t\right\}$ are almost periodic for all real $t$ with at most countably many exceptions. We also compare this notion to other notions of almost periodicity in use.

*Please note, the wrong video feed was captured for this lecture so the writing on the blackboard is not legible.*

**Jun, 21**: A race problem arising from elliptic curves

Speaker:

*Kin Ming Tsang*

Given an elliptic curve $E/\mathbb{Q}$, we can consider its trace of Frobenius, denoted as $a_p(E)$, where $p$ is a prime. We will discuss the race problem arising from these ap values and the general strategy in attacking these problems.

**Jun, 21**: Ramanujan sums and the Hardy–Littlewood prime tuple conjecture

Speaker:

*Shivani Goel*

In 1999, Gadiyar and Padma discovered a simple heuristic to derive the generalized twin prime conjecture using an orthogonality principle for Ramanujan sums originally discovered by Carmichael. We derive a limit formula for higher convolutions of Ramanujan sums, generalizing an old result of Carmichael. We then apply this in conjunction with the general theory of arithmetical functions of several variables to give a heuristic derivation of the Hardy–Littlewood formula for the number of prime $k$-tuples less than $x$.

**Jun, 21**: Number theory versus random matrix theory: the joint moments story

Speaker:

*Andrew Pearce-Crump*

It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height $T$, is $\frac{1}{2}(e^2 −5)\log T$ as $T\rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer

**Jun, 21**: Zero-free regions of the Riemann zeta-function

Speaker:

*Andrew Yang*

A zero-free region is a subset of the complex plane where the Riemann zeta-function does not vanish. Such regions have historically been used to further our understanding of prime-number distributions. In the classical approach, we first assume that a zero exists off the critical line, then arrive at an inequality involving its real and imaginary parts. One notable aspect of the classical argument is that it does not require any knowledge about the relationship between the zeroes. However, it is well known that the location of a hypothetical zero depends strongly on the behaviour of nearby zeroes—for instance, N. Levinson showed in 1969 that if zeroes of the zeta-function are well-spaced near the 1-line, then we can obtain a zero-free region stronger than any that are currently known. In this talk we will discuss some ideas on how one might incorporate information about distributions of hypothetical zeroes to improve existing zero-free regions.

**Jun, 21**: Spacing statistics of the Farey sequence

Speaker:

*Bittu*

The Farey sequence $\mathcal{F}_Q$ of order $Q$ is the ascending sequence of fractions $\frac{a}{b}$ in the unit interval $(0, 1]$ with $gcd(a, b) = 1$ and $0 < a \leq b \leq Q$. The study of the Farey fractions is of major interest because of their role in problems related to Diophantine approximation. Also, there is a connection between the distribution of Farey fractions and the Riemann hypothesis, which further motivates their study. In this talk, we will discuss the distribution of Farey fractions with some divisibility constraints on denominators by studying their pair-correlation measure. This is based on joint work with Sneha Chaubey.

**Jun, 21**: Unconditional comparative prime number theory over function fields

Speaker:

*Alexandre Bailleul*

In classical comparative prime number theory, it is customary to assume some kind of linear independence hypothesis about the zeros of the underlying L-functions. These hypotheses are completely out of reach of current methods. However, in the function field case, it is sometimes possible to prove them, or at least to show they hold generically. In this talk I will present recent results in comparative prime number theory over function fields that establish infinite families of “irreducible polynomial races” which we can study unconditionally. Some of those results are joint work with L. Devin, D. Keliher, and W. Li.

**Jun, 20**: Oscillation results for the summatory functions of fake \mu

Speaker:

*Chi Hoi (Kyle) Yip*

Recently Martin, Mossinghoff, and Trudgian investigated comparative number theoretic results for a family of arithmetic functions called “fake $\mu$’s”. In their paper, they focused on the bias and oscillation of the summatory function of a fake $\mu$ at the $\sqrt{x}$ scale, while acknowledging that a function with no bias at this scale could well see one at a smaller scale. In this spirit, I will discuss some new oscillation results for the summatory functions of general fake $\mu$’s. This is joint work with Greg Martin.

*Please note, this recording is incomplete due to a problem with the room system.*

**Jun, 20**: On the generalised Dirichlet divisor problem

Speaker:

*Chiara Bellotti*

In this talk we present new unconditional estimates on $\Delta k(x)$, the remainder term associated with the generalised divisor function, for large $k$. By combining new estimates of exponential sums and Carlson’s exponent, we show that $\Delta k(x) \ll x^{1−1.224(k−2.36)^{−2/3}}$ for $k \geq 58$ and $\Delta k(x) \ll x^{1−1.889k^{−2/3}}$ for all sufficiently large fixed $k$. This is joint work with Andrew Yang.

