# 5/2/22 Through 5/23/22

### Number Theory Seminar: Randomness of Sequences

 Date May 3, 2022 Time 11:00 am Location via Zoom (contact Kevin Ford) Speaker Niclas Technau, Caltech Contact Kevin Ford Email ford126@illinois.edu Sponsor n/a

The first part of this talk surveys different notions of randomness of one-dimensional sequences. A focal point will be on what is not known about local statistics which go beyond uniform distribution modulo one. The second part is reporting on joint work with Christopher Lutsko and Athanasios Sourmelidis. By employing Fourier analytic tools, this work provides a better understanding of how random slowly growing monomial sequences are - showing that their correlations are Poissonian.

### Graph Theory and Combinatorics Seminar: Rational exponents for generalized extremal problems

 Date May 3, 2022 Time 1:00 pm Location 345 AH Speaker Anastasia Halfpap (University of Montana) Contact Sean English Sponsor N/A

Fix a target graph H and a family F of forbidden graphs. The generalized extremal number ex(n, H, F) is the maximum number of H-copies possible in an n-vertex graph which avoids F. Note that when H is an edge, ex(n, H, F) is the ordinary extremal number ex(n, F). After the systematic study of generalized extremal numbers was initiated by Alon and Shikhelman in 2016, the area has received substantial attention. In addition to explicit computation of ex(n, H, F) for specific choices of H, F, many questions in extremal graph theory (e.g., supersaturation, stability) naturally extend to the generalized setting.

In 2015, Bukh and Conlon applied the random algebraic method to show that, for any rational r in the interval [1,2], there is a family such that ex(n, F) = Theta(n^r).  Analogously, for a fixed target graph H and a rational number r within an appropriate interval, we may ask whether it is possible to find a forbidden family for which ex(n, H, F) = Theta(n^r). In this talk, we present results on this question for some specific target graphs H, focusing on the case where H is a triangle, for which we show that all rational exponents in [1,3] are realizable. Joint work with Sean English and Bob Krueger.

### Canary Professorship: Candidate Presentation

 Date May 3, 2022 Time 4:00 pm - 5:00 pm Location via Zoom Speaker Jacob Bedrossian (Maryland) Contact Jared Bronski Email bronski@illinois.edu Sponsor n/a

Title: Chaos and turbulence in stochastic fluid mechanics: What is it and how could we study it?

Abstract: In this survey-style talk I discuss the (old) idea of studying turbulence in stochastically-forced fluid equations. I will discuss  efinitions of chaos, anomalous dissipation, and various other predictions by physicists that can be phrased as mathematically precise conjectures in this context. Then, I will discuss some recent work by my collaborators and I on various aspects, namely (1) a straightforward characterization of anomalous dissipation that implies the classical Kolmogorov 4/5 law for 3d NSE (joint with Michele Coti Zelati, Sam Punshon-Smith, and Franziska Weber); (2) the study of "Lagrangian chaos" and exponential mixing of scalars and how it leads to a proof of anomalous dissipation and of the power spectrum predicted by Batchelor in 1959 for the simpler problem of Batchelor-regime passive scalar turbulence (joint with Alex Blumenthal and Sam Punshon-Smith); (3) the more recent proof of "Eulerian chaos" for Galerkin truncations of the Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith).

For other details, please see email sent on behalf of Jared Bronski dated April 27 (sent by P. Currid)

### Graduate Commutative Algebra & Algebraic Geometry Seminar: Condensation and the Algebraic  Geometry of Topological Algebras

 Date May 3, 2022 Time 4:00 pm Location 241 Altgeld Hall Speaker Brian Shin Contact Likun Xie Email likunx2@illinois.edu Sponsor n/a
A key foundational input to the success of modern algebraic geometry is the niceness of the category of abelian groups. In particular, it is Grothendieck Abelian with a very nice tensor product. Seeing as many of our favorite algebras come with interesting topologies (e.g. the fields of real and p-adic numbers), one might hope these nice properties persist upon passing to topological abelian groups. Unfortunately, what is typically one's first guess at a definition fails dramatically: the category of topological abelian groups is *terrible*, and there are now many tensor products, each having its own set of pros and cons. These failures make the category of topological abelian groups unsuitable for a foundation of topologo-algebraic geometry. How are we to study algebraic geometry while incorporating the topologies on our rings? In this expository talk, we'll give a brief introduction to Clausen--Scholze's answer to this question: condensed mathematics and analytic geometry.

