### Special Pre-enrollment Program in Mathematics at Rutgers

**SPP 2019 - A special program for entering PhD graduate students**

The lectures, problem sessions and Written Exams will be in classroom Hill 705. Breakfast and Lunch, when provided, will be available in the lounge, Hill 703.

The format of the program will be two three-hour modules each day, on analysis related and algebra related topics respectively. Prof. Chris Mirek will teach the analysis topics from 9:30am to 11:30am, with Graduate Assistant Chloe Wawrzyniak taking over from 11:30am to 12:30pm. Prof. Anders Buch will teach the algebra topics from 2:00pm to 4:00pm, with Graduate Assistant (TBA) taking over from 4:00pm to 5:00pm. (These graduate student and faculty member assignments are from last year, still finalizing for 2019.)

A few talks by current graduate students will also be included in the afternoon session of Friday, August 23, to give an idea of the dissertation research being done at Rutgers.

It is also an opportunity for new students to meet each other and to get acquainted with some faculty members and continuing graduate students.

**The lectures and problem sessions will be in Hill 705. Breakfast and Lunch, when provided, will be available in the lounge, Hill 703.**

### Monday, August 19

- 9:00 – 9:30AM : Breakfast (
**provided**) - 9:30 – 11:20AM : Analysis Lecture
- 11:30 – 12:30PM : Analysis Problem Session
- 12:30 – 1:00PM : Lunch (
**provided**) and Break - 1:00 – 1:15pm: Meet with Technology Staff in Room 705 - Emails, computers, photos.
- 1:15 - 2:00PM : Administrative Stuff (keys, photos, etc.)
- 2:00 – 3:50PM : Algebra Lecture
- 4:00 – 5:00PM : Algebra Problem Session

### Tuesday, August 20

- 9:30 – 11:20AM : Analysis Lecture
- 11:30 – 12:30PM : Analysis Problem Session
- 12:30 – 2:00PM : Lunch and Break
- 2:00 – 3:50PM : Algebra Lecture
- 4:00 – 5:00PM : Algebra Problem Session

### Wednesday, August 21

- 9:30 – 11:20AM : Analysis Lecture
- 11:30 – 12:30PM : Analysis Problem Session
- 12:30 – 2:00PM : Payroll Onboarding Process for TA's (Hill Room 313); Lunch and Break
- 2:00 – 3:50PM : Algebra Lecture
- 4:00 – 5:00PM : Algebra Problem Session

### Thursday, August 22

- 10:30am-2:00pm: For International Students: Social Security Applications - Trip from 180 College Ave to SSA - will take you to the Social Security Office, 550 Jersey Avenue, Suite 200 New Brunswick, NJ
- 12:30 – 2:00PM : Lunch and Break
- 2:00 – 3:50PM : Algebra Lecture
- 4:00 – 5:00PM : Algebra Problem Session

### Friday, August 23

- 9:30 – 11:20AM : Analysis Lecture
- 11:30 – 12:30PM : Analysis Problem Session
- 12:30 – 2:00PM : Lunch (
**provided**) meeting with current students : "What every Math grad student should know" - 2:00 – 3:50PM : Algebra Lecture
**4:00 – 4:20PM : Student Research Glimpse #1 Speaker :****Erik de Amorim**- Title: The Hartree limit of a bosonic atom in its center-of-mass frame
- Abstract: The study of the equilibrium energy and configuration of a many-body system in Quantum Mechanics, like an atom with its nucleus and N electrons, involves finding the minimizer of a certain energy functional in an appropriate space of wavefunctions. When N goes to infinity, with the assumption that the bodies are bosons (which implies certain symmetry properties in the space where the minimization takes place), a mean-field approximation is often used that basically says that each body independently follows the probability distribution of the same wavefunction, and then the equilibrium energy converges to the minimizer of the so-called 1-body Hartree functional. Rigorous justifications for this approximation exist in different circumstances, but they are often done under the Born-Oppenheimer approximation: assuming that the nucleus of the system sits immovable at the origin. In this presentation I will explain in an elementary manner (no Physics background required) how to get rid of this assumption, which, for quantum mechanical models, turns out to be a nontrivial task.
**4:30 – 4:50PM : Student Research Glimpse #2 Speaker :****Chloe Wawrzyniak**- Title: Polynomial Convexity: Easy to State, Tough to Prove
- Abstract: A compact set X in C^n is polynomially convex if for every x outside X, there is a polynomial P such that |P(x)| is strictly larger than the supremum of |P| over X. A natural question is to classify which sets are polynomially convex. In one complex dimension, the answer is well-known. But in dimension 2 and higher, this question is still an area of active research. Many of the proofs in this area are long, technical, and require extra background. So, we will spend most of our time playing with examples while assuming some major results. Time permitting, I will discuss Eva Kallins problem - a tantalizingly easy-to-state open question in the field.
**5:00 – 5:20PM : Student Research Glimpse #3 Speaker : Chengxi Wang**- Title: Strong exceptional collections of line bundles on the projective space.
- Abstract: In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry. Hilbert's syzygy theorem concern the relations between the generators of an ideal, or, more generally, a module. In this presentation, I will talk about projective spaces, Hilbert syzygy theorem and strong exceptional collections of line bundles on the projective space.
- 5:30 – 630PM : Board Games
- 6:30PM : Graduate Student Welcome Meeting (Food and drinks provided)

### Monday, August 26

- 9:30am-11:20am - Analysis Lecture
- 11:30AM – 12:30 : Analysis Problem Session
- Noon – 1:00PM : Lunch and Break
- 1:00 – 3:00PM : New Student Orientation presented by the Graduate School at Rutgers Student Center Multipurpose Room, College Avenue Campus, 126 College Avenue, New Brunswick, NJ

### Tuesday, August 27

- 9:00 – 10:50AM : Problem Session related to the Analysis Written Quals;
- 11:00am-12noon: Algebra Problem Session
- 12:30 – 2:30PM : Lunch and Break
- 3:00 - 5:00 PM:
**Algebra****Written Quals**

### Wednesday, August 28

- 8:30 – 12:00PM :
**For International Students (only)**- Teaching Assistant Orientation, Rutgers Student Center Multipurpose Room, College Avenue Campus, 126 College Avenue, New Brunswick, NJ - 12:00 – 2:30PM : Lunch & Break
- 3:00 – 5:00PM :
**Complex Variables and Advanced Calculus****Written Quals**

### Thursday, August 29

- 8:00 – 1:15PM :
**For Domestic & International Students**- Teaching Assistant Orientation, Rutgers Student Center Multipurpose Room, College Avenue Campus, 126 College Avenue, New Brunswick, NJ - 1:15 – 2:30PM : Lunch & Break
- 3:00 – 5:00PM :
**Real Variables and Elementary Point-Set Topology****Written Quals**

### Friday, August 30

- 12:00 – 1:00PM : Math Department Welcome Lunch (
**provided**) - 1:00 – 1:30PM : Introduction to Library Resources - Room 705
- 1:30 – 4:00PM : Administrative Stuff Free Time

**Tuesday, September 4 : Classes Begin!**