**Jun, 20**: Average value of $\pi(t) - li(t)$

Speaker:

*Daniel Johnston*

Central to comparative number theory is the study of the difference $\Delta(t) = \pi(t) − li(t)$, where $\pi(t)$ is the prime counting function and $li(t)$ is the logarithmic integral. Prior to a celebrated 1914 paper of Littlewood, it was believed that $\Delta < 0$ for all $t > 2$. We now know however that $\Delta(t)$ changes sign infinitely often, with the first sign change occuring before 10320. Despite this, it still appears that $\Delta(t)$ is negative “on average”, in that integrating $\Delta (t)$ from $t = 2$ onwards yields a negative value. In this talk, we will explore this idea in detail, discussing links with the Riemann hypothesis and also extending such ideas to other differences involving arithmetic functions.

**Jun, 20**: Moments in the Chebotarev density theorem

Speaker:

*Florent Jouve*

In joint work with Régis de la Bretéche and Daniel Fiorilli, we consider weighted

moments for the distribution of Frobenius substitutions in conjugacy classes of

Galois groups of normal number field extensions. The question is inspired by work

of Hooley and recent progress by de la Bretéche–Fiorilli in the case of moments for

primes in arithmetic progressions. As in their work, the results I will discuss are

conditional on the Riemann Hypothesis and confirm that the moments considered

should be Gaussian. Time permitting, I will address a different notion of moments

that can be considered in the same context and that leads to non-Gaussian families

for particular Galois group structures.

**Jun, 20**: The Riemann hypothesis via the generalized von Mangoldt function

Speaker:

*Saloni Sinha*

Based on work previously done by Gonek, Graham, and Lee, we show that the Riemann Hypothesis (RH) can be reformulated in terms of certain asymptotic estimates for twisted sums with k-fold convolution of von Mangoldt function and the generalized von Mangoldt function. For each $k \in\mathbb{N}$, we study two types of twisted sums:

1. $\sum_{n\leq x} \Lambda^k(n)n^{-iy}$, where $\Lambda^k(n) = \underbrace{\Lambda\star\cdots\Lambda}_\text{k copies}$

2. $\sum_{n\leq x} \Lambda_k(n)n^{-iy}$, where $\Lambda_k(n) :=\sum_{d|n}\mu(d)\left(\log{\frac{n}{d}}\right)^k$.

Where $\Lambda$ is the von Mangoldt function and $\mu$ is the Möbius function, and establish similar connections with RH.

**Jun, 19**: A simple proof of the Wiener–Ikehara Tauberian theorem

Speaker:

*Jagannath Sahoo*

The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series. In this talk, we present a simple and elegant proof of the Wiener–Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method allows us to derive a version of the WienerIkehara theorem with an error term. This is joint work with Prof. M. Ram Murty and Prof. Akshaa Vatwani.

**Jun, 19**: Shifting the ordinates of zeros of the Riemann zeta function

Speaker:

*William Banks*

Let $y\neq 0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_* > 0$ (depending on $y$ and $C$) such that for every $T>T_*$, both

$$

\zeta(\frac{1}{2}+i\gamma) = 0 \qquad \mbox{and} \qquad

\zeta(\frac{1}{2} + i(\gamma + y))\neq 0

$$

hold for at least one $\gamma$ in the interval $[T, T(1+\epsilon]$, where $\epsilon := T^{-C/\log\log T}$.

**Jun, 19**: Distribution of Gaussian primes and zeros of L-functions

Speaker:

*Lucile Devin*

In this talk, we are interested in the following question: among primes that can be written as a sum of two squares $p = a^2 + 4b^2$ with $a > 0$, how is the congruence class of a distributed? This will lead us to study the distribution of values of Hecke characters from the point of view of Chebyshev’s bias, as well as the distribution of zeros of the associated L-functions and in particular their vanishing at $1/2$.

**Jun, 18**: Quantitative upper bounds related to an isogeny criterion for elliptic curves

Speaker:

*Tian Wang*

Let $E_1$ and $E_2$ be two non-CM elliptic curves defined over a number field $K$. By an isogeny theorem due to Kulkarni, Patankar, and Rajan, the two curves are geometrically isogenous if and only if the density of primes for which their Frobenius field coincide is positive. In this talk, we present a quantitative upper bounds of this criterion that improves the result of Baier–Patankar and Wong. The strategy relies on effective versions of the Chebotarev Density Theorem. This is joint work with Alina Cojocaru and Auden Hinz.

**Jun, 18**: Joint Distribution of primes in multiple short intervals

Speaker:

*Sun Kai Leung*

Assuming the Riemann hypothesis (RH) and the linear independence conjecture (LI), we show that the weighted count of primes in multiple disjoint short intervals has a multivariate Gaussian logarithmic limiting distribution with weak negative correlation. As a consequence, we derive short-interval counterparts for many important works in the literature of the Shanks–Rényi prime number race, including a sharp phase transition from all races being asymptotically unbiased to the existence of biased races. Our result remains novel, even for primes in a single moving interval, especially under a quantitative formulation of the linear independence conjecture (QLI).