### Department Colloquium: On sphere packings and the hard sphere model

 Date May 4, 2022 Time 4:00 pm Location 245 Altgeld Speaker Will Perkins, UIC Contact Jozsef Balogh Email jobal@illinois.edu Sponsor n/a
The classic sphere packing problem is to determine the densest possible packing of non-overlapping congruent spheres in Euclidean space.  The problem is trivial in dimension 1, straightforward in dimension 2, but a major challenge or mystery in higher dimensions, with the only other solved cases being dimensions 3, 8, and 24.  The hard sphere model is a classic model of a gas from statistical physics, with particles interacting via a hard-core pair potential.  It is believed that this model exhibits a crystallization phase transition in dimension 3, giving a purely geometric explanation for freezing phenomena in nature, but this remains an open mathematical problem. The sphere packing problem and the hard sphere model are closely linked through the following rough rephrasing of the phase transition question: do typical sphere packings at densities just below the maximum density align with a maximum packing or are they disordered?

I will present results on high-dimensional sphere packings and spherical codes and new bounds for the absence of phase transition at low densities in the hard sphere model.  The techniques used take the perspective of algorithms and optimization and can be applied to problems in extremal and enumerative combinatorics as well.

Lunch is 11:50am at Mandarin Woks, if you  wish to attend, let Jozsef Balogh know before 11am, Dinner plans to be discussed after the lecture.

### IRisk Lab Spring 2022 Final Presentations

 Date May 5, 2022 Time 9:00 am - 2:00 pm Location 1090 Lincoln Hall Contact Zhiyu (Frank) Quan Email ASRM-advising@illinois.edu Sponsor Illinois Risk Lab

You are invited to attend the IRisk Lab's final presentations event during Reading Day, May 5. The event will be held in 1090 Lincoln Hall.

The IRisk Lab is an industry-academic collaboration hub that facilitates integration of discovery-based learning experiences for students, and showcases state-of-the-art research in all areas of risk analysis and advanced analytics.

Whether your are presenting or just want to learn more about the IRisk Lab, come celebrate this semester's research projects.

Event Agenda

• 9:00 - 9:30 Luyan Sales Data Analysis Project
• 9:30 - 10:00 Smart Contract of Distributed Insurance Project
• 10:00 - 10:30 AXIS Systemic Cyber Threats Project
• 10:30 - 11:00 Automated Machine Learning (AutoML) Project
• 11:00 - 11:30 Insurtech COUNTRY Financial Loss Model Project
• 11:30 - 12:00 Insurtech Innovation via Natural Language Processing (NLP) Project
• 12:00 - 12:30 Building an NLP-Powered Repository and Search Tool for Cyber Risk Literature Project
• 12:30 - 14:00 Lunch & Networking: pizza and soft drinks will be provided.

The organizers highly recommend that you take a COVID test and know the result before joining this event.

If you have any questions about the event, please get in touch with professor Zhiyu (Frank) Quan.

### IGL Open House

 Date May 5, 2022 Time 1:00 pm - 4:00 pm Location Illini Union 314B Contact Madie Farris Email mfarris3@illinois.edu Sponsor IGL
Join the IGL at our fall open house and poster session and see what everyone in the IGL has been working on this year! All are welcome to stop by anytime 1-4 PM.

### Conversation Series: Felix Leditzky

 Date May 5, 2022 Time 1:00 pm Location 245 AH Speaker Felix Leditzky Contact Vesna Stojanoska Email vesna@illinois.edu Sponsor N/A
One more in the series of informal interviews with faculty in the math department, discussing things like one's career path, challenges, professional inspiration, advice for mathematicians at various career stages, etc.

All are welcome!

(Email Vesna for zoom link.)

### Algebra, Geometry & Combinatorics: A Murnaghan-Nakayama rule for Grothendieck polynomials of Grassmannian type

 Date May 5, 2022 Time 3:00 pm Location 347 Altgeld Hall Speaker Khanh Nguyen Duc Contact Gidon Orelowitz Email gidono2@illinois.edu Phone 917-371-1855 Sponsor N/A

The Grothendieck polynomials appearing in the K-theory of Grassmannians are analogs of Schur polynomials. We establish a version of the Murnaghan-Nakayama rule for Grothendieck polynomials of the Grassmannian type. This rule allows us to express the product of a Grothendieck polynomial with a power sum symmetric polynomial into a linear combination of other Grothendieck polynomials.