**Jun, 18**: Explicit estimates for the Mertens function

Speaker:

*Nicol Leong*

We prove explicit estimates of $1/\zeta(s)$ of various orders, and use an improved version of the Perron formula to get explicit estimates for the Mertens function $M(x)$ of order $O(x)$, $O(x/\log^k x)$, and $O(x\log x exp(−\sqrt{\log{x}})$. These estimates are good for small, medium, and large ranges of $x$, respectively.

**Jun, 18**: Counting “supersingularity” in arithmetic statistics

Speaker:

*Wanlin Li*

Supersingularity is a notion to describe certain elliptic curves defined over a field with positive characteristic $p > 0$. Supersingular elliptic curves possess many special properties, such as larger endomorphism rings, extremal point counts, and special p-torsion group scheme structures. This notion was then generalized to higherdimensional abelian varieties. A global function field is associated with an algebraic curve defined over a finite field; the supersingularity of the Jacobian would affect the prime distribution of this function field. In this talk, I want to discuss the effect of supersingularity on prime distribution for function fields and introduce some perspectives to study this phenomenon.

**Jun, 18**: A Mertens function analogue for the Rankin–Selberg L-function

Speaker:

*Amrinder Kaur*

Let f be a self-dual Maass form for $SL(n, Z)$. We write $L_f (s)$ for the Godement–Jacquet L-function associated to $f$ and $L_{f\times f} (s)$ for the Rankin–Selberg L-function of $f$ with itself. The inverse of $L_{f\times f} (s)$ is defined by

$$

\frac{1}{L_{f\times f}(s)} := \sum_{m=1}^\infty \frac{c(m)}{m^s}, \mathfrak{R}(s) > 1.

$$

It is well known that the classical Mertens function $M(x) := \sum_{m\leq x} \mu(m)$ is related to

$$

\frac{1}{\zeta(s)} = \sum_{m=1}^\infty \frac{\mu(m)}{m^s}, \mathfrak{R}(s) > 1.

$$

We define the analogue of the Mertens function for $L_{f\times f} (s)$ as $\widetilde{M}(x) := \sum_{m\leq x} c(m)$ and obtain an upper bound for this analogue $\widetilde{M}(x)$, similar to what is known for the Mertens function $M(x)$. In particular, we prove that $\widetilde{M}(x) \ll_f x \exp(−A\sqrt{\log{x}}$ for sufficiently large $x$ and for some positive constant $A$. This is a joint work with my Ph.D. supervisor Prof. A. Sankaranarayanan.

**Jun, 18**: Oscillations in Mertens’ product theorem for number fields

Speaker:

*Ethan Lee*

The content of this talk is based on joint work with Shehzad Hathi. First, I will give a short but sweet proof of Mertens’ product theorem for number fields, which generalises a method introduced by Hardy. Next, when the number field is the rationals, we know that the error in this result changes sign infinitely often. Therefore, a natural question to consider is whether this is always the

case for any number field? I will answer this question (and more) during the talk. Furthermore, I will present the outcome of some computations in two number fields: $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{13})$.

**Jun, 18**: The influence of the Galois group structure on the Chebyshev bias in number fields

Speaker:

*Mounir Hayani*

In 2020, Fiorilli and Jouve proved an unconditional Chebyshev bias result for a Galois extension of number fields under a group theoretic condition on its Galois group. We extend their result to a larger family of groups. This leads us to characterize abelian groups enabling extreme biases. In the case of prime power degree extensions, we give a simple criterion implying extreme biases and we also investigate the corresponding Linnik-type question.

**Jun, 18**: Remarks on Landau–Siegel zeros

Speaker:

*Debmalya Basak*

One of the central problems in comparative prime number theory involves understanding primes in

arithmetic progressions. The distribution of primes in arithmetic progressions are sensitive to real zeros near $s = 1$ of L-functions associated to primitive real Dirichlet characters. The Generalized Riemann Hypothesis implies that such L-functions have no zeros near $s = 1$. In 1935, Siegel proved the strongest known upper bound for the largest such real zero, but his result is vastly inferior to what is known unconditionally for other L-functions. We exponentially improve Siegel’s bound under a mild hypothesis that permits real zeros to lie close to $s = 1$. Our hypothesis can be verified for almost all primitive real characters. Our work extends to other families of L-functions. This is joint work with Jesse Thorner and Alexandru Zaharescu.

**Jun, 17**: The distribution of analytic ranks of elliptic curve over prime cyclic number fields

Speaker:

*Gyeongwon Oh*

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $C_l$ be the family of prime cyclic extensions of degree $l$ over $\mathbb{Q}$. Under GRH for elliptic L-functions, we give a lower bound for the probability for $K \in C_l$ such that the difference $r_K(E) − r_\mathbb{Q}(E)$ between analytic rank is less than a for $a \asymp l$. This result gives conjectural evidence that the Diophantine Stability problem suggested by Mazur and Rubin holds for most of $K \in C_l$.

**Jun, 17**: Joint distribution of central values and orders of Sha groups of quadratic twists of an elliptic curve

Speaker:

*Peng-Jie Wong*

As a refinement of Goldfeld’s conjecture, there is a conjecture of Keating–Snaith asserting that $\log L(1/2,E_d)$ for certain quadratic twists $E_d$ of an elliptic curve $E$ behaves like a normal random variable. In light of this, Radziwill and Soundararajan conjectured that the distribution of $\log(|Sha(E_d)|/\sqrt{|d|}$ is approximately Gaussian for these $E_d$, and proved that the conjectures of Keating–Snaith and theirs are both valid “from above”. More recently, under GRH, they further established a lower bound for the involving distribution towards Keating–Snaith’s conjecture. In this talk, we shall discuss the joint distribution of central values and orders of Sha groups of $E_d$ and how to adapt Radziwill–Soundararajan’s methods to study upper bound and lower bounds for such a joint distribution if time allows.

**Jun, 17**: Prime Number Error Terms

Speaker:

*Nathan Ng*

In 1980 Montgomery made a conjecture about the true order of the error term in the prime number theorem. In the early 1990s Gonek made an analogous conjecture for the sum of the Mobius function. In 2012 I further revised Gonek’s conjecture by providing a precise limiting constant. This was based on work on large deviations of sums of independent random variables. Similar ideas can be applied to any prime number error term. In this talk I will speculate about the true order of prime number error terms.

**Jun, 17**: The Shanks–Rényi prime number race problem

Speaker:

*Youness Lamzouri*

Let $\pi(x; q, a)$ be the number of primes $p\leq x$ such that $p \equiv a (\mod q)$. The classical Shanks–Rényi prime number race problem asks, given positive integers $q \geq 3$ and $2 \leq r \leq \phi(q)$ and distinct reduced residue classes $a_1, a_2, . . . , a_r$ modulo $q$, whether there are infinitely many integers $n$ such that $\pi (n; q, a1) > \pi(n; q, a2) > \cdots > \pi(n; q, ar)$. In this talk, I will describe what is known on this problem when the number of competitors $r \geq 3$, and how this compares to the Chebyshev’s bias case which corresponds to $r = 2$.

**Jun, 17**: Fake mu's: Make Abstracts Great Again!

Speaker:

*Tim Trudgian*

The partial sums of the Liouville function $\lambda(n)$ are "often" negative, and yet the partials sums of the Möbius function $\mu(n)$ are positive or negative "roughly equally". How can this, be, given that $\mu(n)$ and $\lambda(n)$ are so similar? I shall discuss some problems in this area, some joint work with Greg Martin and Mike Mossinghoff, and a possible application to zeta-zeroes.

**May, 23**: Introduction to unbalanced optimal transport and its efficient computational solutions

Speaker:

*Laetitia Chapel*

Optimal transport operates on empirical distributions which may contain acquisition artifacts, such as outliers or noise, thereby hindering a robust calculation of the OT map. Additionally, it necessitates equal mass between the two distributions, which can be overly restrictive in certain machine learning or computer vision applications where distributions may have arbitrary masses, or when only a fraction of the total mass needs to be transported. Unbalanced Optimal Transport addresses the issue of rebalancing or removing some mass from the problem by relaxing the marginal conditions. Consequently, it is often considered to be more robust, to some extent, against these artifacts compared to its standard balanced counterpart. In this presentation, I will review several divergences for relaxing the marginals, ranging from vertical divergences like the Kullback-Leibler or the L2-norm, which allow for the removal of some mass, to horizontal ones, enabling a more robust formulation by redistributing the mass between the source and target distributions. Additionally, I will discuss efficient algorithms that do not necessitate additional regularization on the OT plan.

**Apr, 18**: Generic Representations and ABV packets for p-adic Groups

Speaker:

*Sarah Dijols*

After a brief introduction on the theory of p-adic groups complex representations, I will explain why tempered and generic Langlands parameters are open. I will further derive a number of consequences, in particular for the enhanced genericity conjecture of Shahidi and its analogue in terms of ABV packets. This is a joint work with Clifton Cunningham, Andrew Fiori, and Qing Zhang.

**Apr, 11**: Hypergeometric functions through the arithmetic kaleidoscope

Speaker:

*Ling Long*

The classical theory of hypergeometric functions, developed by generations of mathematicians including Gauss, Kummer, and Riemann, has been used substantially in the ensuing years within number theory, geometry, and the intersection thereof. In more recent decades, these classical ideas have been translated from the complex setting into the finite field and p

-adic settings as well.

In this talk, we will give a friendly introduction to hypergeometric functions, especially in the context of number theory.

**Apr, 10**: Bounds on the Number of Solutions to Thue Equations

Speaker:

*Greg Knapp*

In 1909, Thue proved that when $F(x,y)$ is an irreducible, homogeneous, polynomial with integer coefficients and degree at least 3, the inequality $\left\| F(x,y) \right\| \leq h$ has finitely many integer-pair solutions for any positive $h$. Because of this result, the inequality $\left\| F(x,y) \right\| \leq h$ is known as Thue’s Inequality. Much work has been done to find sharp bounds on the size and number of integer-pair solutions to Thue’s Inequality, with Mueller and Schmidt initiating the modern approach to this problem in the 1980s. In this talk, I will describe different techniques used by Akhtari and Bengoechea; Baker; Mueller and Schmidt; Saradha and Sharma; and Thomas to make progress on this general problem. After that, I will discuss some improvements that can be made to a counting technique used in association with “the gap principle” and how those improvements lead to better bounds on the number of solutions to Thue’s Inequality.