### Spring 2022 Faculty meeting

 Date May 5, 2022 Time 4:00 pm Contact Rebecca Bishop Email rjbishop@illinois.edu Phone 217-254-4244 Sponsor Vera Hur

The Spring regular meeting of the Department. Please pre-register with Becky Bishop to receive zoom link.

rjbishop@illinois.edu

### Graduate Student Homotopy Theory Seminar: Calculus for Algebraic Topologists

 Date May 6, 2022 Time 3:00 pm Location 341 Altgeld Hall Speaker Johnson Tan Contact Doron Grossman-Naples Email doronlg2@illinois.edu Sponsor n/a
Early on in our mathematical studies, we learn that instead of studying a problem directly it is useful to study a linearization of the problem. For example, to understand a smooth map between manifolds we can look at the resulting linear map between the tangent spaces at a point. In this talk we will be looking at a categorification of this idea through the lens of Goodwillie Calculus. In particular, given a map between sufficiently nice infinity categories we will define what it means for such a map to be "linear" and furthermore how one can approximate by such maps. Time permitting, we will explore further generalizations of this idea.

### AWM Ice Cream Social

 Date May 7, 2022 Time 3:00 pm - 6:00 pm Location Main Quad (Alternate location in case of inclement weather: 245 AH) Registration Registration Contact Yansy Perez Email yansyp2@illinois.edu Sponsor Association of Women in Math
The Association of Women in Mathematics (AWM) will be hosting an Ice Cream Social on Saturday, May 7th, 2022, from 3-6 pm on the Main Quad! In case of rain, we will be in 245 AH.There will be ice cream, (non-alcoholic) drinks, and games! Feel free to also bring your own snacks/dessert to share.

Take this time to hang out with others in our department, celebrate the end of the school year, and congratulate our graduating class! You may also bring your family or a guest!

### Symplectic and Poisson geometry seminar: Toric Hamiltonian actions in a Poisson context

 Date May 9, 2022 Time 3:00 pm Location 347 Altgeld Hall Speaker Maarten Mol Contact Joey Palmer Email jpalmer5@illinois.edu Sponsor n/a
This talk concerns toric Hamiltonian actions with momentum maps taking values in regular Poisson manifolds of compact type, or more precisely: toric Hamiltonian actions of regular and proper symplectic groupoids. Examples of these include symplectic toric manifolds, proper Lagrangian fibrations and proper isotropic realizations of Poisson manifolds of compact type. The aim will be to explain the classification of such toric actions in terms of "Delzant polytopes" in the leaf space of the symplectic groupoid.

### Graduate Commutative Algebra and Algebraic Geometry Seminar: Introduction to Geometric Invariant Theory II

 Date May 10, 2022 Time 4:00 pm Location 241 Altgeld Hall Speaker Zijing Ye Contact Likun Xie Email likunx2@illinois.edu Sponsor n/a

This talk is the second of a series of talks on this topic. In the first talk, I introduced the quotient scheme and linearization of invertible sheaves. In the second talk, I will use them to introduce the stability of points under a group action. Then I will show how the 1-parameter subgroups determine the properties of our group action and stableness. Cookies will be provided as usual.

### Algebraic Geometry Seminar: Virasoro constraints for moduli of sheaves

 Date May 16, 2022 Time 2:00 pm Location 243 Altgeld Hall Speaker Rahul Pandharipande, ETH Zurich Contact Sheldon Katz Email katzs@illinois.edu Sponsor n/a

Descendent classes on moduli spaces of sheaves are defined via the Chern characters of the universal sheaf.  I will present several conjectures and results concerning Virasoro constraints for integrals of the descendent classes.

### Combinatorics Colloquium: The upper tail for triangles in random graphs

 Date May 18, 2022 Time 11:00 am Location Altgeld Hall (room # TBD) Speaker Wojciech Samotij, Tel Aviv University Contact Jozsef Balogh Email jobal@illinois.edu Sponsor n/a

Abstract: Let $X$ denote the number of triangles in the random graph $G_{n,p}$.  The problem of determining the asymptotics of the logarithimic upper tail probability of $X$, that is, $\log \Pr(X > (1+\delta)\mathbb{E}[X])$, for every fixed positive $\delta$ has attracted considerable attention of both the combinatorics and the probability communities.  We shall present an elementary solution to this problem, obtained recently in a joint work with Matan Harel and Frank Mousset.  The crux of our approach is a simple probabilistic argument, inspired by the work of Janson, Oleszkiewicz and Ruci\’nski, that reduces the estimation of this upper tail probability to a counting problem.

### Elliptic Cohomology and Conformal Field Theory

 Date May 19, 2022 Time 2:00 pm Location 343 Altgeld Hall Speaker Doron Grossman-Naples Contact Doron Grossman-Naples Email doronlg2@illinois.edu Sponsor n/a