**Apr, 4**: Density functional theory and multi-marginal optimal transport: Introduction

Speaker:

*Yair Shenfeld*

Density functional theory (DFT) is one of the workhorses of quantum chemistry and material science. In principle, the joint probability of finding a specific electron configuration in a material is governed by a Schrödinger wave equation. But numerically computing this joint probability is computationally infeasible, due to the complexity scaling exponentially in the number of electrons. DFT aims to circumvent this difficulty by focusing on the marginal probability of one electron. In the last decade, a connection was found between DFT and a multi-marginal optimal transport problem with a repulsive cost. I will give a brief introduction to this topic, including some open problems, and recent progress.

**Mar, 25**: Zeros of linear combinations of Dirichlet L-functions on the critical line

Speaker:

*Jérémy Dousselin*

Fix $N\geq 1$ and let $L_1, L_2, \ldots, L_N$ be Dirichlet L-functions with distinct, primitive and even Dirichlet characters. We assume that these functions satisfy the same functional equation. Let $F(s)∶= c_1L_1(s)+c_2L_2(s)+\ldots+c_NL_N(s)$ be a linear combination of these functions ($c_j \in\mathbb{R}^*$ are distinct). $F$ is known to have two kinds of zeros: trivial ones, and non-trivial zeros which are confined in a vertical strip. We denote the number of non-trivial zeros $\rho$ with $\mathfrak{I}(\rho)\leq T$ by $N(T)$, and we let $N_\theta(T)$ be the number of these zeros that are on the critical line. At the end of the 90's, Selberg proved that this linear combination had a positive proportion of zeros on the critical line, by showing that $\kappa F∶=\lim \inf T (N_\theta(2T)−N_\theta(T))/(N(2T)−N(T))\geq c/N^2$ for some $c>0$. Our goal is to provide an explicit value for $c$, and also to improve the lower bound above by showing that $\kappa_F \geq 2.16\times 10^{-6}/(N \log N)$, for any large enough $N$.

**Mar, 25**: Zeros of linear combinations of Dirichlet L-functions on the critical line

Speaker:

*Jérémy Dousselin*

Fix $N\geq 1$ and let $L_1, L_2, \ldots, L_N$ be Dirichlet L-functions with distinct, primitive and even Dirichlet characters. We assume that these functions satisfy the same functional equation. Let $F(s)∶= c_1L_1(s)+c_2L_2(s)+\ldots+c_NL_N(s)$ be a linear combination of these functions ($c_j \in\mathbb{R}^*$ are distinct). $F$ is known to have two kinds of zeros: trivial ones, and non-trivial zeros which are confined in a vertical strip. We denote the number of non-trivial zeros $\rho$ with $\frac{F}(\rho)$\leq T$ by $N(T)$, and we let $N_\theta(T)$ be the number of these zeros that are on the critical line. At the end of the 90's, Selberg proved that this linear combination had a positive proportion of zeros on the critical line, by showing that $\kappa F∶=\lim \inf T (N_\theta(2T)−N_\theta(T))/(N(2T)−N(T))\geq c/N^2$ for some $c>0$. Our goal is to provide an explicit value for $c$, and also to improve the lower bound above by showing that $\kappa F \geq 2.16\times 10^{-6}/(N \log N)$, for any large enough $N$.

**Mar, 21**: Pro-p Iwahori Invariants

Speaker:

*Emanuele Bodon*

Let $F$ be the field of $p$-adic numbers (or, more generally, a non-

archimedean local field) and let $G$ be $\mathrm{GL}_n(F)$ (or, more generally,

the group of $F$-points of a split connected reductive group). In the

framework of the local Langlands program, one is interested in studying

certain classes of representations of $G$ (and hopefully in trying to match

them with certain classes of representations of local Galois groups).

In this talk, we are going to focus on the category of smooth representations

of $G$ over a field $k$. An important tool to investigate this category is

given by the functor that, to each smooth representation $V$, attaches its

subspace of invariant vectors $V^I$ with respect to a fixed compact open

subgroup $I$ of $G$. The output of this functor is actually not just a $k$-

vector space, but a module over a certain Hecke algebra. The question we are

going to attempt to answer is: how much information does this functor preserve

or, in other words, how far is it from being an equivalence of categories? We

are going to focus, in particular, on the case that the characteristic of $k$

is equal to the residue characteristic of $F$ and $I$ is a specific subgroup

called "pro-$p$ Iwahori subgroup".

**Mar, 18**: The fourth moment of quadratic Dirichlet L-functions

Speaker:

*Quanli Shen*

I will discuss the fourth moment of quadratic Dirichlet L-functions where we prove an asymptotic formula with four main terms unconditionally. Previously, the asymptotic formula was established with the leading main term under generalized Riemann hypothesis. This work is based on Li's recent work on the second moment of quadratic twists of modular L-functions. It is joint work with Joshua Stucky.

**Mar, 14**: Analogues of the Hilbert Irreducibility Theorem for integral points on surfaces

Speaker:

*Simone Coccia*

We will discuss conjectures and results regarding the Hilbert

Property, a generalization of Hilbert's irreducibility theorem to arbitrary

algebraic varieties. In particular, we will explain how to use conic fibrations

to prove the Hilbert Property for the integral points on certain surfaces,

such as affine cubic surfaces.

**Mar, 13**: On extremal orthogonal arrays

Speaker:

*Sho Suda*

An orthogonal array with parameters \((N,n,q,t)\) (\(OA(N,n,q,t)\) for short) is an \(N\times n\) matrix with entries from the alphabet \(\{1,2,...,q\}\) such that in any of its \(t\) columns, all possible row vectors of length \(t\) occur equally often. Rao showed the following lower bound on \(N\) for \(OA(N,n,q,2e)\):

\[ N\geq \sum_{k=0}^e \binom{n}{k}(q-1)^k, \]

and an orthogonal array is said to be complete or tight if it achieves equality in this bound. It is known by Delsarte (1973) that for complete orthogonal arrays \(OA(N,n,q,2e)\), the number of Hamming distances between distinct two rows is \(e\). One of the classical problems is to classify complete orthogonal arrays.

We call an orthogonal array \(OA(N,n,q,2e-1)\) extremal if the number of Hamming distances between distinct two rows is \(e\). In this talk, we review the classification problem of complete orthogonal arrays with our contribution to the case \(t=4\) and show how to extend it to extremal orthogonal arrays. Moreover, we give a result for extremal orthogonal arrays which is a counterpart of a result in block designs by Ionin and Shrikhande in 1993.

**Mar, 8**: Interactions between topology and algebra: advances in algebraic K-theory

Speaker:

*Teena Gerhardt*

The field of algebraic topology has exposed deep connections between topology and algebra. One example of such a connection comes from algebraic K-theory. Algebraic K-theory is an invariant of rings, defined using tools from topology, that has important applications to algebraic geometry, number theory, and geometric topology. Algebraic K-groups are difficult to compute, but advances in algebraic topology have led to many recent computations which were previously intractable. In this talk I will introduce algebraic K-theory and its applications, and discuss recent advances in this field.

**Mar, 4**: Reproductive value, prevalence, and perturbation theory of Perron vectors

Speaker:

*Eric Foxall*

In a linear population model that has a unique “largest” eigenvalue and is suitably irreducible, the corresponding left and right (Perron) eigenvectors determine the long-term relative prevalence and reproductive value of different types of individuals, as described by the Perron-Frobenius theorem and generalizations. It is therefore of interest to study how the Perron vectors depend on the generator of the model. Even when the generator is a finite-dimensional matrix, there are several approaches to the corresponding perturbation theory. We explore an approach that hinges on stochasticization (re-weighting the space of types to make the generator stochastic) and interprets formulas in terms of the corresponding Markov chain. The resulting expressions have a simple form that can also be obtained by differentiating the renewal-theoretic formula for the Perron vectors. The theory appears well-suited to the study of infection spread that persists in a population at a relatively low prevalence over an extended period of time, via a fast-slow decomposition with the fast/slow variables corresponding to infected/non-infected compartments, respectively. This is joint work with MSc student Tareque Hossain.

**Mar, 4**: Primes in arithmetic progressions to smooth moduli

Speaker:

*Julia Stadlmann*

The twin prime conjecture asserts that there are infinitely many primes p for which p+2 is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013 Zhang achieved a breakthrough, showing that there exists some positive integer h for which p and p+h are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the q-van der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli.

**Feb, 29**: The Distribution of Logarithmic Derivatives of Quadratic L-functions in Positive Characteristic

Speaker:

*Félix Baril Boudreau*

To each square-free monic polynomial $D$ in a fixed polynomial ring $\mathbb{F}_q[t]$, we can associate a real quadratic character $\chi_D$, and then a Dirichlet $L$-function $L(s,\chi_D)$. We compute the limiting distribution of the family of values $L'(1,\chi_D)/L(1,\chi_D)$ as $D$ runs through the square-free monic polynomials of $\mathbb{F}_q[t]$ and establish that this distribution has a smooth density function. Time permitting, we discuss connections of this result with Euler-Kronecker constants and ideal class groups of quadratic extensions. This is joint work with Amir Akbary.

**Feb, 26**: L-functions in Analytic Number Theory: Biitu

Speaker:

*Bittu*

**Feb, 14**: Hilbert Class Fields and Embedding Problems

Speaker:

*Abbas Maarefparvar*

The class number one problem is one of the central subjects in algebraic number theory that turns back to the time of Gauss. This problem has led to the classical embedding problem which asks whether or not any number field $K$ can be embedded in a finite extension $L$ with class number one. Although Golod and Shafarevich gave a counterexample for the classical embedding problem, yet one may ask about the embedding in 'Polya fields', a special generalization of class number one number fields. The latter is the 'new embedding problem' investigated by Leriche in 2014. In this talk, I briefly review some well-known results in the literature on the embedding problems. Then, I will present the 'relativized' version of the new embedding problem studied in a joint work with Ali Rajaei.

**Feb, 9**: Consecutive sums of two squares in arithmetic progressions

Speaker:

*Vivian Kuperberg*

In 2000, Shiu proved that there are infinitely many primes whose last digit is 1 such that the next prime also ends in a 1. However, it is an open problem to show that there are infinitely many primes ending in 1 such that the next prime ends in 3. In this talk, we'll instead consider the sequence of sums of two squares in increasing order. In particular, we'll show that there are infinitely many sums of two squares ending in 1 such that the next sum of two squares ends in 3. We'll show further that all patterns of length 3 occur infinitely often: for any modulus q, every sequence (a mod q, b mod q, c mod q) appears infinitely often among consecutive sums of two squares. We'll discuss some of the proof techniques, and explain why they fail for primes. Joint work with Noam Kimmel.

**Feb, 8**: A Conjecture of Mazur predicting the growth of Mordell--Weil ranks in Z_p-extensions

Speaker:

*Debanjana Kundu*

Let \(p\) be an odd prime. We study Mazur's conjecture on the growth of the Mordell--Weil ranks of an elliptic curve \(E/\mathbb{Q}\) over an imaginary quadratic field in which \(p\) splits and \(E\) has good reduction at \(p\). In particular, we obtain criteria that may be checked through explicit calculation, thus allowing for the verification of Mazur's conjecture in specific examples. This is joint work with Rylan Gajek-Leonard, Jeffrey Hatley, and Antonio Lei.

**Feb, 8**: Gromov-Wasserstein Alignment: Statistical and Computational Advancements via Duality

Speaker:

*Ziv Goldfeld*

The Gromov-Wasserstein (GW) distance quantifies dissimilarity between metric measure (mm) spaces and provides a natural alignment between them. As such, it serves as a figure of merit for applications involving alignment of heterogeneous datasets, including object matching, single-cell genomics, and matching language models. While various heuristic methods for approximately evaluating the GW distance from data have been developed, formal guarantees for such approaches—both statistical and computational—remained elusive. This work closes these gaps for the quadratic GW distance between Euclidean mm spaces of different dimensions. At the core of our proofs is a novel dual representation of the GW problem as an infimum of certain optimal transportation problems. The dual form enables deriving, for the first time, sharp empirical convergence rates for the GW distance by providing matching upper and lower bounds. For computational tractability, we consider the entropically regularized GW distance. We derive bounds on the entropic approximation gap, establish sufficient conditions for convexity of the objective, and devise efficient algorithms with global convergence guarantees. These advancements facilitate principled estimation and inference methods for GW alignment problems, that are efficiently computable via the said algorithms.

**Feb, 7**: Moments of higher derivatives related to Dirichlet L-functions

Speaker:

*Samprit Ghosh*

The distribution of values of Dirichlet L-functions \(L(s, \chi)\) for variable \(χ\) has been studied extensively and has a vast literature. Moments of higher derivatives has been studied as well, by Soundarajan, Sono, Heath-Brown etc. However, the study of the same for the logarithmic derivative \(L'(s, \chi)/ L(s, \chi)\) is much more recent and was initiated by Ihara, Murty etc. In this talk we will discuss higher derivatives of the logarithmic derivative and present some new results related to their distribution and moments at s=1.

**Feb, 1**: Collision of orbits under the action of a Drinfeld module

Speaker:

*Dragos Ghioca*

We present various results and conjectures regarding unlikely intersections of orbits for families of Drinfeld modules. Our questions are motivated by the groundbreaking result of Masser and Zannier (from 15 years ago) regarding torsion points in algebraic families of elliptic curves.

**Jan, 31**: A discrete mean value of the Riemann zeta function and its derivatives

Speaker:

*Ertan Elma*

In this talk, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis, we obtain a lower bound for the 2kth moment of all the derivatives of the Riemann zeta function evaluated at its nontrivial zeros. This is based on a joint work with Kübra Benli and Nathan Ng.

**Jan, 29**: Fourier optimization and the least quadratic non-residue

Speaker:

*Emily Quesada-Herrera*

We will explore how a Fourier optimization framework may be used to study two classical problems in number theory involving Dirichlet characters: The problem of estimating the least character non-residue; and the problem of estimating the least prime in an arithmetic progression. In particular, we show how this Fourier framework leads to subtle, but conceptually interesting, improvements on the best current asymptotic bounds under the Generalized Riemann Hypothesis, given by Lamzouri, Li, and Soundararajan. Based on joint work with Emanuel Carneiro, Micah Milinovich, and Antonio Ramos.

**Jan, 25**: Sums of proper divisors with missing digits

Speaker:

*Kübra Benli*

Let $s(n)$ denote the sum of proper divisors of a positive integer $n$. In 1992, Erdös, Granville, Pomerance, and Spiro conjectured that if \(\square\) is a set of integers with asymptotic density zero then the preimage set \(s^{−1}(\square)\) also has asymptotic density zero. In this talk, we will discuss the verification of this conjecture when \(\square\) is the set of integers with missing digits (also known as ellipsephic integers) by giving a quantitative estimate on the size of the set \(s^{-1}(\square)\). This talk is based on the joint work with Giulia Cesana, C\'{e}cile Dartyge, Charlotte Dombrowsky and Lola Thompson.

**Jan, 24**: Projective Planes and Hadamard Matrices

Speaker:

*Hadi Kharaghani*

It is conjectured that there is no projective plane of order 12. Balanced splittable Hadamard matrices were introduced in 2018. In 2023, it was shown that a projective plane of order 12 is equivalent to a balanced multi-splittable Hadamard matrix of order 144. There will be an attempt to show the equivalence in a way that may require little background.

**Jan, 24**: Introduction to agent-based evolutionary game theory

Speaker:

*Luis R. Izquierdo*

Evolutionary game theory is a discipline devoted to studying populations of individuals that are subject to evolutionary pressures, and whose success generally depends on the composition of the population. In biological contexts, individuals could be molecules, simple organisms or animals, and evolutionary pressures often take the form of natural selection and mutations. In socioeconomic contexts, individuals could be humans, firms or other institutions, and evolutionary pressures often derive from competition for scarce resources and experimentation.

In this talk I will give a very basic introduction to agent-based evolutionary game theory, a bottom-up approach to modelling and analyzing these systems. The defining feature of this modelling approach is that the individual units of the system and their interactions are explicitly and individually represented in the model. The models thus defined can be usefully formalized as stochastic processes, whose dynamics can be explored using computer simulation and approximated using various mathematical theories.

**Jan, 22**: Mean values of long Dirichlet polynomials

Speaker:

*Winston Heap*

We discuss the role of long Dirichlet polynomials in number theory. We first survey some applications of mean values of long Dirichlet polynomials over primes in the theory of the Riemann zeta function which includes central limit theorems and pair correlation of zeros. We then give some examples showing how, on assuming the Riemann Hypothesis, one can compute asymptotics for such mean values without using the Hardy-Littlewood conjectures for additive correlations of the von-Mangoldt functions.

**Jan, 18**: Equidistribution of some families of short exponential sums

Speaker:

*Théo Untrau*

Exponential sums play a role in many different problems in number theory. For instance, Gauss sums are at the heart of some early proofs of the quadratic reciprocity law, while Kloosterman sums are involved in the study of modular and automorphic forms. Another example of application of exponential sums is the circle method, an analytic approach to problems involving the enumeration of integer solutions to certain equations. In many cases, obtaining upper bounds on the modulus of these sums allow us to draw conclusions, but once the modulus has been bounded, it is natural to ask the question of the distribution of exponential sums in the region of the complex plane in which they live. After a brief overview of the motivations mentioned above, I will present some results obtained with Emmanuel Kowalski on the equidistribution of exponential sums indexed by the roots modulo p of a polynomial with integer coefficients.

**Jan, 16**: Explicit bounds for $\zeta$ and a new zero free region

Speaker:

*Chiara Bellotti*

In this talk, we prove that |ζ(σ+it)|≤ 70.7 |t|4.438(1-σ)^{3/2} log2/3|t| for 1/2≤ σ ≤ 1 and |t| ≥ 3, combining new explicit bounds for the Vinogradov integral with exponential sum estimates. As a consequence, we improve the explicit zero-free region for ζ(s), showing that ζ(σ+it) has no zeros in the region σ ≥ 1-1/(53.989 (log|t|)2/3(log log|t|)1/3) for |t| ≥ 3.

**Jan, 10**: Phase dynamics of cyclic reptilian tooth replacement

Speaker:

*Laurent MacKay*

For over a century, scientists have studied striking spatiotemporal patterns during the continual tooth replacement of reptiles. Aside from the compelling aesthetics of this phenomenon, it is thought that understanding the underlying mechanisms may provide the insight required to trigger adult tooth replacement in humans. Theoretical frameworks have long been proposed to understand the rules behind the observed spatiotemporal order, but have only been analyzed mathematically more recently. Starting from Edmund's observations in crocodiles and proposed theory of replacement waves, we show how a simple model consisting of a row of non-interacting phase oscillators predicts several experimental observations. Next, inspired by the hypothesis put forth by Osborn, we consider a variation of the phase model with ODEs that account for mutual inhibition between tooth sites, and use continuation methods to thoroughly search parameter space for experimentally validated solutions. We then extend the model to a PDE that explicitly accounts for the diffusion of inhibitory signals between teeth, yielding some novel solution types. Using continuation methods once again, we delineate parameter regimes with solutions that closely resemble experimental observations in leopard geckos.